Symbolic Computation of Conserved Densities and Fluxes for...

Symbolic Computation of

Conserved Densities and Fluxes for

Nonlinear Systems of Differential-Difference

Equations

by

Holly Eklund

Copyright c© 2003 Holly Eklund,

All rights reserved.

ii

A thesis submitted to the Faculty and the Board of Trustees of the Colorado

School of Mines in partial fulfillment of the requirements for the degree of Masters of

Science (Mathematical and Computer Sciences).

Golden, Colorado

Date

Signed:Holly Eklund

Approved:Dr. Willy HeremanMathematical and ComputerSciencesThesis Advisor

Golden, Colorado

Date

Dr. Graeme FairweatherHead DepartmentMathematical and ComputerSciences

iii

ABSTRACT

Two algorithms are presented to find conserved densities and fluxes of nonlin-

ear systems of differential-difference equations. Both algorithms utilize the scaling

properties of lattice equations to reduce the problem to a calculus and linear algebra

problem. The two algorithms are illustrated for the Kac-van Moerbeke, Toda, and

Ablowitz-Ladik lattices. The first method leads to a three step algorithm which uti-

lizes the dilation invariance of the conservation laws to construct the form of the den-

sity. For this method, the discrete Euler operator or discrete variational derivative is

an advantageous tool. The algorithm is implemented in Mathematica. The package is

called DDEDensityFlux.m. The key applications are to analyze the discretizations

of the Korteweg-de Vries (KdV), and modified Korteweg-de Vries (mKdV) lattices.

A combination of the KdV and mKdV lattices is also considered. The second method

leads to a five step algorithm that is primarily useful in determining fluxes. Both of

the algorithms presented could be used to investigate the integrability of semi-discrete

lattices.

iv

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 DIFFERENTIAL-DIFFERENCE EQUATIONS . . . . 6

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Example: The Kac-van Moerbeke lattice . . . . . . . . . . . . . . . . 112.4 Dilation Invariance: The concept behind our algorithms . . . . . . . . 132.5 Density-flux pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Tool 1: Equivalence criterion . . . . . . . . . . . . . . . . . . . . . . . 192.7 Tool 2: The discrete Euler operator (variational derivative) . . . . . . 20

Chapter 3 THE FIRST MATHEMATICAL METHOD AND AL-GORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Steps of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 4 IMPLEMENTATION AND SOFTWARE . . . . . . . . 29

4.1 Existing software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 The new code DDEDensityFlux.m . . . . . . . . . . . . . . . . . . . . 314.3 Using the software: A sample session . . . . . . . . . . . . . . . . . . 324.4 Options for the user . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Chapter 5 COMPUTATION OF CONSERVATION LAWS . . . . 42

5.1 The Kac-van Moerbeke lattice . . . . . . . . . . . . . . . . . . . . . . 425.2 The Toda lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 The Ablowitz-Ladik lattice . . . . . . . . . . . . . . . . . . . . . . . . 55

v

5.3.1 Introducing a first weighted parameter (α) . . . . . . . . . . . 555.3.2 Introducing a second weighted parameter (β) . . . . . . . . . 565.3.3 Introducing Shifts . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 6 A SECOND METHOD TO DETERMINE DENSITIESAND FLUXES . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1 The second algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 The Kac-van Moerbeke lattice . . . . . . . . . . . . . . . . . . . . . . 716.3 The Toda lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.4 The Ablowitz-Ladik lattice . . . . . . . . . . . . . . . . . . . . . . . . 75

Chapter 7 APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . 81

7.1 Discretization of the combined KdV-mKdV equation . . . . . . . . . 817.2 Discretization of the Korteweg-de Vries equation . . . . . . . . . . . . 917.3 Discretization of the modified Korteweg-de Vries equation . . . . . . 96

Chapter 8 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . 101

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Appendix A DATA FILES . . . . . . . . . . . . . . . . . . . . . . . . . . 107

vi

LIST OF TABLES

5.1 Ranks of terms in density candidate (R=3) for the KvM lattice . . . 44

5.2 Ranks of terms in density candidate (R=3) for the Toda lattice . . . 50

5.3 Ranks of terms in density candidate (R=12) for the AL lattice . . . . 57

5.4 Ranks of terms in density candidate (R=12) for the AL lattice . . . . 61

7.1 Ranks of terms in density candidate (R=12) for the combined equation 83






7.7 Ranks of terms in density candidate (R=1) for the KdV discretization 92

7.8 Ranks of terms in density candidate (R=1) for the KdV discretization 94

7.9 Ranks of terms in density candidate (R=12) for the mKdV discretization 97



vii

ACKNOWLEDGMENTS

I greatly appreciate the help of Dr. Willy Hereman, my thesis advisor for his

clear explanations, helpful input, and patience throughout.

Without the support of the National Science Foundation Computer Science, En-

gineering and Mathematics Scholarships program, and the National Science Founda-

tion research award CCR-9901929, this research would not have been possible.

Many thanks go to my thesis committee members, Dr. Barbara Moskal and Dr.

Paul Martin for all of their comments and suggestions.

Finally, my sincere gratitude goes to my fiance Jeffrey Bellman and my family

for their support and encouragement throughout.

viii

1

Chapter 1

INTRODUCTION

Differential-difference equations (DDEs) have been the focus of many nonlinear

studies since the original work of Fermi, Pasta, and Ulam in the nineteen sixties

[9]. Nonlinear DDEs describe many interesting physical phenomena including vi-

brations of particles in lattices and currents in electrical networks, Langmuir waves,

and interactions between competing populations. Also, DDEs play an important role

in queuing problems and discretizations in solid state physics and quantum fields.

Lastly, they are used in numerical simulations of nonlinear PDEs [1].

Recently, there has been a renewed interest in DDEs (see e.g. [40] for a review of

the literature). DDEs are semi-discretized as the single space variable is discretized,

and time is kept continuous. This is in contrast to their fully discretized counterparts,

called difference equations, in which there currently is also a great deal of interest

(see e.g. [7, 21, 27, 28]).

In this thesis we focus on one aspect of the integrability of DDEs, namely the com-

putation of polynomial conserved densities and associated fluxes via direct methods

which can be implemented in computer algebra systems. The first few conservation

laws of a DDE may have a physical meaning, for instance conserved momentum and

energy. Additional ones may facilitate the study of both quantitative and qualitative

properties of solutions [23]. Also, the existence of a sequence of conserved densities

predicts integrability of DDEs [13]. However, the absence of conserved densities does

not preclude integrability. The integrable DDEs could indeed be disguised with a co-

2

ordinate transformation so that the transformed equation no longer admits conserved

densities of polynomial type.

The main result of this thesis is two-fold. First, I added several new aspects to the

previously existing method used in condens.m [10], InvariantsSymmetries.m [12]

and diffdens.m [11]. Second, I implemented the new components in Mathematica.

The new package DDEDensityFlux.m is much more reliable than previous versions,

and it calculates conservation laws for new, more complicated systems of DDEs.

In this thesis, I describe two novel methods [20] to construct families of conserved

densities and apply them to specific examples.

(1) The first method relies heavily on the notion of dilation invariance. It is

shown in this thesis that conservation laws for DDEs are dilation invariant. We uti-

lize this fact and construct the form of the density, ρn. Then, we may either use a

“shifting” technique or the discrete Euler operator to find the unknown constants.

Using the “shifting” technique to determine the constants, gives the flux, Jn, auto-

matically. Using the discrete Euler operator to find the unknown constants simplifies

the calculation of the flux, Jn, but does not determine it directly. The first method

leads to the first algorithm which was implemented in Mathematica as DDEDensi-

tyFlux.m.

(2) The second method, suggested by Hickman [20], is more theoretical and is

primarily useful in determining fluxes. It leads to a five step algorithm that uses

repeated decomposition of the identity operator I to find pieces in and outside the

image (Im) of the operator ∆ = D − I, where D is the up-shift operator. First,

we determine the furthest negative shifted variable in expression E (i.e. Dt ρn after

replacement from the system). The expression E is then split into two parts, A(j) and

A(j+1), where A(j) contains all terms that are independent of the lowest shifted variable

3

and A(j+1) has terms dependent on the lowest shifted variable. We then down-shift

A(j) and add D−1A(j) to A(j+1). This process is repeated until D−1A(j) + A(j+1) = 0.

The constraint determines the unknown coefficients leading to the final form of the

density, ρn, and the flux, Jn.

The techniques described in this thesis are applicable to complicated nonlinear

systems of DDEs. Yet, to keep the ideas transparent and avoid lengthy computation

(which are best performed with Maple, Mathematica, or muPAD) we use the Kac-van

Moerbeke (KvM), Toda, and Ablowitz-Ladik (AL) lattices to illustrate our methods.

Like the algorithms and Mathematica codes (for densities and generalized symmetries)

in [14, 15, 17, 19], the methods in the present paper are restricted to polynomial

densities and fluxes.

There is a vast body of work on DDEs, including investigations of integrability

criteria via the computation of densities, generalized and master symmetries, recur-

sion operators, etc. For nonlinear DDEs several solution methods and integrability

tests are applicable. The solution methods include symmetry reduction [29], and

an extension of the spectral transform method [26]. Adaptations of the singular-

ity confinement approach [33], the Wahlquist-Estabrook method [8], and the master

symmetry technique [6] allow one to test integrability of DDEs. The most compre-

hensive integrability study of DDEs was done by Levi and colleagues [26, 30], Yamilov

[46, 47] and co-workers [4, 5, 6, 36, 37, 38]. Their papers provide a classification of

semi-discrete equations possessing infinitely many local conservation laws. Using the

formal symmetry approach, they derive the necessary and sufficient conditions for the

existence of local conservation laws, and provide an algorithm to construct them.

In contrast to these algorithms, in this thesis we present new direct algorithms

that allow one to compute conserved densities and fluxes of DDEs. Our algorithms are

4

fairly straightforward. They only rely on algebra, calculus, and a tool from variational

calculus. Therefore, they can be implemented in computer algebra languages. The

first algorithm is implemented in Mathematica as DDEDensityFlux.m.

DDEDensityFlux.m is based on previously existing methods used in con-

dens.m, InvariantsSymmetries.m, and diffdens.m. The program condens.m

automatically carries out the lengthy symbolic computations for the construction of

conserved densities [13]. The package InvariantsSymmetries.m is a straightforward

algorithm for the symbolic computation of generalized (higher-order) symmetries of

nonlinear evolution equations and lattice equations [15, 16]. The code diffdens.m

calculates conserved densities for several well-known lattice equations [17, 14].

The thesis is organized as follows.

Chapter 2 covers preliminary material about conservation laws of DDEs. Here

we utilize equivalence criteria to simplify our calculations. An analogy to a result

for PDEs, the discrete Euler operator (or variational derivative) is introduced as a

valuable tool to test conserved densities. We prove the necessary and sufficient con-

dition for a function of a discrete variable (and its shifts) to be the total difference of

another function of discrete variables. A few simple examples illustrate the concepts.

Chapter 3 describes the first algorithm for determining densities and fluxes. The

first algorithm is organized into three steps.

In Chapter 4, we address existing related algorithms. In addition, we explain how

to use the developed software called DDEDensityFlux.m. Chapter 4 also includes

a description of how the computation of the fluxes and the discrete Euler operator

were implemented.

The KvM, Toda, and AL lattices are used as examples to illustrate the three

step algorithm in Chapter 5. The importance of introducing shifts in the density

5

(specifically for complicated examples) is demonstrated for the AL lattice. The chap-

ter concludes with applications of the Euler operator to find densities and fluxes of

the KvM, Toda, and AL lattices.

In Chapter 6 we focus on the computation of the associated fluxes. We illustrate

how the second method could be used to compute densities as well as fluxes.

We revisit the KvM, Toda, and AL lattices to illustrate the method.

We show some additional applications of the first algorithm in Chapter 7. Here

we use our methods to compute densities and fluxes for discretizations of the Korteweg-

de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations, as well as a

combined equation.

We draw some conclusions in Chapter 8.

In summary, the original contribution of this thesis is two-fold: improved algo-

rithms to determine conserved densities and fluxes of DDEs, and their implementation

in Mathematica. We offer the scientific community a symbolic package called DDE-

DensityFlux.m that carries out the tedious calculations of conserved densities and

fluxes of nonlinear DDEs. The software and data files are available from various

sources.

6

Chapter 2

DIFFERENTIAL-DIFFERENCE EQUATIONS

In this chapter we review preliminary material [20] about densities and fluxes of

nonlinear DDEs. We also show a few simple examples.

2.1 Definitions

Definition 2.1 Differential-difference equations (DDEs) are equations that are con-

tinuous in time and discretized in space. They are of the following form:

un = f(un−l, un−l+1, . . . , un, . . . , un+m−1, un+m) (2.1)

with

∂ f∂ un−l

∂ f∂ un+m

6= 0,

where n is an arbitrary integer. In general, f is a nonlinear vector-valued function of

a finite number of dynamical variables, each un is a vector-valued function of t, and

un is the usual time derivative of un(t).

The index n may lie in Z, or the un+k may be periodic, un+k = un+k+N . The

integer l is the furthest negative shift, and m is the furthest positive shift of any

variable in (2.1). If l = m = 0 then the equation is local and reduces to a system of

ordinary differential equations.

7

There are two notations for DDEs. The index n, may be omitted identifying

un ≡ u0, un±p ≡ u±p, etc. However, for clarity we commit to including the index n.

Definition 2.2 The up-shift operator D is defined by

D un+k = un+k+1. (2.2)

Definition 2.3 Its inverse, the down-shift operator, is given by

D−1 un+k = un+k−1. (2.3)

Thus we have un+k = Dk un. The action of D and D−1 is extended to functions by

acting on their arguments. For example,

D g(un−p, un−p+1, . . . , un+q) = g(D un−p, D un−p+1, . . . , D un+q)

= g(un−p+1, un−p+2, . . . , un+q+1).

In particular, we have

D(

∂∂ un+k

g(un−p, un−p+1, . . . , un+q))

= ∂∂ un+k+1

g(un−p+1, un−p+2, . . . , un+q+1).

In order to standardize notation, we consider p and q to be positive integers. Identify

p with the furthest negative shift of any variable in the system, and q with the furthest

positive shift of any variable in the system. Moreover, for equations of type (2.1), the

shift operator commutes with the time derivative

D(

ddt un

)= D f(un−l, un−l+1, . . . , un, . . . , un+m−1, un+m)

= f(un−l+1, un−l+2, . . . , un+1, . . . , un+m, un+m+1)

8

= ddt un+1

= ddt (D un) .

Definition 2.4 Next, we define the (forward) difference operator, ∆ = D− I, by

∆ un+k = (D− I) un+k = un+k+1 − un+k,

where I is the identity operator.

This operator takes the role of a spatial derivative on the shifted variables as many

examples of DDEs arise from the discretizations of a PDE in (1 + 1) variables [35].

The difference operator extends to functions and we have

∆ g(un−p, un−p+1, . . . , un+q) = (D− I) [g(un−p, un−p+1, . . . , un+q)]

= D [g(un−p, un−p+1, . . . , un+q)]− I [g(un−p, un−p+1, . . . , un+q)]

= g(un−p+1, un−p+2, . . . , un+q+1)− g(un−p, un−p+1, . . . , un+q).

Definition 2.5 For any function g = g(un−p, un−p+1, . . . , un+q), the total time deriva-

tive Dt g is computed as

Dtg(un−p, un−p+1, . . . , un+q) = ( ∂ g∂ un−p

)un−p + · · ·+ ( ∂ g∂ un

)un + · · ·

+( ∂ g∂ un+q

)un+q

= ( ∂ g∂ un−p

)D−pun + · · ·+ ( ∂ g∂ un

)D0un + · · ·+ ( ∂ g∂ un+q

)Dqun

= (q∑

k=−p

∂ g∂ un+k

Dk)un

= (q∑

k=−p

∂ g∂ un+k

Dk)f(un−l, un−l+1, . . . , un+m−1, un+m) (2.4)

9

on solutions of (2.1).

A simple calculation shows that the shift operator D commutes with Dt. Note that

we consider only autonomous functions and systems, i.e f and g do not explicitly

depend on t. Hence, ∂ f∂ t = 0 and ∂ g

∂ t = 0.

2.2 Conservation laws

Definition 2.6 A function ρn = ρn(un−p, un−p+1, . . . , un+q) is a (conserved) density

of (2.1) if there exists a function Jn = Jn(un−r, un−r+1, . . . , un+s), called the (associ-

ated) flux, such that

Dt ρn + ∆ Jn = 0 (2.5)

or equivalently,

Dt ρn = −∆ Jn

= −(D− I)Jn

= [Jn − Jn+1]

is satisfied on the solutions of (2.1). (2.5) is called a local conservation law.

Any shift of a density, Dkρn, is trivially a density since

Dt Dk ρn + ∆ Dk Jn = Dk (Dt ρn + ∆ Jn) = Dt 0 = 0.

The associated flux is Dk Jn.

10

Constants of motion for (2.1) are easily obtained from densities and their shifts.

Indeed, for any density ρn with corresponding flux Jn, consider

Ω =q∑

k=−p

Dkρn. (2.6)

The total time derivative of Ω is

Dt Ω = −q∑

k=−p

∆DkJn

= −q∑

k=−p

(Dk+1 −Dk

)Jn

= −(Dq+1 −D−p

)Jn.

Applying appropriate boundary conditions, for example,

limn→∞

un = 0, limn→∞

un+p = 0, (2.7)

one gets the conservation law

Dt

∞∑k=−∞

Dkρn

= − limq→∞

Dq+1Jn + limp→∞

D−pJn = 0.

For a periodic chain, where uk = uk+N , after summing over a period, one obtains

Dt

(N∑

k=0

Dkρn

)= −DN+1Jn + D0Jn = −Jn + Jn = 0.

In either case, Ω is a constant of motion of (2.1) since Ω does not change with time.

11

Definition 2.7 A density which is a total difference,

ρn = ∆ Fn, (2.8)

(so that Dt ρn = ∆ Dt Fn and therefore Jn = −Dt Fn is an associated flux), is called

trivial.

These densities lead to trivial conservation laws since

Ω =q∑

k=−p

Dk∆ Fn = Dq+1Fn −D−pFn

holds identically, not just on solutions of (2.1).

2.3 Example: The Kac-van Moerbeke lattice

Example 2.1

Consider the semi-discrete KvM lattice

un = un (un+1 − un−1), (2.9)

where as usual, un = dun/dt. Note that (2.9) is equivalent to the notationally simpler

u0 = u0(u1 − u−1).

However, as previously mentioned, for the sake of illustration we shall commit to the

first standard notation. Eq. (2.9) is often referred to as a Volterra lattice [45, 47],

although it is a special case of the two-component Volterra system [22, 36].

12

It arises in the study of Langmuir oscillations in plasmas, population dynamics,

quantum field theory, polymer science, and appears in the context of matrix factor-

ization (see references in [40]).

Eq. (2.9) appears in the literature in other forms, including

Rn = 12(e−Rn−1 − e−Rn+1), (2.10)

and

wn = wn(w2n+1 − w2

n−1), (2.11)

which relate to (2.9) by simple transformations [40]. However, (2.9) most conveniently

illustrates this algorithm.

It has the following pairs of conserved densities and fluxes [17] of rank ≤ 3:

ρ(1)n = un, J (1)

n = −un−1un, (2.12)

ρ(2)n = 1

2un

2 + unun+1, (2.13)

J (2)n = −un−1un

2 − un−1unun+1, (2.14)

ρ(3)n = 1

3un

3 + unun+1(un + un+1 + un+2), (2.15)

J (3)n = −(un−1un

3 + 2un−1un2un+1 + un−1unun+1

2 + un−1unun+1un+2). (2.16)

Then clearly (2.5) holds since

Dt ρ(1)n = un = un (un+1 − un−1) = [J (1)

n − J(1)n+1] = −∆ J (1)

n

and

Dt ρ(2)n = unun + unun+1 + unun+1

13

= un2 (un+1 − un−1) + unun+1 (un+1 − un−1) + unun+1 (un+2 − un)

= [J (2)n − J

(2)n+1]

= −∆ J (2)n .

This holds for all density flux pairs of (2.1).

2.4 Dilation Invariance: The concept behind our algorithms

Many definitions and key ideas are given here. These concepts will be used

repeatedly throughout this thesis.

Definition 2.8 Key to our methods is the concept of dilation symmetry or (scaling)

(t, u, v) → (λat, λbu, λcv), (2.17)

where λ is an arbitrary parameter.

Note that a, b, and c could be fractions. If a 6= 0, without loss of generality we may

set w( ddt

) = 1. Therefore, a = 1 and u corresponds to b derivatives with respect to t.

We denote this by u ∼ db

dtb, and v corresponds to c derivatives with respect to t, or

v ∼ dc

dtc. The terms on the right hand side of (2.9) both have rank R = 2, since each

of these (monomial) terms is quadratic [24].

Recall the KvM lattice given by (2.9). The equation is invariant under a dilation

(scaling) symmetry. Indeed, (2.9) is invariant under (t, un) → (λ−1t, λun). Therefore,

un corresponds to one derivative with respect to t. We denote this un ∼ d/dt. We

say the weight of un is one. By choice, we set w(t) = −1 or w( ddt

) = −1.

Definition 2.9 The weight, w, of a variable is equal to the number of derivatives

with respect to t the variable carries.

14

Weights of dependent variables and parameters are non-negative, rational, and in-

dependent of n. Note that weights may be fractions. The weight of each term in a

single equation is equal to the weights of the other terms in the same equation. It

is always legitimate to consider w(un) 6= 0 and w(vn) 6= 0, since zero weights would

lead to a trivial case.

Definition 2.10 The rank of a monomial is defined as the total weight of the mono-

mial. Once an equation is made scaling invariant, all the terms (monomials) in a

particular equation have the same rank. This property is called uniformity in rank.

A system is uniform in rank if every equation is uniform in rank. Note that the ranks

of the various equations may differ from each other. By definition, (2.9) is uniform

in rank and the rank is 2. Conservation laws are also uniform in rank. This property

will be addressed in further detail shortly.

Definition 2.11 An equation is scaling invariant iff weights can be assigned to each

term in the equation so that the total weights of all the terms are equal.

A system of DDEs is scaling invariant is every equation is scaling invariant. If a

system is scaling invariant, there is a consistent system of equations for the unknown

weights. So, we may assign the appropriate resulting weights to the variables. Note

that different equations in the vector equation (2.1) may have different ranks.

For more complicated DDEs, it is also convenient to consider the case when

w( ddt

) = 0, i.e. a = 0. Using this extra scale lets us to group terms in ρn according to

w( ddt

) = 0. This allows for much simpler calculations because for any given density

that is uniform in rank under Scale 1, we may separate it into pieces which are uniform

in rank under Scale 0. We now give the formal definitions of the two scales.

15

Definition 2.12 When computing the weights in a system of DDEs, one may choose

w( ddt

) = 1, and calculate the weights of the variables accordingly. We will call this

Scale 1. In other words, Scale 1 corresponds to the choice w( ddt

) = 1.

Definition 2.13 When computing the weights in a system of DDEs, one may choose

w( ddt

) = 0, and calculate the weights of the variables accordingly. We will call this

Scale 0. In other words, Scale 0 corresponds to the choice w( ddt

) = 0.

For systems that are not scaling invariant, we use the following trick: We introduce

one (or more) auxiliary parameter(s), and treat them as dependent variables with an

appropriate scaling. When introducing the parameter(s), it is vital to assume that no

parameter equals 0. Setting parameters zero would make some terms vanish and this

would alter the problem entirely. By introducing auxiliary parameters and assigning

weights to them, we can make each term in a single equation of the same weight

as the other terms. This process creates a modified, but scaling invariant system of

DDEs. Furthermore, by extending the action of the dilation symmetry to the space

of independent and dependent variables and parameters, we are able to apply our

first algorithm to any polynomial system of DDEs. However, this comes at a great

cost because by introducing such auxiliary parameter(s), the resulting ρn and Jn are

no longer linearly independent.

2.5 Density-flux pairs

Example 2.2

Recall the density flux pair in (2.15) and (2.16), namely:

ρ(3)n = 1

3un

3 + unun+1(un + un+1 + un+2), (2.18)

J (3)n = −(un−1un


2 + un−1unun+1un+2). (2.19)

16

We can verify that (2.18) and (2.19) indeed obey

Dt ρ(3)n = [J (3)

n − J(3)n+1]

since

Dt (13un

3 + unun+1(un + un+1 + un+2))

= un2un + 2ununun+1 + un

2 ˙un+1 + unun+12 + 2unun+1 ˙un+1 + unun+1un+2

+unun+1un+2 + unun+1 ˙un+2

= −un−1un3 − 2un−1un

2un+1 + unun+13 − un−1unun+1

2 + 2unun+12un+2

−un−1unun+1un+2 + unun+1un+22 + unun+1un+2un+3

= −(un−1un3 + 2un−1un

2un+1 + un−1unun+12 + un−1unun+1un+2)

+(unun+13 + 2unun+1

2un+2 + unun+1un+22 + unun+1un+2un+3)

= [J (3)n − J

(3)n+1].

In general, for any conservation law Dt ρn + ∆Jn = 0,

rank(J (3)n ) = w(Dt ) + rank(ρ(3)

n )

= 1 + rank(ρ(3)n ).

The reason for the uniformity in rank of the conservation law is obvious: In computing

Dt ρn we use the scaling invariant DDE to replace all time derivatives. In doing so,

the conservation law “inherits” the scaling symmetry of the given DDE. Therefore,

ρn, Jn, and the terms in the conservation law must all be uniform in rank. So, we

will use the fact that (2.5) is uniform in rank when computing conserved densities

17

and fluxes. Indeed, in Step 2 of our first algorithm we will build a candidate ρn as a

linear combination of monomials of a given rank (say R).

Example 2.3

Now consider the Toda lattice [18, 41]

yn = exp (yn−1 − yn)− exp (yn − yn+1). (2.20)

In (2.20), yn is the displacement from equilibrium of the nth particle with unit mass

under an exponential decaying interaction force between nearest neighbors.

With the change of variables,

un = yn, vn = exp (yn − yn+1),

lattice (2.20) can be written in algebraic form

un = vn−1 − vn, vn = vn(un − un+1). (2.21)

We can compute a couple of conservation laws for (2.21) by hand. Indeed,

un = Dt ρn = vn−1 − vn = [Jn − Jn+1]

with Jn = vn−1. We denote this first pair by

ρ(1)n = un, J (1)

n = vn−1.

18

After some work, we obtain a second pair:

ρ(2)n = 1

2un

2 + vn, J (2)n = unvn−1.

Key to our method is the observation that (2.5) and (2.21), together with the

above densities and fluxes, are invariant under the dilation symmetry

(t, un, vn) → (λ−1t, λun, λ2vn), (2.22)

where λ is an arbitrary parameter. The result of this dimensional analysis can be

stated as follows: un corresponds to one derivative with respect to t; for short, un ∼ ddt

.

Similarly, vn ∼ d2

dt2. Scaling invariance, which is a special Lie-point symmetry, is an

intrinsic property of many integrable nonlinear DDEs.

For scaling invariant systems such as (2.9) and (2.21), it suffices to consider the

dilation symmetry on the space of independent and dependent variables.

Certainly, we may verify

Dt ρ(2)n = [J (2)

n − J(2)n+1],

since

Dt [12un

2 + vn] = unun + vn

= un(vn−1 − vn) + vn(un − un+1)

= unvn−1 − un+1vn

= [J (2)n − Jn+1

(2)].

19

2.6 Tool 1: Equivalence criterion

Definition 2.14 Two densities ρn, ρn are called equivalent if ρn − ρn ∈ Im ∆,

i.e. ρn − ρn = ∆Fn for some Fn.

Equivalent densities, denoted as ρn ∼ ρn, yield the same conservation law. Note that

(2.8) expresses that ρn ∼ 0.

It is easy to verify that compositions of D and D−1 define an equivalence relation

on monomials. The equivalence criterion will be used in Step 3 of our first algorithm.

Definition 2.15 In the algorithms in this thesis, we will use the following equivalence

criterion: if two monomials, m1 and m2, are equivalent, m1 ≡ m2, then m1 −m2 =

∆Mn for some polynomial Mn that depends on un and its shifts.

For example, m1 = un−2un ≡ un−1un+1 = m2 since

un−2un − un−1un+1 = ∆Mn

= Mn+1 −Mn

= −un−1un+1 − (−un−2un), (2.23)

with Mn = −un−2un.

Definition 2.16 We call the main representative of an equivalence class, the mono-

mial of that class with n as lowest label on u (or v).

For example, unun+2 is the main representative of the class with elements un−2un,

un−1un+1, un+1un+3, etc. We use lexicographical ordering to resolve conflicts. That

is, unvn+2 (not un−2vn) is the main representative in the class with elements un−3vn−1,

un+2vn+4, etc.

20

Simply stated, all shifted monomials are equivalent. For example, un−1vn+1 ≡

un+2vn+4 ≡ un−3vn−1, with unvn+2 as the main representative. This equivalence

relation holds for any function of the dependent variables, but for the construction of

conserved densities we will apply it only to monomials.

2.7 Tool 2: The discrete Euler operator (variational derivative)

Completely integrable PDEs and DDEs exhibit unique analytic properties. For

instance, most completely integrable differential equations have infinitely many sym-

metries and conserved densities. For PDEs, in the computation of ρn and Jn, one has

to determine whether or not E = Dtρn can be written as −DxJn. To verify whether

or not Jn exists, one can use a tool from variational calculus. It is well-known that a

(continuous) function g(u(x), u′(x), u′′(x), ..., un(x)) can be integrated with respect to

x if and only if (iff) the variational derivative, Lu, of g vanishes. Formally, g = Dx h

for some function h(u(x), u′(x), u′′(x), ..., um(x)) iff Lu(g) = 0. Here, Dx refers to

total differentiation with respect to x and Lu is the continuous Euler operator (vari-

ational derivative). In particular, a function is a conserved density of a PDE iff its

time-derivative is in the kernel (Ker) of the Euler operator (see e.g. [13]).

We now present a discrete analog of this important result. One can test for

complete integrability by examining conservation laws of DDEs. Namely, E = Dt ρn =

−∆Jn, where ρn is the density and Jn is the flux. If E = Dt ρn is a total difference,

then E = Dt ρn can be written as [Jn−Jn+1], and if E = Dt ρn is not a total difference,

then the nonzero terms must vanish identically. The discrete Euler operator (discrete

variational derivative) is one tool to test if a discrete expression is a total difference.

To verify whether or not E is a total difference we will now introduce the discrete

analog of the Euler operator, denoted Lun .

21

Definition 2.17 The discrete Euler operator or discrete variational derivative, Lu,

is identified with the following equation:

Lun(g) =q∑

k=−p

D−k ∂ g∂ un+k

= Dp ∂ g∂ un−p

+ Dp−1 ∂ g∂ un−p+1

+ . . . + D0 ∂ g∂ un

+ . . . + D−q ∂ g∂ un+q

.(2.24)

Note that we can rewrite the Euler operator as

Lun(g) = ∂∂ un

q∑k=−p

D−k g

, (2.25)

and thatq∑

k=−p

Dk∆ = Dq+1 −D−p.

If we know that ρn is a density, then E = Dt ρn is a total difference because E =

Dtρn = −∆Jn, where Jn is the flux associated with ρn. In summary, if Lun(E) = 0,

then E = Dt ρn equals −∆Jn for some Jn, and vice versa. So, the Euler operator is

most useful in our first algorithm. It will be utilized in Step 3 of our first algorithm.

Example 2.4

Recall the semi-discrete KvM lattice identified by (2.9) . It is well known that

ρ(3)n = 1

3un

3 + unun+1(un + un+1 + un+2), (2.26)

J (3)n = −(un−1un


2 + un−1unun+1un+2) (2.27)

are a density-flux pair for (2.9). If we replace the correct numerical constants by

22

arbitrary constants, then

ρ(3)n = c1 un

3 + c2 un2un+1 + c3 unun+1

2 + c4 unun+1un+2 (2.28)

is the form of the density candidate. We compute

Dt ρ(3)n = 3c1un

2un + c2un2un+1 + 2c2unun+1un + 2c3unun+1un+1

+c3un+12un + c4unun+1un+2 + c4unun+2un+1 + c4un+1un+2un.(2.29)

After replacing the time derivatives (using (2.9)) we get

E = Dt ρ(3)n = 3c1un

3un+1 − 3c1un−1un3 + c2un

2un+1un+2 − c2un3un+1

+2c2un2un+1

2 − 2c2un−1un2un+1 + 2c3unun+1

2un+2 − 2c3un2un+1

2

+c3unun+13 − c3un−1unun+1

2 + c4unun+1un+2un+3 + c4unun+1un+22

+c4un2un+1un+2 − c4un−1unun+1un+2. (2.30)

Applying the discrete Euler operator to E gives

Lun(E) =q∑

k=−p

D−k ∂ E∂ un+k

= Dp ∂ E∂ un−p

+ Dp−1 ∂ E∂ un−p+1

+ . . . + D0 ∂ E∂ un

+ . . . + D−q ∂ E∂ un+q

= (9c1un2un+1 − 3c2un

2un+1 − 9c1un−1un2 + 2c2unun+1un+2

−2c4unun+1un+2 + 4c2unu2n+1 − 4c3unu

2n+1 − 4c2un−1unun+1

+2c3un+12un+2 + c3un+1

3 − c3un−1un+12 + c4un+1un+2un+3

+c4un+1un+22 − c4un−1un+1un+2) + D−1(3c1un

3 − c2un3 + c2un

2un+2

−c4un2un+2 + 4c2un

2un+1 − 4c3un2un+1 − 2c2un−1un

2 + 4c3unun+1un+2

23

+3c3unun+12 − 2c3un−1unun+1 + c4unun+2un+3 + c4unun+2

2

−c4un−1unun+2) + D−2(c2un2un+1 − c4un

2un+1 + 2c3unun+12

+c4unun+1un+3 + 2c4unun+1un+2 − c4un−1unun+1) + D−3(c4unun+1un+2)

+D(−3c1un3 − 2c2un

2un+1 − c3unun+12 − c4unun+1un+2)

= 9c1un2un+1 − 3c2un

2un+1 − 9c1un−1un2 + 2c2unun+1un+2

−2c4unun+1un+2 + 4c2unun+12 − 4c3unun+1

2 − 4c2un−1unun+1

+2c3un+12un+2 + c3un+1

3 − c3un−1un+12 + c4un+1un+2un+3 + c4un+1un+2

2

−c4un−1un+1un+2 + 3c1un−13 − c2un−1

3 + c2un−12un+1 − c4un−1

2un+1

+4c2un−12un − 4c3un−1

2un − 2c2un−2un−12 + 4c3un−1unun+1 + 3c3un−1un

2

−2c3un−2un−1un + c4un−1un+1un+2 + c4un−1un+12 − c4un−2un−1un+1

+c2un−22un−1 − c4un−2

2un−1 + 2c3un−2un−12 + c4un−2un−1un+1

+2c4un−2un−1un − c4un−3un−2un−1 + c4un−3un−2un−1 − 3c1un+13

−2c2un+12un+2 − c3un+1un+2

2 − c4un+1un+2un+3.

Grouping like terms gives

Lun(E) = (9c1 − 3c2)un2un+1 + (−9c1 + 3c3)un−1un

2 + (2c2 − 2c4)unun+1un+2

+(4c2 − 4c3)unun+12 + (−4c2 + 4c3)un−1unun+1 + (2c3 − 2c2)un+1

2un+2

+(−3c1 + c3)un+13 + (−c3 + c4)un−1un+1

2 + (c4 − c4)un+1un+2un+3

+(c4 − c3)un+1un+22 + (−c4 + c4)un−1un+1un+2 + (3c1 − c2)un−1

3

+(c2 − c4)un−12un+1 + (4c2 − 4c3)un−1

2un + (−2c2 + 2c3)un−2un−12

+(−2c3 + 2c4)un−2un−1un + (−c4 + c4)un−2un−1un+1

+(c2 − c4)un−22un−1 + (−c4 + c4)un−3un−2un−1. (2.31)

24

So, ρn will be a true density if E = Dt ρn is a total difference. Hence, Lun(E) = 0,

which leads to the system

−3c1 + c3 = 0,

c3 − c4 = 0,

3c1 − c2 = 0,

c2 − c4 = 0,

c2 − c3 = 0. (2.32)

Solving this linear system gives 3c1 =c2 =c3 =c4. If we let c1 = 13, then c2 =c3 =c4 =1.

So, application of the discrete variational derivative to E = Dt ρn and setting

the resulting expression equal to zero provides the linear system of ci. Hence, once

the form of ρn is known, we have reduced the problem of determining the constants

to simple calculus and linear algebra.

25

Chapter 3

THE FIRST MATHEMATICAL METHOD AND ALGORITHM

3.1 The method

The first algorithm requires solving the “total-difference” condition for the un-

known density (ρn) and then constructing the associated flux (Jn). It can be described

in three basic steps. First, it is necessary to determine the weights of the variables

according to both Scale 1 and Scale 0. More formal definitions for Scale 1 and Scale

0 were discussed in Chapter 2. Then, we must construct the form of the density

candidate. Finally, we determine the unknown coefficients in the density and the as-

sociated flux. Computation of the unknown coefficients can be done in two different

ways. One may choose to compute the coefficients by using a “shifting” routine. Or,

one may choose to compute the coefficients by employing the discrete Euler operator.

3.2 Steps of the algorithm

Step 1: Determine the weights of the variables according to both Scales

As mentioned previously in Chapter 2, (2.1) is either scaling invariant or can

be made scaling invariant by introducing auxiliary weighted parameters. Therefore,

every monomial in (2.1) has the same rank R. So, we get two different systems for the

weights of the dependent variables. In Chapter 2, we used one scale, Scale 1, where

t is replaced by λ−1t. This scale corresponds to w( ddt

) = 1. One can also consider the

26

case where t is left unscaled, i.e. t is replaced by λ0t. Hence, w(t) = 0 or w( ddt

) = 0.

We call this Scale 0. The multiple-scale approach was suggested by Sanders [34]. We

may then solve the two systems for the weights of the dependent variables.

Step 2: Construct the form of the density

This step involves finding the building blocks (monomials) of a polynomial den-

sity with a prescribed rank R.

Recall that all terms in the density must have the same rank R. Since we may

introduce parameters with weights, the fact that the density is a sum of monomials

of uniform rank does not necessarily imply that the density must be uniform in rank

with respect to the dependent variables.

Let V be the list of all the variables with positive weights, including the pa-

rameters with weight. The following procedure is used to determine the form of the

density of rank R:

• Form the set G of all monomials of rank R or less by taking all appropriate

combinations of different powers of the variables in V .

• For each monomial in G, introduce the appropriate number of derivatives with

respect to t so that all the monomials exactly have weight R. Gather in set H

all the terms that result from computing the various derivatives.

• Identify the monomials that belong to the same equivalence classes and replace

them by the main representatives. Call the resulting simplified set I, which

consists of the building blocks of the density with desired rank R.

• Linear combinations of the elements in I with constant coefficients, ci, gives

the form of polynomial density of rank R.

27

Using the linear combinations, construct a table of the terms in ρn based on Scale 1,

where w( ddt

) = 1. Adjacent to the terms, record the rank of each term according to

Scale 1 and also Scale 0. We may now group the terms by rank according to Scale

0. The new ρn candidate is a linear combination of any one of these groups of terms

with constants ci for i = 1, 2, ...n. Note that for simple examples, using the multiple

scale technique may not eliminate terms.

Step 3: Determine the unknown coefficients

The following “shifting” procedure simultaneously determines the constants, ci,

and the form of the flux Jn:

• Compute Dt ρn and use (2.1) to remove all t derivatives. Once we have replaced

from (2.1) we represent Dt ρn as E.

• Prior to replacement from (2.1), ρn depends on un, un+1, . . . , un+q−1, un+q, so

ρn(un, un+1, . . . , un+q−1, un+q), (3.1)

where q is a positive integer as before, and q is furthest positive shift of any

variable in ρn. However, after replacement from (2.1), Dt ρn = E depends on

un−p, un−p+1, . . . , un+q−1, un+q, thus

E(un−p, un−p+1, . . . , un+q−1, un+q), (3.2)

where p and q are positive integers as before, and p is the furthest negative shift

of any variable in E, and q is furthest positive shift of any variable in E. For

simplicity of notation, we will replace p → p and q → q. So, p always refers to

the furthest negative shift of any variable and q represents the furthest positive

shift of any variable.

28

• Use the equivalence criterion to modify E. The goal is to introduce the main

representatives and to identify the terms that match the pattern [Jn−Jn+1]. In

order to do so, we may add and subtract up-shifted or down-shifted monomials

repeatedly to E until all main representatives are introduced. For example, if

E = c1un−1unvn−1, then we may add and subtract up-shifted or down-shifted

monomials to get E = c1unun+1vn +[c1un−1unvn−1− c1unun+1vn]. Clever group-

ing ensures that E matches the pattern [Jn − Jn+1], where the flux is the first

piece in the pattern [Jn − Jn+1].

Definition 3.1 If there are terms in E that do not match the pattern [Jn −

Jn+1], then these terms form the obstruction. They must vanish identically (i.e.

the coefficients for any combination of the components of un and their shifts

must be zero).

• We know from (2.5) that E = Dt ρn = −∆Jn = [Jn − Jn+1]. Setting the

obstruction equal to zero leads to a linear system S in the unknowns, ci.

• If S has parameters, careful analysis leads to conditions on these parameters

guaranteeing the existence of densities. See [13] for a description of this com-

patibility analysis. Note that if we artificially introduced parameters to ensure

scaling invariance, then there may be freedom in many of the constant coeffi-

cients in ρn. We set such arbitrary coefficients equal to 1 one at a time when

determining the density.

The first algorithm is implemented in Mathematica as DDEDensityFlux.m.

A description of how the code was developed based on previous work is described in

Chapter 4. We also explain how the software is used, and various implementation

issues that were encountered.

29

Chapter 4

IMPLEMENTATION AND SOFTWARE

As noted previously, the methods used in the new code DDEDensityFlux.m

are based on some existing methods. These methods were already incorporated in

condens.m, diffdens.m, and InvariantsSymmetries.m which were implemented in

Mathematica.

4.1 Existing software

An algorithm for the symbolic computation of polynomial conserved densities

for systems of nonlinear evolution equations is presented in [13]. The algorithm is

implemented in Mathematica and is called condens.m [10]. The code is tested on

several well-known partial differential equations from soliton theory. For systems with

parameters, condens.m can be used to determine the conditions on these parameters

so that a sequence of conserved densities might exist. As with DDEs, the existence

of a large number of conservation laws is a predictor for integrability of the system.

A straightforward algorithm for the symbolic computation of generalized (higher-

order) symmetries of nonlinear evolution equations and lattice equations is presented

in [15] and [16]. The scaling properties of the evolution or lattice equations are used

to determine the polynomial form of the generalized symmetries. The coefficients

of the symmetry can be found by solving a linear system. The method applies to

polynomial systems of PDEs of first-order in time and arbitrary order in one space

variable. Likewise, lattices must be of first order in time but may involve arbitrary

30

shifts in the discretized space variable.

The algorithm InvariantsSymmetries.m [12] is implemented in Mathematica

and can be used to test the integrability of both nonlinear evolution equations and

semi-discrete lattice equations. With InvariantsSymmetries.m, generalized sym-

metries are obtained for several well-known systems of evolution and lattice equations

in [15] and [16]. For PDEs and lattices with parameters, the code allows one to deter-

mine the conditions on these parameters so that a sequence of generalized symmetries

exists. The existence of a sequence of such symmetries is a predictor for integrability.

The software package InvariantsSymmetries.m was used by Sakovich (Insti-

tute of Physics, National Academy of Sciences, Minsk, Belarus) and Tsuchida (Dept.

of Physics, University of Tokyo, Tokyo, Japan) in the investigation of the integrability

of coupled nonlinear Schrodinger equations.

The new algorithm DDEDensityFlux.m is based on diffdens.m [11] described

in [17] and [14]. The code diffdens.m computes conserved densities of nonlinear

differential-difference equations.

The code diffdens.m for conservation laws of DDEs was successfully used by

Tsuchida, Ujino and Wadati in a study of integrable semi-discretizations of the cou-

pled modified Korteweg- de Vries equations [42, 43] and the study of integrable semi-

discretizations of the coupled nonlinear Schrodinger equations [44].

Although condens.m, InvariantsSymmetries.m, and diffdens.m are pre-

cursors to the new algorithm DDEDensityFlux.m, there are many notable im-

provements to the new code. The additions included in DDEDensityFlux.m make

the software much more reliable and able to calculate conservation laws for a much

broader class of DDEs.

31

4.2 The new code DDEDensityFlux.m

New to DDEDensityFlux.m is the implementation of the direct computation

of the flux. This is a great improvement to the package because now the software is

completely self-testing. In fact, the code automatically verifies (2.5).

The implementation of the discrete Euler operator to solve for the unknown

constants is also new. The previous method of solving for such constants involved

a “shifting” technique that is also described in this thesis. Utilizing the discrete

Euler operator gives a much more mathematical calculation of conserved densities.

However, as mentioned earlier, the discrete Euler operator does not solve for the flux

directly.

In addition, earlier versions of this package only consider a single-scale approach.

The multi-scale approach is an improvement since now we can simplify the density

candidate by separating it into pieces that are uniform in rank according to Scale 0.

Therefore, we can calculate densities and fluxes for much more complicated systems

of DDEs.

This version of the software includes implementation of a new way to form the

density including shifts on the dependent variable. Computation of some conserva-

tion laws is greatly simplified by introducing shifts on the dependent variable when

building up the density candidate. The new code allows for two different choices to

determine the shifted dependent variables. The two ways to consider the shifts are

explained more in detail in Introducing Shifts in Chapter 5.

Finally, DDEDensityFlux.m has an improved linear systems analyzer. New

to the code is the simplification of the linear system before it is analyzed. In other

words, it eliminates any duplicate results for the ci and uses any ci equal to zero when

analyzing the rest of the equations in the linear system. So, the new code may now

32

handle more complicated linear systems for the unknown constants, ci.

So, the ideas used in condens.m, InvariantsSymmetries.m, and diffdens.m

are utilized in our new software package DDEDensityFlux.m. Although some of

the algorithm was previously developed and implemented in diffdens.m, the new

version of DDEDensityFlux.m includes many important improvements. These

improvements not only make the DDEDensityFlux.m more reliable than previous

versions, but they also make it possible to compute conservation laws for much more

complicated DDEs. Some of these more complex systems are investigated in Chapter

7 of this thesis.

DDEDensityFlux.m is written and implemented in Mathematica. The pro-

gram DDEDensityFlux.m and all data files are available via anonymous FTP from

mines.edu. The login name is anonymous and the password is your email address. The

files are in the subdirectory pub/papers/math cs dept/software/DDEDensityFlux.m.

The software is also available from the scientific section of Hereman’s homepage with

URL: http://www.mines.edu/fs home/whereman/.

DDEDensityFlux.m automatically carries out the tedious calculations needed

to determine conserved densities and fluxes for nonlinear systems of DDEs. The code

is completely menu driven which makes it very easy to use. We now proceed with an

example of how the software is used.

4.3 Using the software: A sample session

Consider the example for determining ρn and Jn of the KvM lattice of rank R=3.

In[1]:= << DDEDensityFlux.m

Reads in the code << DDEDensityFlux.m, and produces the following menu:

*** MENU INTERFACE *** (page: 1)

33

-------------------------------------------------------------

1) Kac-van Moerbeke Equation (d kdv.m)

2) Modified KdV Equation (with parameter) (d mkdv.m)

3) Modified (quadratic) Volterra Equation (d molvol.m)

4) Ablowitz-Ladik Discretization of NLS Equation (d ablnl1.m)

5) Toda Lattice (d toda.m)

6) Standard Discretization of NLS Equation (d stdnls.m)

7) Herbst/Taha Discretization of KdV Equation (d diskdv.m)

8) Herbst/Taha Discretization of mKdV Equation (d dimkdv.m)

9) Herbst/Taha Discretization

of combined KdV-mKdV Equation (d herbs1.m)

10) Self-Dual Network Equations (d dual.m)

nn) Next Page

tt) Your System

qq) Exit Program

-------------------------------------------------------------

Taking option nn shows the remaining data cases

*** MENU INTERFACE *** (page: 2)

-------------------------------------------------------------

11) Parameterized Toda Lattice (d ptoda.m)

12) Generalized Toda Lattice-1 (d gtoda1.m)

13) Generalized Toda Lattice-2 (d gtoda2.m)

14) Henon System (d henon.m)

nn) Next Page

34

tt) Your System

qq) Exit Program

-------------------------------------------------------------

Selecting choice 1 in the menu, the code continues with

****************************************************************

WELCOME TO THE MATHEMATICA PROGRAM

by UNAL GOKTAS, WILLY HEREMAN, AND HOLLY EKLUND

FOR THE COMPUTATION OF CONSERVED DENSITIES AND FLUXES.

Version 3 released on January 10, 2003

Copyright 1998-2003

****************************************************************

Working with the data file for the

Kac-van Moerbeke (or Volterra) Equation.

Equation 1 of the system with 1 equation(s):

(u1,n)′ = u1,n(−(u1,−1+n) + u1,1+n)

Note that u1,n is representative of un. If there were a u2,n, it would represent of vn,

and u3,n would represent wn, etc.

LINEAR SYSTEM FOR THE WEIGHTS CORRESPONDING TO w(d/dt)=0:

zeroweightu[1]==2 zeroweightu[2]

SOLUTION OF THE SCALING EQUATIONS for w(d/dt)=0:

zeroweightu[1] → 0

LINEAR SYSTEM FOR THE WEIGHTS CORRESPONDING TO w(d/dt)=1:

35

2 weightu[1]==1+weightu[1]

SOLUTION OF THE SCALING EQUATIONS for w(d/dt)=1:

weightu[1] → 1

Program determines the weights of the variables (and parameters)

corresponding to w(d/dt)=1.

For the given system:

* weight of u[1] and all its shifts is 1.

The rank of the rho should be an integer multiple of the lowest

weight of the DEPENDENT variable(s). Fractional weights are

allowed.

ENTER YOUR CHOICE: 1

Enter the rank of rho: 3

This produces the following density-flux pair:

****************************************************************

This is the density, rho:

13(u1,n)3 + (u1,n)2(u1,1+n) + (u1,n)(u1,1+n

2) + (u1,n)(u1,1+n)(u1,2+n)

The density has no free coefficients.

Saving the density in an output file.

****************************************************************

36

The corresponding flux J is:

−(u1,−1+n)(u1,n3)− 2(u1,−1+n)(u1,n

2)(u1,1+n)−

(u1,−1+n)(u1,n)(u1,1+n2)− (u1,−1+n)(u1,n)(u1,1+n)(u1,2+n)

The flux has no free coefficients.

****************************************************************

Apart from having access to Mathematica, users need to have the program and

the data files ready in the same directory. One may create a data file similar to those

given in the Appendix, and import it by choosing the tt option.

4.4 Options for the user

Our software package leaves a great deal of flexibility to the user. In the data

file, like those in the Appendix, the user must assign True or False values to force

flags. Depending on the posed problem and the desired result, the user may opt from

the following choices:

• Scale Options

forcesinglescale = True, and forcemultiplescale = False:

The software only uses Scale 1 when determining ρn and Jn.

forcesinglescale = False, and forcemultiplescale = True:

The software uses Scale 1 and Scale 0 when determining ρn and Jn.

forcesinglescale = False, and forcemultiplescale = False:

37

The software decides whether to use only Scale 1 when determining ρn and Jn,

or both Scale 1 and Scale 0 depending on necessity.

• Density Form Options

forceshiftsinrho = True:

The software includes extra shifts when computing the form of ρn.

forceshiftsinrho = False:

The software does not use additional shifts when computing the form of ρn.

• Shifting Options

forcemaximumshiftrhsdde = True, and forcemaximumshiftpowers = False:

The software builds up the form of ρn with shifts depending on the furthest

shift on any variable occurring in the DDE system. This is explained in more

detail as the first method in Introducing Shifts in Chapter 5.

forcemaximumshiftrhsdde = False, and forcemaximumshiftpowers = True:

The software builds up the form of ρn with shifts depending on the highest

power on any of the monomials. This is explained in more detail as the second

method in Introducing Shifts in Chapter 5.

forcemaximumshiftrhsdde = False, and forcemaximumshiftpowers = False:

The software sets the highest possible shift on any monomial to be 1 by default.

• Option for Determining Obstruction and System of Constants

forcediscreteeuler = True, forceshifting = False:

38

The software applies the discrete Euler operator to determine the obstruction

and linear system for the constants, ci. This will be illustrated for the KvM,

Toda, and AL lattices in Chapter 5.

forceshifting = True, forcediscreteeuler = False:

The software uses the “shifting” technique to compute the obstruction and

linear system for the constants, ci. This will also be illustrated for the KvM,

Toda, and AL lattices in Chapter 5.

• Options for Simplifying System of Constants

forcestripparameters = True:

The software automatically removes powers of the non-zero parameters. For

example, it removes coefficients like α, β3, but not (α2 − β) or (α + β)4.

forceextrasimplifications = True:

The software applies any ci = 0 to the remaining equations in the system.

Thus, we have produced a flexible Mathematica package DDEDensityFlux.m to

compute conserved densities and fluxes for DDEs where the user may pick and choose

from the above options.

4.5 Implementation issues

Recall that our algorithm is a simple three step algorithm to compute conserved

densities and fluxes for DDEs. New to our algorithm is the computation of fluxes and

the implementation of the discrete Euler operator option. The calculation of fluxes

for DDEs is implemented in the following manner:

39

1. The software takes E and forms a list of the terms,

(i.e. c[7]u[1][-3+n,t]u[1][n,t]u[2][-3+n,t],. . . ).

It considers each term separately.

2. It then separates each term into lists according to the dependent variables,

(i.e.u[1][-3+n,t]u[1][n,t], u[2][-3+n,t]).

3. The software considers the furthest negative shift in the first dependent variable,

(i.e. -3).

Note that this value could be zero or it could be positive.

4. It lists the terms and their shifts appropriately depending on the furthest neg-

ative shift,

(i.e. c[7]u[1][-3+n,t]u[1][n,t]u[2][-3+n,t], c[7]u[1][-2+n,t]u[1][1+n,t]u[2][-2+n,t],

c[7]u[1][-1+n,t]u[1][2+n,t]u[2][-1+n,t], c[7]u[1][n,t]u[1][3+n,t]u[2][n,t].

Note that for a zero shift, only the term itself would appear in this list. Note

that for a positive shift, the list of terms would be shifted in the opposite (down)

direction.

5. The software lists terms that go in the obstruction. These are the main repre-

sentatives of the previous list,

(i.e. c[7]u[1][n,t]u[1][3+n,t]u[2][n,t]).

6. The rest of the terms listed go into the flux list, or Jn list,

(i.e. c[7]u[1][-3+n,t]u[1][n,t]u[2][-3+n,t], c[7]u[1][-2+n,t]u[1][1+n,t]u[2][-2+n,t],

c[7]u[1][-1+n,t]u[1][2+n,t]u[2][-1+n,t]).

40

7. Then, the software considers the next term in E. It repeats the above process

and adds the appropriate parts to the obstruction or appends the appropriate

part to the Jn list.

8. Finally, the software takes the Jn list and adds each term together to determine

the flux, Jn. The terms in the obstruction list must vanish. So, this will

determine the system for the constants.

The calculation of the fluxes is key. Computing densities as well as fluxes allows for

automated verification of (2.5). This makes our software completely self-testing.

As previously mentioned, the discrete Euler operator, or the discrete variational

derivative, may be chosen by the user as a method to determine the linear system for

the constants. It is implemented in the following way:

1. The software considers each dependent variable separately,

(i.e. u[1][n,t] first, u[2][n,t] second, etc.).

2. It takes the partial derivative of E with respect to the positive shifted terms.

3. The software then applies the appropriate number of down-shifts.

4. It computes the partial derivative of E with respect to the negative shifted

terms.

5. The software applies the appropriate number of up-shifts to this result.

6. It takes the partial derivative of E with respect to the non-shifted terms.

7. Then, the software adds the above results.

8. The software groups the unknown coefficients of like monomials,

(i.e. c1unun+1 + c3unun+1 → (c1 + c3)unun+1) .

41

9. The coefficients of these monomials are set equal to zero, leading to a subsystem

for the constants.

10. The process is repeated for each dependent variable.

11. The subsystems are joined to produce the global system for the constants.

To illustrate our first algorithm we proceed with a few simple examples in Chap-

ter 5. As previously noted, once we have computed the E in Step 3, there is an

alternative way to compute the obstruction and form a linear system S in the un-

knowns, ci. Instead of using the “shifting” procedure, one can use the Euler operator

(or variational derivative). The advantage to utilizing the Euler operator is that it

computes the obstruction directly, and gives us the linear system to determine the

unknowns, ci. However, the Euler operator does not calculate the flux simultaneously

as the “shifting” procedure does. Instead, once the unknowns (ci) are determined,

we replace them in the density candidate from Step 2, and computation of the Jn is

greatly simplified. Examples illustrating use of the Euler operator follow each leading

example.

42

Chapter 5

COMPUTATION OF CONSERVATION LAWS

We will use the first algorithm to compute densities and fluxes for three examples.

First, we will return to the KvM lattice. Next, we will return to the Toda lattice.

Finally, we will examine completely new DDE. This third example is a parameterized

system known as the Ablowitz-Ladik lattice. Application of the Euler operator follows

each of these leading examples.

5.1 The Kac-van Moerbeke lattice

Example 5.1 (The Kac-van Moerbeke lattice)

Again consider the KvM lattice [25, 31, 32] given by

un = un (un+1 − un−1). (5.1)

The KvM lattice most conveniently illustrates the first algorithm. Recall that it is

invariant under the dilation symmetry

(t, un) → (λ−1t, λun). (5.2)

Hence, un corresponds to one derivative with respect to t, i.e. un ∼ ddt

.

43


Scale 1: When w( ddt

) = 1, all terms in (5.1) have the same rank if

w(un) + 1 = 2w(un).

Thus, w(un) = 1, which agrees with the dilation symmetry (5.2).


) = 0, the terms in (5.1) have the same rank when

w(un) + 0 = 2 w(un), thus, w(un) = 0 which is the trivial case. So, as expected, we

will see that Scale 0 for this simple case will not improve the algorithm.


Let us find the form of density with rank R=3 of (5.1). Forming all monomials

of un with rank 3 or less yields the list G = un3 , un

2, un. Introducing the necessary

t-derivatives, leads to

H = un3, un

2un+1, un−1un2, unun+1

2, un−1unun+1, unun+1un+2, un−2un−1un, un−12un.

Using un−2un−1un ≡ un−1unun+1 ≡ unun+1un+2, un−1un2 ≡ unun+1

2 and

un−12un ≡ un

2un+1, we obtain the list

I = un3, un

2un+1, unun+12, unun+1un+2.

Recall from (2.28) that linear combination of the terms in I with constant coefficients,

ci, gives

ρ(3)n = c1 un

3. + c2 un2un+1 + c3 unun+1

2 + c4 unun+1un+2 (5.3)

In Table 5.1, the terms are listed in the first column. The rank of the term according

to Scale 0 is listed in the second column. The rank of the term according to Scale 1

44

is listed in the third column.

Term Scale 0 Scale 1

c1 un3 0 3

c2 un2 un+1 0 3

c3 un un+12 0 3

c4 un un+1 un+2 0 3

Table 5.1. Ranks of terms in density candidate (R=3) for the KvM lattice

All terms have the same rank (0) according to Scale 0. Therefore, we are unable

to split up the ρn candidate into different choices according to Scale 0. We can see

that for this example, the process of considering various scales was trivial since all of

the terms should be included. In more complicated cases, the use of Scale 0 will be

beneficial. We merely demonstrated the process here for clarity.


If we take the derivative of ρn in (5.3) with respect to time we get

Dt ρ(3)n = 3c1un

2un + c2un2un+1 + 2c2unun+1un + 2c3unun+1un+1

+c3un+12un + c4unun+1un+2 + c4unun+2un+1 + c4un+1un+2un, (5.4)

where

un = unun+1 − un−1un,

un+1 = un+1un+2 − un+1un,

un+2 = un+2un+3 − un+2un+1. (5.5)

45

The goal is to introduce the main representatives. Using substitution of (5.5) into

(5.4) and using the equivalence criterion

un−1un2 ≡ unun+1,

un−1unun+1 ≡ un−2un−1un ≡ unun+1un+2, (5.6)

we get




+2c2un2un+1


2un+2 − 2c3un2un+1

2




After shifting and regrouping we obtain

Dt ρn = (3c1 − c2)un3un+1 + (c3 − 3c1)unun+1

3 + 2(c2 − c3)un2un+1

2

+2(c3 − c2)unun+12un+2 + (c2 − c4)un

2un+1un+2 + (c4 − c3)unun+1un+22

+[(−3c1un−1un3 − 2c2un−1un

2un+1 − c3un−1unun+12 − c4un−1unun+1un+2)

−(−3c1unun+13 − 2c3unun+1

2un+2 − c3unun+1un+22

−c4unun+1un+2un+3)]. (5.8)

The monomials outside the square brackets in (5.8) must vanish. This yields

S = 3c1 − c2 = 0, 3c1 − c3 = 0, c2 − c3 = 0, c2 − c4 = 0, c3 − c4 = 0. (5.9)

Choosing c1 = 13, one has c2 = c3 = c4 = 1.

46

The flux is determined by the first part of the square brackets. So,

Jn = −(3c1un−1un3 + 2c2un−1un

2un+1 + c3un−1unun+12 + c4un−1unun+1un+2).(5.10)

Therefore, replacing the values for the constants in (5.3) gives

ρn = 13un

3 + unun+1(un + un+1 + un+2). (5.11)

Replacing the values for the constants in (5.10) results in

Jn = −(un−1un3 + 2un−1un

2un+1 + un−1unun+12 + un−1unun+1un+2). (5.12)

Analogously, for (5.1) we computed the densities of rank ≤ 5 :

ρ(1)n = un, (5.13)

ρ(2)n = 1

2un

2 + unun+1, (5.14)

ρ(3)n = 1

3un

3 + unun+1(un + un+1 + un+2), (5.15)

ρ(4)n = 1

4un

4 + un3un+1 + 3

2un

2un+12 + unun+1

2(un+1 + un+2)

+unun+1un+2(un + un+1 + un+2 + un+3), (5.16)

ρ(5)n = 1

5un

5 + unun+1(un3 + un+1

3) + 2un2un+1

2(un + un+1)

+unun+1un+2(un2 + unun+2 + un+1un+3) + 3unun+1

2un+2

(un + un+1 + un+2) + unun+1un+22(un+2 + un+3)

+unun+1un+2un+3(un + un+1 + un+2 + un+3 + un+4). (5.17)

We now return to the application of the discrete Euler operator to the KvM lattice,

previously discussed in Chapter 2.

47

Example 5.2 (The Kac-van Moerbeke lattice revisited)

To reiterate, there is an alternative way to form the linear system of ci using the

variational derivative (or Euler operator) which becomes effective in Step 3 of our

first algorithm. We have already computed Dt ρ(3)n for the KvM lattice (5.1) in Step

3 of our first algorithm. Recall from Chapter 2, that after replacement from (5.1),

Dt ρ(3)n reduced to




+2c2un2un+1


2un+2 − 2c3un2un+1

2




. Applying the Euler operator or variational derivative to (5.18) gives

−3c1 + c3 = 0,

c3 − c4 = 0,

3c1 − c2 = 0,

c2 − c4 = 0,

c2 − c3 = 0. (5.19)

Hence, if c1 = 13, then c2 = c3 = c4 = 1. Inserting the values for the constants into

(5.3) gives

ρn = 13un

3 + unun+1(un + un+1 + un+2). (5.20)

48

The computation of Jn of rank R=4 will be simplified since we already determined

the constants. In order to compute Jn we take the total derivative of ρn, replace

t-derivatives using (5.1), and group terms with un−1 together and terms with un

together. This results in

Dt ρn = (−un−1un3 − 2un−1un

2un+1 − un−1unun+12 − un−1unun+1un+2)

−(−unun+13 − 2unun+1

2un+2 − unun+1un+22

−unun+1un+2un+3). (5.21)

By (2.5) it is evident that Jn matches the first part of the pattern in (5.21). Therefore,



5.2 The Toda lattice

Example 5.3 (The Toda lattice)

The Toda lattice illustrates how our first algorithm works for multi-component DDEs.

Recall

un = vn−1 − vn, vn = vn(un − un+1). (5.23)

As mentioned in Chapter 2, it is invariant under

(t, un, vn) → (λ−1t, λun, λ2vn). (5.24)

49


Requiring uniformity in rank for each equation in (5.23) allows one to compute

the weights of the dependent variables.

Scale 1: If w( ddt

) = 1, then

w(un) + 1 = w(vn),

w(vn) + 1 = w(un) + w(vn).

So, w(un) = 1, w(vn) = 2, which is consistent with (5.24).


) = 0, then w(un) = 0 and w(vn) = 0 which is again trivial.


As an example, let us compute the form of the density of rank R=3. we first list

all monomials in un and vn of rank 3 or less: G = un3, un

2, unvn, un, vn. Next, for

each monomial in G, we introduce enough t-derivatives, so that each term exactly has

weight 3. Thus, using (5.23),

d0

dt0 (un3) = un

3,d0

dt0 (unvn) = unvn,

d

dt(un

2) = 2unun = 2unvn−1 − 2unvn,d

dt(vn) = vn = unvn − un+1vn,

d2

dt2 (un) =d

dt(un) =

d

dt(vn−1 − vn) = un−1vn−1 − unvn−1 − unvn + un+1vn.

Gathering any resulting terms gives set H = un3, unvn−1, unvn, un−1vn−1, un+1vn.

Next, we identify members that belong to the same equivalence classes and replace

them by the main representatives. For example, since unvn−1 ≡ un+1vn both are re-

50

placed by unvn−1. Doing so, H is replaced by I = un3, unvn−1, unvn, which contains

the building blocks of the density. Linear combination of the monomials in I with

constant coefficients, ci, gives the form of the density:

ρn = c1 un3 + c2 unvn−1 + c3 unvn. (5.25)

As before, listing the various ranks of the monomials gives Table 5.2.


c1 un3 0 3

c2 un vn−1 0 3c3 un vn 0 3

Table 5.2. Ranks of terms in density candidate (R=3) for the Toda lattice

So, again Scale 0 does not allow any simplification. All terms must be included

when building up ρn.


Now we determine the coefficients c1 through c3 by requiring that (2.5) holds.

During this step we also compute the unknown flux Jn using the “shifting” procedure.

Taking the total derivative of (5.25), and replacing all un+p from (5.23) gives

E = Dt ρn = 3c1un2vn−1 − c2un

2vn−1 + c3un2vn − 3c1un

2vn

+c3vn−1vn − c2vn−1vn + c2un−1unvn−1 + c2vn−12

−c3unun+1vn − c3vn2. (5.26)

51

After shifting and regrouping we get,

Dt ρn = (3c1 − c2)un2vn−1 + (c3 − 3c1)un

2vn

+(c3 − c2)vnvn+1 + [(c3 − c2)vn−1vn − (c3 − c2)vnvn+1]

+c2unun+1vn + [c2un−1unvn−1 − c2unun+1vn]

+c2vn2 + [c2vn−1

2 − c2vn2]− c3unun+1vn − c3vn

2. (5.27)

Next, we group the terms outside of the square brackets and move the pairs inside

the square brackets to the bottom. We rearrange the latter terms so that they match

the pattern [Jn − Jn+1]. Hence,

Dt ρn = (3c1 − c2)un2vn−1 + (c3 − 3c1)un

2vn

+(c3 − c2)vnvn+1 + (c2 − c3)unun+1vn + (c2 − c3)vn2

+[(c3 − c2)vn−1vn + c2un−1unvn−1 + c2vn−12

−(c3 − c2)vnvn+1 + c2unun+1vn + c2vn2]. (5.28)

The terms outside the square brackets must all vanish, yielding

S = 3c1 − c2 = 0, 3c1 − c3 = 0, c2 − c3 = 0. (5.29)

The solution is 3c1 = c2 = c3. Since densities can only be determined up to a multi-

plicative constant, we choose c1 = 13, c2 = c3 = 1.

The terms inside the square brackets determine that

Jn = (c3 − c2)vn−1vn + c2un−1unvn−1 + c2vn−12. (5.30)

52

Substituting the values for the constants into (5.25) and (5.30) gives

ρn = 13un

3 + un(vn−1 + vn) (5.31)

and

Jn = un−1unvn−1 + vn−12. (5.32)

Analogously, we computed conserved densities of rank ≤ 5 for (5.23). They are:

ρ(1)n = un, (5.33)

ρ(2)n = 1

2un

2 + vn, (5.34)

ρ(3)n = 1

3un

3 + un(vn−1 + vn), (5.35)

ρ(4)n = 1

4un

4 + un2(vn−1 + vn) + unun+1vn + 1

2vn

2 + vnvn+1, (5.36)

ρ(5)n = 1

5un

5 + un3(vn−1 + vn) + unun+1vn(un + un+1)

+unvn−1(vn−2 + vn−1 + vn) + unvn(vn−1 + vn + vn+1). (5.37)

Ignoring irrelevant shifts in n, these densities agree with the results in [18].

Example 5.4 (The Toda lattice revisited)

Instead of using the “shifting” method, we alternatively proceed by using the Euler

operator (or variational derivative) as was done previously for (5.1). However, the

Toda lattice is a two component system so we must use the variational derivative

for each component separately. First, we consider the un component, then the vn

component. By applying the Euler operator for un to (5.26) we get the following:

Lun(E) =q∑

k=−p

D−k ∂ E∂ un+k

53


+ Dp−1 ∂ E∂ un−p+1

+ . . . + D0 ∂ E∂ un

+ . . . + D−q ∂ E∂ un+q

,

where p is the furthest negative shift on any un term and q is the furthest positive

shift on any un term. Here, p = 1 and q = 1 . So,

Lun(E) = 6c1unvn−1 − 2c2unvn−1 + 2c3unvn − 6c1unvn − c3un+1vn

−c3un−1vn−1 + c2un+1vn.

Grouping like terms gives

Lun(E) = (6c1 − 2c2)unvn−1 + (2c3 − 6c1)unvn + (c2 − c3)un−1vn−1

+(c3 − c2)un+1vn. (5.38)

As before, we require Lun(E) ≡ 0. This produces the following linear system:

3c1 − c2 = 0,

3c1 − c3 = 0,

c2 − c3 = 0. (5.39)

Then, by applying the Euler operator for vn to (5.26) we get:

Lvn(E) =q∑

k=−p

D−k ∂ E∂ vn+k

= Dp ∂ E∂ vn−p

+ Dp−1 ∂ E∂ vn−p+1

+ . . . + D0 ∂ E∂ vn

+ . . . + D−q ∂ E∂ vn+q

,

54

This time, p = 1 and q = 0, hence

Lvn(E) = c3un2 − 3c1un

2 + c3vn−1 − c2vn−1 − c3unun+1 − 2c3vn + 3c1un+12

−c2un+12 + c3vn+1 − c2vn+1 + c2unun+1 + 2c2vn.

Grouping terms gives

Lvn(E) = (c3 − 3c1)un2 + (c3 − c2)vn−1 + (−c3 + c2)unun+1 + (−2c3 + 2c2)vn

+(3c1 − c2)un+12 + (c3 − c2)vn+1. (5.40)

Setting Lvn(E) ≡ 0 results again in (5.39), which coincides with S in (5.29). Solving

this system gives 3c1 = c2 = c3.

If we set c1 = 13

then c2 = c3 = 1. Substituting these values into (5.25) gives

ρn = 13un

3 + un(vn−1 + vn). (5.41)

Again, now that we know the constants, computation of the Jn is greatly simplified.

Taking the total derivative of ρn, replacing from (2.21), and grouping terms gives

Dt ρn = (un−1unvn−1 + vn−12)− (unun+1vn + vn

2)

= [Jn − Jn+1]. (5.42)

So,

Jn = un−1unvn−1 + vn−12. (5.43)

Next, we will examine a completely new parameterized DDE.

55

5.3 The Ablowitz-Ladik lattice

Example 5.5 (The Ablowitz-Ladik lattice)

In [2, 3], Ablowitz and Ladik studied properties of the following integrable

discretizations of the Nonlinear Schrodinger (NLS) equation:

i un = un+1 − 2un + un−1 ± u∗nun(un+1 + un−1), (5.44)

where u∗n is the complex conjugate of un. We continue with the + sign; the other case

is analogous. Instead of splitting un into its real and imaginary parts, we treat un

and vn = u∗n as independent variables and augment (5.44) with its complex conjugate

equation. Absorbing i in the scale on t, we get

un = un+1 − 2un + un−1 + unvn(un+1 + un−1),

vn = −(vn+1 − 2vn + vn−1)− unvn(vn+1 + vn−1). (5.45)

Since vn = u∗n, we have w(vn) = w(un).

5.3.1 Introducing a first weighted parameter (α)

Neither of the equations in (5.45) is uniform in rank. To circumvent this problem

we introduce an auxiliary parameter α with weight, and replace (5.45) by

un = α(un+1 − 2un + un−1) + unvn(un+1 + un−1),

vn = −α(vn+1 − 2vn + vn−1)− unvn(vn+1 + vn−1). (5.46)

56


Scale 1: Uniformity in rank requires that for w( ddt

) = 1,

w(un) + 1 = w(α) + w(un) = 2w(un) + w(vn),

w(vn) + 1 = w(α) + w(vn) = 2w(vn) + w(un), (5.47)

which yields w(vn) = 1− w(un), w(α) = 1, or, un2 ∼ vn

2 ∼ α ∼ ddt

.

Scale 0: However, since w( ddt

) = 0, makes w(vn) = −w(un), and requires that w(α) =

0, the candidate expression for ρn could not depend on α as it does when w( ddt

) = 1.

5.3.2 Introducing a second weighted parameter (β)

Therefore, we must introduce yet another parameter β when using multiple

scales. This transforms (5.46) into

un = α(βun+1 − 2βun + βun−1) + unvn(βun+1 + βun−1),

vn = −α(βvn+1 − 2βvn + βvn−1)− unvn(βvn+1 + βvn−1). (5.48)

The computations now proceed as in the previous examples. So,

w(d

dt) + w(un) = w(α) + w(β) + w(un) = 2w(un) + w(β) + w(vn),

w(d

dt) + w(vn) = w(α) + w(β) + w(vn) = 2w(vn) + w(β) + w(un). (5.49)

Scale 1: For Scale 1, where w( ddt

) = 1,

w(α) + w(β) = w(un) + w(β) + w(vn) = 1,

w(α) + w(β) = w(vn) + w(β) + w(un) = 1. (5.50)

57

Therefore, w(α) = 1−w(β) and w(un) = 1−w(β)−w(vn). There is too much freedom

in these weight values. In order to proceed, we pick w(vn) = 14

and w(β) = 12. This

implies w(un) = 14

and w(α) = 12.

Scale 0: For Scale 0, where w( ddt

) = 0,

w(α) + w(β) = w(un) + w(β) + w(vn) = 0,

w(α) + w(β) = w(vn) + w(β) + w(un) = 0. (5.51)

So, w(α) = −w(β) and w(un) = −w(β)− w(vn).


Let us find the form of density with rank R=12

of (5.48). Forming all monomials

of un with rank R=12

or less yields the list G = un2, unvn, vn

2, un, vn. In this case,

we cannot introduce any t-derivatives. Therefore, H = un2, unvn, vn

2 = I each of

rank R=12. So, I = H = un

2, unvn, vn2. A linear combination of the terms in I

with constant coefficients, ci, gives

ρn = c1 un2 + c2 unvn + c3 vn

2. (5.52)

Table 5.3 lists the terms without shifts in the density in the first column, and their

respective ranks according to both Scale 0 and Scale 1 in the columns that follow.


c1 un2 −2w(β)− 2w(vn) 1

2

c2 un vn −w(β) 12

c3 vn2 2w(vn) 1

2

Table 5.3. Ranks of terms in density candidate (R=12) for the AL lattice

58

So, there are three different choices, where each of the monomials are uniform in rank

according to Scale 0.

• Choice 1 is a scalar multiple of the term of rank −w(β) .

• Choice 2 is a scalar multiple of the term of rank −2w(β)− 2w(vn) .

• Choice 3 is a scalar multiple of the term of rank −2w(vn) .

Considering each of these choices separately simplifies the computations since we may

now address each case separately.


Choice 1 gives ρn = c1unvn. So,

Dt ρn = c1unvn + c1unvn.

Again, replacing the un from the (5.48) and performing the necessary shifts gives

c1 = 0. However, we could have considered Choice 2, or Choice 3. Choice 2 gives

ρn = c1un2. So,

Dt ρn = 2c1unun.

Replacing the un from (5.48) and performing the necessary shifts gives c1 = 0. Finally,

Choice 3 gives ρn = c1vn2. So,

Dt ρn = 2c1vnvn.

This again gives that c1 = 0. Therefore, we get only trivial densities resulting from

any of these Choices.

59

Note, however, that if we were to consider a higher rank, namely rank R = 32,

we would get the result that

ρn = c1unvn−1 + c2unvn+1. (5.53)

This comes at a great cost however. Considering higher rank equations means the

inclusion of many more terms, and the calculations become much more tedious.

5.3.3 Introducing Shifts

A better method for calculating densities and fluxes of (5.48) is to consider the

shifts of the variables in the list V and G in Step 2 of the first algorithm. One may

introduce shifts in two different ways.

• The first method of introducing the necessary shifts is implemented in the fol-

lowing manner: One examines the right hand side of (2.1) and determines the

highest shift, r, on any variable in the system. Note that this shift, r, could

either be in the positive or negative direction. Then, the set V includes all

shifted monomials up to n + r. Consider the following hypothetical example:

un = vn−1 − vn+1,

vn = vn−3un+2 − vn−4un+1. (5.54)

Then the furthest shifted variable in (5.54) is vn−4. Therefore, r = 4. So, for

this example, when constructing ρn of rank 2,

V = un, un+1, un+2, un+3, un+4, vn.

60

Therefore, taking all appropriate combinations gives

G = un2, unun+1, unun+2, unun+3, unun+4, un, vn.

• The second method of introducing the necessary shifts is more effective. We will

use this method when calculating densities. It works in the following way: To

construct ρn of rank R, with w(un) = w, we begin by computing power, p = Rw.

Then, when building set G, we not only include upn, but also unun+1 . . . un+p−1.

For (5.54), and ρn of rank R=2, we have p = 21

= 2. Therefore, for this artificial

example,

G = un2, unun+1, un, vn.

Returning to (5.48), and using the second method for shifts,

G = un2, unvn, vn

2, un, vn, unun+1, un−1vn, vnvn+1, vn−1vn, unvn−1, unvn+1,

un+1vn, un−1vn, un+1vn+1, un−1vn−1.

We would not introduce any t-derivatives since we are looking for a density of rank

R=12. So,

H = un2, unvn, vn

2, un, vn, unun+1, un−1vn, vnvn+1, vn−1vn, unvn−1, unvn+1,

un+1vn, un−1vn, un+1vn+1, un−1vn−1.

61

Clearly,

unun+1 ≡ un−1un,

vnvn+1 ≡ vn−1vn,

un+1vn ≡ unvn−1,

un−1vn ≡ unvn+1,

unvn ≡ un+1vn+1 ≡ un−1vn−1.

Therefore, I = un2, unun+1, vn

2, vnvn+1, unvn, unvn−1, unvn+1, and

ρn = c1un2 + c2unun+1 + c3vn

2 + c4vnvn+1 + c5unvn + c6unvn−1 + c7unvn+1. (5.55)

We list the ranks of the various terms with shifts according to Scale 0 and Scale 1 in

Table 5.4.


c1 un2 −2w(β)− 2w(vn) 1

2

c2 un un+1 −2w(β)− 2w(vn) 12

c3 vn2 2w(vn) 1

2

c4 vn vn+1 2w(vn) 12

c5 un vn −w(β) 12

c6 un vn−1 −w(β) 12

c7 un vn+1 −w(β) 12

Table 5.4. Ranks of terms in density candidate (R=12) for the AL lattice

Again there are three different choices according to Scale 0.

62

• Choice 1 is a linear combination of the 3 terms of rank −w(β) .

• Choice 2 is a linear combination of the 2 terms of rank −2w(β)− 2w(vn) .

• Choice 3 is a linear combination of the 2 terms of rank 2w(vn).

Choice 2 and Choice 3 do not result in any densities or fluxes. Choice 1 is a linear

combination of the last three terms. We will continue with this density candidate of

rank equal to −w(β) according to Scale 0,

ρn = c1unvn−1 + c2unvn + c3unvn+1. (5.56)

Taking the necessary derivatives gives,

E = Dt ρn = β(−αc1unvn−2 + αc1un−1vn−1 − αc2unvn−1 + αc1un+1vn−1

−c1un−1unvn−2vn−1 + αc2un−1vn − αc1unvn − αc3unvn + αc2un+1vn

−c2un2vn−1vn + c1unun+1vn−1vn + c2un−1unvn

2 + c2unun+1vn2

+αc3un−1vn+1 − αc2unvn+1 + αc3un+1vn+1 + c3un−1unvnvn+1

−c2un2vnvn+1 − αc3unvn+2 − c3unun+1vn+1vn+2). (5.57)

After shifting and regrouping we get

Dt ρn = β(−c2un2vnvn+1 − c2un

2vn−1vn + c2unun+1vn2 + c2unun+1vn+1

2

+[(−c1un−1unvn−2vn−1 − c1αunvn−2 + c1αun−1vn−1 − c2αunvn−1

+c2αun−1vn + c2un−1unvn2 − c3αunvn + c3αun−1vn+1 + c3un−1unvnvn+1)

−(−c1unun+1vn−1vn − c1αun+1vn−1 + c1αunvn − c2αun+1vn

+c2αunvn+1 + c2unun+1vn+12 − c3αun+1vn+1 + c3αunvn+2

63

+c3unun+1vn+1vn+2)]). (5.58)

Since the terms outside the square brackets must vanish, we can therefore only de-

termine that c2 = 0. Substituting values for the constants into (5.56) gives

ρn = c1unvn−1 + c3unvn+1. (5.59)

Clearly then, there is freedom in the constant coefficients here. Setting c1 = 1,

ρn = unvn−1 + c3unvn+1. (5.60)

So, the part of the density with coefficient c3 is

ρn = unvn+1, (5.61)

and the part of the density without a free coefficient is

ρn = unvn−1. (5.62)

We know Jn must match the first part of the square brackets in (5.58). Substituting

c2 = 0 gives

Jn = β(−c1un−1unvn−2vn−1 − c1αunvn−2 + c1αun−1vn−1 − c3αunvn + c3αun−1vn+1

+c3un−1unvnvn+1). (5.63)

64

In addition we list some conserved densities of (5.45), which correspond with those

in [2]:

ρ(1)n = c1(

12un

2vn−12 + unun+1vn−1vn + unvn−2)

+ c2(12un

2vn+12 + unun+1vn+1vn+2 + unvn+2), (5.64)

ρ(2)n = c1[

13un

3vn−13 + unun+1vn−1vn(unvn−1 + un+1vn + un+2vn+1)

+ unvn−1(unvn−2 + un+1vn−1) + unvn(un+1vn−2 + un+2vn−1) + unvn−3]

+ c2[13un

3vn+13 + unun+1vn+1vn+2(unvn+1 + un+1vn+2 + un+2vn+3)

+ unvn+2(unvn+1 + un+1vn+2) + unvn+3(un+1vn+1 + un+2vn+2)

+unvn+3]. (5.65)

If defined on an infinite interval, as shown in [2], scheme (5.44) admits infinitely many

independent conserved densities. Although it is a constant of motion, we cannot derive

the Hamiltonian of (5.44),

H = −i∑n

[u∗n(un−1 + un+1)− 2 ln(1 + unu∗n)], (5.66)

since it has a logarithmic term [1].

Example 5.6 (The Ablowitz-Ladik lattice revisited)

For (5.57) in Step 3 we use the Euler operator to determine the constants ci.

Considering the terms involving un,

Lun(E) =q∑

k=−p

D−k ∂ E∂ un+k


+ Dp−1 ∂ E∂ un−p+1

+ . . . + D0 ∂ E∂ un

+ . . . + D−q ∂ E∂ un+q

,

65

where p is the furthest shift in the negative direction and q is the furthest shift in the

positive direction. So, p = 1 and q = 1, and

Lun(E) = βc2un−1vn−12 − 2βc2unvn−1vn + βc2un−1vn

2 + βc2un+1vn2

−2βc2unvnvn+1 + βc2un+1vn+12.

As before, we demand that Lun(E) ≡ 0. Therefore, c2 = 0

Then, by applying the Euler operator to (5.57), for the terms involving vn we

get

Lvn(E) =q∑

k=−p

D−k ∂ E∂ un+k


+ Dp−1 ∂ E∂ un−p+1

+ . . . + D0 ∂ E∂ un

+ . . . + D−q ∂ E∂ un+q

,

where p is the furthest negative shift on any vn and q is the furthest positive shift on

vn. With p = 2 and q = 2, we get

Lvn(E) = −βc2un−12vn−1 − βc2un

2vn−1 + 2βc2un−1unvn + 2βc2unun+1vn

−βc2un2vn+1 − βc2un+1

2vn+1.

Since Lvn(E) must vanish identically, c2 = 0. Note that the result is consistent with

the previous investigation, and

ρn = c1unvn−1 + c3unvn+1. (5.67)

66

To compute Jn, we can take the total derivative of ρn given in (5.67), replace from

(5.48) and group terms. Hence,

Dt ρn = β(−c1un−1unvn−2vn−1 − c1αunvn−2 + c1αun−1vn−1 − c3αunvn

+c3αun−1vn+1 + c3un−1unvnvn+1)− (−c1unun+1vn−1vn

−c1αun+1vn−1 + c1αunvn − c3αun+1vn+1 + c3αunvn+2

+c3unun+1vn+1vn+2). (5.68)

Since Jn must match the first part of the pattern in (5.68),

Jn = β(−c1un−1unvn−2vn−1 − c1αunvn−2 + c1αun−1vn−1 − c3αunvn + c3αun−1vn+1

+c3un−1unvnvn+1), (5.69)

which is identical to (5.63).

67

Chapter 6

A SECOND METHOD TO DETERMINE DENSITIES AND FLUXES

We know that the densities ρn and fluxes Jn are related by (2.5). In principle,

to compute Jn = −∆−1Dt ρn, one needs to invert the operator ∆ = D− I.

Working with the formal inverse,

∆−1 = D−1 + D−2 + D−3 + . . . , (6.1)

would be impractical. Instead, we present a five step algorithm which circumvents

the above infinite formal series.

Assume that ρn is a valid density. The idea is to recast Dt ρn into the form

−∆Jn, leading to the desired flux Jn. As suggested by Hickman [20], this can be done

as follows. Using

I = (D− I + I) D−1 = ∆ D−1 + D−1, (6.2)

one writes

Dt ρn = E = ∆ D−1E + D−1 E = −∆Jn,

where the form −D−1E will become part of Jn and D−1E is the obstruction.

6.1 The second algorithm

Step 1: Determine the lowest shift

Starting from a valid density ρn, we first compute Dt ρn. Then, we replace all

68

time derivatives, and finally use the equivalence relation to obtain the expression E.

By examining the subscripts of the shifted terms of E, we determine the lowest

shift (p). Note that p is the furthest negative shift on any variable in the system. As

before, p is positive.

Step 2: Splitting E

The expression E is split into

E = A(0) + A(1), (6.3)

where A(0) has all terms independent of the lowest shifted variable un−p that appears

in E, and A(1) has the terms dependent on the lowest shifted variable un−p .

Using (6.2) we write

E = ∆ D−1A(0) + D−1A(0) + A(1).

Now, if D−1A(0) + A(1) = 0, we are finished and Jn = −D−1A(0).

Step 3: Computing D−1A(0)

D−1A(0) comes from a single down-shift on A(0).

Step 4: Computing D−1A(0) + A(1)

Combining the result from the previous step with A(1) gives D−1A(0) + A(1). If

rearranging and grouping terms leads to D−1A(0) + A(1) = 0, then the algorithm

terminates and Jn = −D−1A(0).

Step 5: Repeat as necessary

If D−1A(0) + A(1) 6= 0, then we again split into

D−1A(0) + A(1) = A(2) + A(3), (6.4)

69

where A(2) has terms independent of un−p, and A(3) has terms dependent on un−p.

Applying (6.2) to A(2) we get

E = ∆ D−1A(0) + ∆ D−1A(2) + D−1A(2) + A(3).

If D−1A(2) + A(3) = 0 then the algorithm terminates, and the flux is

Jn = − D−1(A(0) + A(2)

). If not, we repeat the process. This procedure must

eventually terminate since ρn is a valid density.

More importantly, the above procedure may be applied to the problem of com-

puting densities.

Without loss of generality, we can assume that ρn = ρn(un, un+1, . . . , un+q). For

(2.1), we have

E =q∑

k=0

∂ ρn

∂ un+kDkf(un−l, . . . , un+m)

= ∂ ρn

∂ unf(un−l, . . . , un+m) +

q∑k=1

∂ ρn

∂ un+kDkf(un−l, . . . , un+m).

Applying (6.2) to the second term of this equation we obtain

E = ∂ ρn

∂ unf +

(∆ D−1 + D−1

)( q∑k=1

∂ ρn

∂ un+kDkf

)

= ∂ ρn

∂ unf + (∆ + I)

( q∑k=1

∂ D−1 ρn

∂ un+k−1Dk−1f

)

= ∂ ρn

∂ unf + ∆

q−1∑k=0

∂ D−1 ρn

∂ un+kDkf +

q−1∑k=0

∂ D−1 ρn

∂ un+kDkf

=(

∂∂ un

(ρn + D−1ρn

))f + ∆

q−1∑k=0

∂ D−1 ρn

∂ un+kDkf +

q−1∑k=1

∂ D−1 ρn

∂ un+kDkf

with f = f(un−l, . . . , un+m). Next, we repeat this procedure by applying (6.2) to the

70

last term. After a further q − 2 applications, we get

E =(

∂∂ un

(ρn + D−1ρn + D−2ρn

))f + ∆

q−1∑k=0

∂ D−1 ρn

∂ un+kDkf +

q−2∑k=0

∂ D−2 ρn

∂ un+kDkf

+

q−2∑k=1

∂ D−2 ρn

∂ un+kDkf

= · · ·

=

∂∂ un

q∑j=0

D−jρn

f + ∆

q∑j=1

q−j∑k=0

∂ D−jρn

∂ un+kDkf

= Lun(ρn) f + ∆

q∑j=1

q−j∑k=0

∂ D−jρn

∂ un+kDkf

by (2.25).

If Lun(ρn) f = 0 then ρn is a trivial density. For ρn to be a non-trivial density,

we require

Lun(ρn) f = ∆ h (6.5)

for some h with h 6= 0. In this case the associated flux is

J = − h−q∑

j=1

q−j∑k=0

∂ D−jρn

∂ un+kDkf.

We could apply the discrete Euler operator to Lun(ρn) f to determine conditions such

that (6.5) holds. However, we can also repeat the above strategy by splitting this

expression into a part, A(0), that does not depend on the lowest shifted variable and

the remaining terms. We then apply (6.2) repeatedly to A(0) (removing the total

difference terms generated by (6.2) from A(0)) until we obtain a term that involves

both the lowest and highest shifted variables. By noting that a necessary condition

71

for g = g(un−p, . . . , un+q) to be total difference is

∂2

∂ un−p∂ un+qg = 0, (6.6)

we then obtain conditions for (6.5) to hold. This procedure must be repeated until all

the terms in Lun(ρn) f have been rewritten as a total difference. Next, we illustrate

this second algorithm for the same leading examples.

6.2 The Kac-van Moerbeke lattice

Example 6.1 (The Kac-van Moerbeke lattice revisited once more)


From (5.18), we know

E = Dt ρn = 3c1un3un+1 − 3c1un−1un

3 + c2un2un+1un+2 − c2un

3un+1

+2c2un2un+1


2un+2 − 2c3un2un+1

2



+c4un2un+1un+2 − c4un−1unun+1un+2.

Therefore, it is clear that the lowest shift is p = 1.

Step 2: Splitting Dt ρn

We may split the result into two pieces as follows:

Dt ρn = A(0) + A(1)

= (3c1un3un+1 + c2un

2un+1un+2 − c2un3un+1 + 2c2un

2un+12 + 2c3unun+1

2un+2

−2c3un2un+1

2 + c3unun+13 + c4unun+1un+2un+3 + c4unun+1un+2

2

72

−c4un2un+1un+2) + (−3c1un−1un

3 − 2c2un−1un2un+1 − c3un−1unun+1

2

−c4un−1unun+1un+2), (6.7)

where A(0) has terms independent of un−1, and A(1) has terms dependent on un−1.


Taking one down-shift on A(0) gives

D−1A(0) = D−1(3c1un3un+1 + c2un

2un+1un+2 − c2un3un+1 + 2c2un

2un+12

+2c3unun+12un+2 − 2c3un

2un+12 + c3unun+1

3 + c4unun+1un+2un+3

+c4unun+1un+22 − c4un

2un+1un+2)

= 3c1un−13un + c2un−1

2unun+1 − c2un−13un + 2c2un−1

2un2

+2c3un−1un2un+1 − 2c3un−1

2un2 + c3un−1un

3 + c4un−1unun+1un+2

+c4un−1unun+12 − c4un−1

2unun+1. (6.8)


So,

D−1A(0) + A(1) = (3c1un−13un + c2un−1

2unun+1 − c2un−13un + 2c2un−1

2un2

2c3un−1un2un+1 − 2c3un−1

2un2 + c3un−1un

3

+c4un−1unun+1un+2 + c4un−1unun+12 − c4un−1

2unun+1)

+(−3c1un−1un3 − 2c2un−1un

2un+1 − c3un−1unun+12

−c4un−1unun+1un+2). (6.9)

From (6.9) we can see that all monomials depend on un−1. Rearranging the terms

73

produces

D−1A(0) + A(1) = (−3c1 + c3)un−1un3 + (−2c2 + 2c3)un−1un

2un+1

+(c4 − c3)un−1unun+12 + (3c1 − c2)un−1

3un

+(c2 − c4)un−12unun+1 + (2c2 − 2c3)un−1

2un2. (6.10)

It is easily seen that (6.10) equals 0 if 3c1 = c2 = c3 = c4. Therefore, Step 5 is

unnecessary and we do not have to repeat Step 3 and Step 4. Substitution of c1 = 13,

c2 = c3 = c4 = 1 into (5.3) gives

ρn =1

3un

3 + un2un+1 + unun+1

2 + unun+1un+2. (6.11)

Since Jn = −D−1A(0),



6.3 The Toda lattice

Example 6.2 (The Toda lattice revisited once more)



E = Dt ρn = 3c1un2vn−1 − c2un

2vn−1 + c3un2vn − 3c1un

2vn

+c3vn−1vn − c2vn−1vn + c2un−1unvn−1 + c2vn−12

−c3unun+1vn − c3vn2.

74

So, the lowest shift is p = 1.


Splitting the result gives

Dt ρn = A(0) + A(1)

= (c3un2vn − 3c1un

2vn − c3unun+1vn − c3vn2) + (3c1un

2vn−1

−c2un2vn−1 + c3vn−1vn − c2vn−1vn + c2un−1unvn−1 + c2vn−1

2), (6.13)

where A(0) has terms independent of un−1 and vn−1 and A(1) has terms dependent on

un−1 or vn−1.


Applying a single down-shift to A(0) gives

D−1A(0) = c3un−12vn−1 − 3c1un−1

2vn−1 − c3un−1unvn−1 − c3vn−12. (6.14)


Therefore,

D−1A(0) + A(1) = 3c1un2vn−1 − c2un

2vn−1 + c3vn−1vn − c2vn−1vn + c2un−1unvn−1

+c2vn−12 + c3un−1

2vn−1 − 3c1un−12vn−1 − c3un−1unvn−1

−c3vn−12. (6.15)

From (6.15), it is evident that all monomials depend on un−1 or vn−1. Grouping terms

gives

D−1A(0) + A(1) = (3c1 − c2)un2vn−1 + (c3 − c2)vn−1vn + (c2 − c3)un−1unvn−1

75

+(c2 − c3)vn−12 + (c3 − 3c1)un−1

2vn−1. (6.16)

So, D−1A(0) + A(1) = 0 if 3c1 = c2 = c3. Again, the algorithm terminates here and

Step 5 is unnecessary. Substituting c1 = 13, c2 = c3 = 1 back into (5.26) gives

ρn =1

3un

3 + unvn−1 + unvn. (6.17)

Therefore, Jn = −D−1A(0),

Jn = un−1unvn−1 + vn−12. (6.18)

6.4 The Ablowitz-Ladik lattice

Example 6.3 (The Ablowitz-Ladik lattice revisited once more)



E = Dt ρn = β(−αc1unvn−2 + αc1un−1vn−1 − αc2unvn−1 + αc1un+1vn−1

−c1un−1unvn−2vn−1 + αc2un−1vn − αc1unvn − αc3unvn

+αc2un+1vn − c2un2vn−1vn + c1unun+1vn−1vn + c2un−1unvn

2

+c2unun+1vn2 + αc3un−1vn+1 − αc2unvn+1 + αc3un+1vn+1

+c3un−1unvnvn+1 − c2un2vnvn+1 − αc3unvn+2−c3unun+1vn+1vn+2),

with the lowest shift p = 2.

76


We may split the above result into

Dt ρn = A(0) + A(1)

= β[(αc1un−1vn−1 − αc2unvn−1 + αc1un+1vn−1 + αc2un−1vn − αc1unvn

−αc3unvn + αc2un+1vn − c2un2vn−1vn + c1unun+1vn−1vn + c2un−1unvn

2

+c2unun+1vn2 + αc3un−1vn+1 − αc2unvn+1 + αc3un+1vn+1

+c3un−1unvnvn+1 − c2un2vnvn+1 − αc3unvn+2 − c3unun+1vn+1vn+2)

+(−αc1unvn−2 − c1un−1unvn−2vn−1)], (6.19)

where A(0) has terms independent of un−2 and vn−2, and A(1) has terms dependent

on un−2 or vn−2.


One down-shift on A(0) gives

D−1A(0) = β[αc1un−2vn−2 − αc2un−1vn−2 + αc1unvn−2 + αc2un−2vn−1

−αc1un−1vn−1 − αc3un−1vn−1 + αc2unvn−1 − c2un−12vn−2vn−1

+c1un−1unvn−2vn−1 + c2un−2un−1vn−12 + c2un−1unvn−1

2

+αc3un−2vn − αc2un−1vn + αc3unvn + c3un−2un−1vn−1vn

−c2un−12vn−1vn − αc3un−1vn+1 − c3un−1unvnvn+1]. (6.20)


Therefore,

D−1A(0) + A(1) = β[(αc1un−2vn−2 − αc2un−1vn−2 + αc1unvn−2 + αc2un−2vn−1

77

−αc1un−1vn−1 − αc3un−1vn−1 + αc2unvn−1 − c2un−12vn−2vn−1

+c1un−1unvn−2vn−1 + c2un−2un−1vn−12 + c2un−1unvn−1

2

+αc3un−2vn − αc2un−1vn + αc3unvn + c3un−2un−1vn−1vn

−c2un−12vn−1vn − αc3un−1vn+1 − c3un−1unvnvn+1)

+(−αc1unvn−2 − c1un−1unvn−2vn−1)]. (6.21)

It is evident that not all monomials of (6.21) depend on un−2 or vn−2. So, we must

repeat Step 2 through Step 3 and split (6.21) into A(2) + A(3), where A(2) has terms

independent of un−2 and vn−2 and A(3) has terms dependent on un−2 or vn−2.

Step 2 repeated: Splitting D−1A(0) + A(1)

D−1A(0) + A(1) = A(2) + A(3) (6.22)

= β[(−αc1un−1vn−1 − αc3un−1vn−1 + αc2unvn−1

+c2un−1unvn−12 − αc2un−1vn + αc3unvn − c2un−1

2vn−1vn

−αc3un−1vn+1 − c3un−1unvnvn+1) + (αc1un−2vn−2

−αc2un−1vn−2 + αc1unvn−2 + αc2un−2vn−1

−c2un−12vn−2vn−1 + c1un−1unvn−2vn−1 + c2un−2un−1vn−1

2

+αc3un−2vn + c3un−2un−1vn−1vn − αc1unvn−2

−c1un−1unvn−2vn−1)]. (6.23)

Step 3 repeated: Computing D−1A(2)

D−1A(2) = β[−αc1un−2vn−2 − αc3un−2vn−2 + αc2un−1vn−2 + c2un−2un−1vn−22

78

−αc2un−2vn−1 + αc3un−1vn−1 − c2un−22vn−2vn−1 − αc3un−2vn

−c3un−2un−1vn−1vn]. (6.24)

Step 4 repeated: Computing D−1A(2) + A(3)

D−1A(2) + A(3) = β[(−αc1un−2vn−2 − αc3un−2vn−2 + αc2un−1vn−2

+c2un−2un−1vn−22 − αc2un−2vn−1 + αc3un−1vn−1

−c2un−22vn−2vn−1 − αc3un−2vn − c3un−2un−1vn−1vn)

+(αc1un−2vn−2 − αc2un−1vn−2 + αc1unvn−2 + αc2un−2vn−1

−c2un−12vn−2vn−1 + c1un−1unvn−2vn−1 + c2un−2un−1vn−1

2

+αc3un−2vn + c3un−2un−1vn−1vn − αc1unvn−2

−c1un−1unvn−2vn−1)]. (6.25)

Notice again that not all monomials (6.25) depend on un−2 or vn−2. Therefore,

we must repeat Step 2 through Step 4.

Step 2 repeated, second time: Splitting D−1A(2) + A(3)

D−1A(2) + A(3) = A(4) + A(5) = β[(−αc1un−2vn−2 − αc3un−2vn−2 + αc2un−1vn−2

+c2un−2un−1vn−22 − αc2un−2vn−1 − c2un−2

2vn−2vn−1

−αc3un−2vn − c3un−2un−1vn−1vn + αc1un−2vn−2 − αc2un−1vn−2

+αc1unvn−2 + αc2un−2vn−1 − c2un−12vn−2vn−1

+c1un−1unvn−2vn−1 + c2un−2un−1vn−12 + αc3un−2vn

+c3un−2un−1vn−1vn − αc1unvn−2 − c1un−1unvn−2vn−1)

+(αc3un−1vn−1)], (6.26)

79

where A(4) has terms independent of un−2 and vn−2, and A(5) has terms dependent

on un−2 or vn−2.

Step 3 repeated, second time: Computing D−1A(4)

D−1A(4) = αc3un−2vn−2. (6.27)

Step 4 repeated, second time: Computing D−1A(4) + A(5)

D−1A(4) + A(5) = β[(αc3un−2vn−2) + (−αc1un−2vn−2 − αc3un−2vn−2

+αc2un−1vn−2 + c2un−2un−1vn−22 − αc2un−2vn−1

−c2un−22vn−2vn−1 − αc3un−2vn − c3un−2un−1vn−1vn

+αc1un−2vn−2 − αc2un−1vn−2 + αc1unvn−2

+αc2un−2vn−1 − c2un−12vn−2vn−1 + c1un−1unvn−2vn−1

+c2un−2un−1vn−12 + αc3un−2vn + c3un−2un−1vn−1vn

−αc1unvn−2 − c1un−1unvn−2vn−1)]. (6.28)

By examination, all terms of (6.28) depend on un−2 or vn−2. Simple algebra on (6.28)

results in

D−1A(4) + A(5) = β[c2un−2vn−1vn−22 − c2un−1

2vn−2vn−1 + c2un−2un−1vn−12

−c2un−22vn−2vn−1]. (6.29)

So, D−1A(4) + A(5) = 0 if c2 = 0. Therefore, substituting c2 = 0 into (5.56) gives

ρn = c1unvn−1 + c3unvn+1. (6.30)

80

and

Jn = −D−1(A(0) + A(2) + A(4))

= β(−c1αunvn−2 + c1αun−1vn−1 − c1un−1unvn−2vn−1 − c3αunvn

+c3un−1vn+1 + c3αun−1unvnvn+1). (6.31)

The density-flux pair (5.67) and (6.31) agree with (5.59) and (5.63).

81

Chapter 7

APPLICATIONS

In this chapter we consider complicated DDEs for which the computation of den-

sities and fluxes would be nearly impossible by hand. All the results were computed

with the software DDEDensityFlux.m.

7.1 Discretization of the combined KdV-mKdV equation

Example 7.1 (Discretization of the combined KdV-mKdV equation)

Consider the DDE described in [39],

un = −(1 + αh2un + βh2un2)[

1

h3(1

2un+2 − un+1 + un−1 −

1

2un−2)

+α

2h[un+1

2 − un−12 + un(un+1 − un−1) + un+1un+2 − un−2un−1]

+β

2h[un+1

2(un+2 + un)− un−12(un−2 + un)]]. (7.1)

It is an integrable discretization of a combined KdV-mKdV equation,

ut + 6αuux + 6βu2ux + u3x = 0. (7.2)

Discretizations of the KdV and mKdV equations are special cases. Set h = 1 (scaling).

We quickly verify that (7.1) is not uniformity in rank. Therefore, we must introduce

82

auxiliary parameters γ and δ with weights. Eq. (7.1) is replaced by

un = −(γ + αun + βun2)[δ(

1

2un+2 − un+1 + un−1 −

1

2un−2)

+α

2[un+1

2 − un−12 + un(un+1 − un−1) + un+1un+2 − un−2un−1]

+β

2[un+1

2(un+2 + un)− un−12(un−2 + un)]]. (7.3)


Scale 1: Uniformity in rank and w( ddt

) = 1 leads to

w(δ) = w(γ) =1

2,

w(un) =1

2− w(α),

w(β) = −1

2+ 2w(α).


) = 0, all terms have the same rank if

w(δ) = w(γ) = 0,

w(β) = 2w(un),

w(un) = −w(α).

Therefore, in addition to considering Scale 1 and Scale 0, we must also consider

two separate cases for the w(β).

83

Case 1: w(β) = 0

Step 2.1: Construct the form of the density

After replacement from (7.3) and using the equivalence criterion, we find

I = αun, un2, unun+1, unun+2, (7.4)

where all monomials are of rank R=12. In Table 7.1, we list the monomials in I in the

first column, the ranks of the monomials according to Scale 0 in the second column,

and the ranks of the monomials according to Scale 1 in the third column.


c1 αun 0 12

......

...c4 unun+2 0 1

2

Table 7.1. Ranks of terms in density candidate (R=12) for the combined equation

Considering the two separate scale scenarios does not simplify the calculation.

Step 3.1: Determine the unknown coefficients

Because there are only 3 monomials involved, the software quickly determines

that c1 = 1, c2 = 0, and c3 = β. So,

ρn = αun + βunun+1. (7.5)

84


Similarly, if we list all monomials of rank R=34, then

I = α2un, γun, δun, αun2, un

3, αunun+1, un2un+1, unun+1

2, αunun+2, un2un+2,

unun+1un+2, unun+22. (7.6)

Listing the terms and their respective weights as before gives Table 7.2.


c1 α2un 0 34

......

...c12 unun+2

2 0 34


Considering the two separate Scale scenarios clearly does not simplify this calculation.

Therefore, all monomials in I are used to construct the density.


Now, ρn is the sum of 12 terms with coefficients c1 through c12. The software

computes the solution:

c1 =α2 − βδ(c2 + c3)

α2β,

c4 = c5 = c7 = c8 = c9 = c10 = c11 = c12 = 0,

c6 = 1.

Setting δ = γ = 1,

ρn =α

β(αun + βunun+1). (7.7)

85


We find that after replacement from (7.3),

I = α3un, αγun, αδun, α2un

2, γun2, δun

2, . . . , unun+2un+32, unun+3

2, (7.8)

where all monomials are of rank R=1. We list the monomials in I in the first column

of Table 7.3. We list the ranks of the monomials according to Scale 0 in the second

column, and the ranks of the monomials according to Scale 1 in the third column.


c1 α3un 0 1...

......

c45 unun+32 0 1


Again, considering the two separate scale scenarios does not simplify this calcu-

lation.

So, all 45 monomials in I are linearly combined to construct the density.


We get the following solution:

c1 =α2(−1 + βc9) + βδ(1− β(c2 + c3) + c10 + c11)

α2β2,

c4 =α2 − 2βδ(c5 + c6)

2α2β,

c7 = c8 = c13 = c16 = c20 = c21 = c23 = c25 = c26 = c28 = c32 = . . . = c45 = 0,

c12 = c14 = c22 = 1,

86

c15 =β

2,

c17 = −δ(−1 + c18 + c19)

α2,

c29 = −δ(c30 + c31)

α2.

Setting δ = γ = 1 gives

ρn = −α3

β2un +

α

βun +

α3

βc9un +

α

βc10un +

α

βc11un +

α2

2βun

2 + α2c9unun+1

+c10unun+1 + c11unun+1 + αun2un+1 + αunun+1

2 +1

2βun

2un+12 + unun+2

+αunun+1un+2 + βunun+12un+2. (7.9)

So, the part of the density with coefficient c9 = 1 is

ρn = α2(α

βun + unun+1). (7.10)

The part of the density with coefficient c10 or c11 is

ρn =α

βun + unun+1, (7.11)


ρn = −α3

β2un +

α

βun +

α2

2βun

2 + αun2un+1 + αunun+1

2 +1

2βun

2un+12 + unun+2

+αunun+1un+2 + βunun+12un+2.

87

Case 2: w(β) 6= 0


Let w(α) = 38. Again, if we replace from (7.3) and use the equivalence criterion,

we find

I = αun, βun2, βunun+1, βunun+2, βunun+3, un

4, un3un+1, un

2un+12, unun+1

3,

un3un+2, un

2un+1un+2, unun+12un+2, un

2un+22, unun+1un+2

2, unun+23, un

3un+3,

un2un+1un+3, unun+1

2un+3, un2un+2un+3, unun+1un+2un+3, unun+2

2un+3,

un2un+3

2, unun+1un+32, unun+2un+3

2, unun+33, (7.12)

where each monomial is of rank R=12. In the table, we list the terms with their ranks

based on both scales.


c1 αun 0 12

......

...c5 βunun+3 0 1

2

c6 un4 −4w(α) 1

2...

......

c25 unun+33 −4w(α) 1

2


There are two different choices according to Scale 0.

• Choice 1 is a linear combination of the 5 terms of rank 0 .

• Choice 2 is a linear combination of the 20 terms of rank −4w(α).

88

Considering each of these choices separately simplifies the computations somewhat,

since we may now address each case separately.


There is freedom here in the constant coefficients. For Choice 1, the software

does not produce any non-trivial densities. For Choice 2, c1 = 0, c3 = 1, and c2 =

c4 = c5 = 0. So, setting δ = γ = 1 gives

ρn = αun + βunun+1. (7.13)


For rank R=58, we have

I = un5, un

4un+1, un3un+1

2, . . . , γun, δun, αun2, . . . , β2un. (7.14)

Therefore, we obtain Table 7.5.


c1 un5 −5w(α) 5

8...

......

c70 unun+44 −5w(α) 5

8

c71 γun −w(α) 58

......

...c92 βunun+4

2 −w(α) 58

c93 β2un 3w(α) 58


So, there are three choices to consider:

89

• Choice 1 is a linear combination of the 70 terms of rank −5w(α) .

• Choice 2 is a linear combination of the 3 terms of rank −w(α) .

• Choice 3 is a linear combination of the 20 terms of rank 3w(α) .

Considering each of these choices separately simplifies the computations.


Again, there is freedom in the constant coefficients. Choice 1 and Choice 3 do not

result in any non-trivial densities. For Choice 2, c1 = α2

βδ−c2, c3 =c4 =c6 = . . .=c22 =0,

and c5 = 1. After setting δ = γ = 1, we obtain

ρn = α(α

βun + unun+1). (7.15)


For rank R=34

I = un6, un

5un+1, un4un+1

2, . . . , γun2, δun

2, αun3, . . . ,

αβun, β2un

2, β2unun+1. (7.16)

Again, we may list the terms and their respective ranks in Table 7.6.

The table implies the following three different choices:

• Choice 1 is a linear combination of the 252 terms of rank −6w(α) .

• Choice 2 is a linear combination of the 89 terms of rank −2w(α).

• Choice 3 is a linear combination of the 3 terms of rank 2w(α) .

Considering each of these choices separately simplifies the computations.

90


c1 un6 −6w(α) 3

4...

......

c252 unun+55 −6w(α) 3

4

c253 γun2 −2w(α) 3

4...

......

c341 βunun+53 −2w(α) 3

4

c342 αβun 2w(α) 34

c343 β2un2 2w(α) 3

4

c344 β2unun+1 2w(α) 34



Once again, there is freedom in the constant coefficients. For Choice 1 there are

252 terms in the density candidate, ρn, and the code cannot complete the computation

in a reasonable amount of time. For Choice 2, setting δ = γ = 1 gives

ρn =α2

2βun

2 +α2

βunun+1 − unun+1 + αun

2un+1 + αunun+12 +

1

2βun

2un+12

+unun+2 + αunun+1un+2 + βunun+12un+2. (7.17)

For Choice 3, the code determines that c1 = 1, c2 = c4 = c5 = c6 = c7 = 0, and

c3 = 1. Setting δ = γ = 1 gives

ρn = β2(α

βun + unun+1). (7.18)

91

7.2 Discretization of the Korteweg-de Vries equation

Example 7.2 (Discretization of the Korteweg-de Vries equation)

We now consider conserved densities for (7.3), when β = 0.

un = −(γ + αun)[δ(1

2un+2 − un+1 + un−1 −

1

2un−2

+α

2[u2

n+1 − u2n−1 + un(un+1 + un−1) + un+1un+2 − un−2un−1]]. (7.19)

The above equation is a completely integrable discretization of the KdV equation

ut + 6αuux + uxxx = 0.




w(δ) + w(γ) = 1,

w(α) + w(δ) + w(un) = 1,

w(α) + w(γ) + w(un) = 1,

2 w(α) + 2 w(un) = 1.

Simple algebra determines that w(δ) = w(γ) = 12, and w(α) = 1

2− w(un).

Scale 0: Similarly, when w( ddt


w(δ) + w(γ) = 0,

w(α) + w(δ) + w(un) = 0,

w(α) + w(γ) + w(un) = 0,

92

2 w(α) + 2 w(un) = 0.

This implies that w(δ) = w(γ) = 0, and w(α) = −w(un). So, in addition to consid-

ering Scale 1 and Scale 0, we must also consider two individual cases for the w(α).

Case 1: w(α) = 0


As an example, let us compute the form of the density of rank R=1 consider-

ing that w(un) = 12. For this case, use (7.19) to eliminate un terms, and use the

equivalence criterion to find

I = γun, δun, un2, unun+1, (7.20)

where all monomials are of rank R=1. When w(α) = 0 we can list the terms in the

first column of Table 7.7, and their respective ranks according to Scale 0 and Scale 1

in the second and third columns.


c1 γun 0 1...

......

c4 unun+1 0 1

Table 7.7. Ranks of terms in density candidate (R=1) for the KdV discretization

Considering the two separate scale scenarios does not simplify this calculation.

However, because there are only 4 monomials involved, the software would still pro-

duce results.

93


Set δ = γ = 1. There is freedom in the constant coefficients. So, the part of the

density with constant coefficient c1 or c2 is

ρn = un. (7.21)

and

ρn = c1un + c2un + un2 + 2unun+1. (7.22)


ρn = un2 + 2unun+1. (7.23)

Case 2: w(α) 6= 0


Choosing w(un) = 14

implies w(α) = 14. Now,

I = αγun, αδun, α2un

2, α2unun+1, α2unun+2, α

2unun+3, α3un, γun

2, δun2, αun

3,

γunun+1, δunun+1, αun2un+1, αunun+1

2, γunun+2, δunun+2, αun2un+2,

αunun+1un+2, αunun+22, γunun+3, δunun+3, αun

2un+3, αunun+1un+3,

αunun+2un+3, αunun+32, un

4, un3un+1, un

2un+12, unun+1

3, un3un+2,

un2un+1un+2, unun+1

2un+2, un2un+2

2, unun+1un+22, unun+2

3, un3un+3,

un2un+1un+3, unun+1


2un+3,

un2un+3


2, unun+33, (7.24)

where all monomials are of rank R=1. When w(α) 6= 0 we get Table 7.8.

94


c1 αγun 0 1...

......

c6 α2unun+3 0 1c7 α3un −2w(un) 1c8 γun

2 2w(un) 1...

......

c25 αunun+32 2w(un) 1

c26 un4 4w(un) 1

......

...c45 unun+3

3 4w(un) 1

Table 7.8. Ranks of terms in density candidate (R=1) for the KdV discretization

Therefore, there are four different choices according to Scale 0.

• Choice 1 is a linear combination of the 6 terms of rank 0.

• Choice 2 is a scalar multiple of the 1 term of rank −2w(un).

• Choice 3 is a linear combination of the 18 terms of rank 2w(un).


Considering each of these choices separately greatly simplifies calculations.


Again, there is freedom in the constant coefficients. ForChoice 1, ρn has 6 terms.

The computation for δ = γ = 1 leads to

ρn = αun (7.25)

95

if c1 = 1 and c2 = . . . = c6 = 0 or if c2 = 1 and c1 = c3 = . . . = c6 = 0. There is also

a third solution of the linear system, namely c3 = 1, c4 = 2, and c5 = c6 = 0. Setting

δ = γ = 1 then leads to

ρn = αc1un + αc2un + α2un2 + 2α2unun+1. (7.26)

The part of the density without a free coefficient is

ρn = α2un2 + 2α2unun+1. (7.27)

Choice 2 gives c1 = 0 and ρn = α3un. Choice 3 gives c3 =1, c5 =2c1+2c2−c4, c6 =c7 =

c11 = 3, c9 = 3−c8, c10 = c12 = c15 = c16 = c17 = c18 = 0, and c14 =−c13. Set δ = γ = 1.

Then,

ρn = c1un2 + c2un

2 + αun3 + 2c1unun+1 + 2c2unun+1 + 3αun

2un+1 + 3αunun+12

+3unun+2 + 3αunun+1un+2. (7.28)

So, the part of the density with constant coefficient c1 or c2 eventually leads to

ρn = un2 + 2unun+1. (7.29)

The part of the density without a free coefficient is

ρn = αun3 + 3αun

2un+1 + 3αunun+12 + 3unun+2 + 3αunun+1un+2. (7.30)

Choice 4 does not result in any nonzero densities.

96

7.3 Discretization of the modified Korteweg-de Vries equation

Example 7.3 (Discretization of the modified Korteweg-de Vries equation)

Similarly, we computed conserved densities for (7.3) from [39], where α = 0.

un = −(γ + βun2)[δ(

1

2un+2 − un+1 + un−1 −

1

2un−2)

+β

2[u2

n+1(un+2 + un)− un−12(un−2 + un)]], (7.31)

(7.31) is a completely integrable discretization of the modified KdV equation

ut + 6βu2ux + uxxx = 0.




w(δ) + w(γ) = 1,

w(β) + w(δ) + 2w(un) = 1,

w(β) + w(γ) + 2w(un) = 1,

2w(β) + 4w(un) = 1.

Simple algebra determines that w(δ) = w(γ) = 12, and w(β) = 1

2− 2w(un).

Scale 0: Similarly, when w( ddt

) = 0, (7.31) is uniform in rank if

w(δ) + w(γ) = 0,

w(β) + w(δ) + 2w(un) = 0,

97

w(β) + w(γ) + 2w(un) = 0,

2w(β) + 4w(un) = 0.

This implies that w(δ) = w(γ) = 0, and w(β) = −2w(un). Therefore, in addition to

considering Scale 1 and Scale 0, we must consider two cases for the w(β). Case 1:

w(β) = 0


As an example, let us compute the density of rank R=12

with choice w(un) = 14.

So, based on Scale 1, w(β) = 0. For this case,

I = un2, unun+1. (7.32)

All monomials are of rank R=12. When w(β) = 0, we form Table 7.9.


c1 un2 0 1

2

c2 unun+1 0 12

Table 7.9. Ranks of terms in density candidate (R=12) for the mKdV discretization

Considering the two separate scales is irrelevant.


There are only 2 terms in ρn. The software computes that c1 = 0 and c2 = 1,

and therefore

ρn = unun+1. (7.33)

98


As an example, we compute the density of rank R=1 using w(un) = 14

to make

w(β) = 0. Here,

I = γun2, δun

2, un4, . . . , unun+2un+3

2, unun+33. (7.34)

All monomials in I are of rank R=1. When w(β) = 0 we form Table 7.10 as before.

Working with two separate scales does not help.


c1 γun2 0 1

......

...c28 unun+3

3 0 1



For the 28 constant in ρn, the software produces c1 =−c2, c3 =c6 =c8 =c11 =c12 =

c14 = . . . = c16 = c19 = . . . = c28 = 0, c7 = 1, c9 = 2β−c10, c13 = 2, and c17 =−c18. Setting

δ = γ = 1 gives

ρn = c4unun+1 + c5unun+1 + un2un+1

2 +2unun+2

β+ 2unu

2n+1un+2. (7.35)

So, the part of the density with coefficient c4 or c5 is

ρn = unun+1.

99

The part of the density without arbitrary coefficients is

ρn = un2un+1

2 +2unun+2

β+ 2unun+1

2un+2.

Case 2: w(β) 6= 0


For w(β) 6= 0, the user may pick w(un) = 18

making w(β) = 14. Here,

I = βun2, βunun+1, βunun+2, βunun+3, un

4, un3un+1, un

2un+12, unun+1

3, un3un+2,

un2un+1un+2, unun+1

2un+2, un2un+2

2, unun+1un+22, unun+2

3, un3un+3,

un2un+1un+3, unun+1


2un+3,

un2un+3


2, unun+33. (7.36)

All monomials in I are of rank R=12. When w(β) 6= 0 we list the terms and their

respective ranks in Table 7.11.


c1 βun2 0 1

......

...c4 , βunun+3 0 1c5 un

4 4w(un) 1...

......

c24 unun+33 4w(un) 1


There are two different choices based on Scale 0.

100

• Choice 1 is a linear combination of the 4 terms of rank 0.



Choice 1 gives c1 = c3 = c4 = 0 and c2 = 1. After setting δ = γ = 1 we obtain

ρn = βunun+1. (7.37)

Choice 2 results in only trivial densities.

101

Chapter 8

CONCLUSION

The two methods discussed in this thesis allow one to find polynomial-type con-

served densities and fluxes for nonlinear systems of DDEs. These algorithms rely

heavily on the dilation invariance of the given DDEs. Both methods are illustrated

for the KvM, Toda, and AL lattices. In addition, our key applications are discretiza-

tions of the KdV equation, the mKdV equation and a combination of the two.

It is conjectured that nonlinear DDEs with a large (in principle, infinite) number

of conserved densities are completely integrable. Hence, our program can be used to

test the integrability of such DDE systems.

The first algorithm leaves many options to the user. Depending on the posed

problem and desired result, proper selection from these options could simplify cal-

culations. One option is that the user may choose to compute the densities either

using a “shifting” technique or utilizing the discrete Euler operator. Both of these

techniques are implemented.

The second method can be used in two different ways. If the density ρn is

known, one can find the associated flux by splitting Dt ρn into a piece that lies in the

image of the operator ∆ and a piece that does not. Alternatively, sophisticated use

of splitting technique also allows one to simultaneously find densities and fluxes at

increased computational effort. This method is of more theoretical value and was not

implemented.

We offer the scientific community a new symbolic Mathematica package called

102

DDEDensityFlux.m to carry out the tedious calculations of conserved densities

and fluxes for lattice equations. It is effective for systems of DDEs with parameters.

In fact, for such DDEs our program gives conditions on the parameters so that the

system will admit conserved densities of a given rank.

This new package is much more reliable than earlier versions, since it now com-

putes the flux as well as the density. The addition of the flux computation makes the

software completely self-testing. Also, DDEDensityFlux.m now successfully com-

putes conservation laws for much more complex DDEs that were previously unattain-

able.

In the future, we hope to improve the speed of our algorithm, particularly for

more complicated systems. Furthermore, we hope to extend these algorithms to

calculate non-polynomial conserved densities and fluxes.

103

REFERENCES

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[3] M.J. Ablowitz and J.F. Ladik, Stud. Appl. Math. 55 (1976) 213.

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[13] U. Goktas and W. Hereman, J. Symb. Comp. 24 (1997) 591.

[14] U. Goktas and W. Hereman, Physica D 123 (1998) 425.

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[19] W. Hereman, U. Goktas, M. Colagrosso, and A. Miller, Comp. Phys. Comm.

115 (1998) 428.

[20] M. Hickman and W. Hereman, Computation of Densities and Fluxes of Nonlinear

Differential-Difference Equations, Proc. Roy. Soc. London A (2003), in press.

[21] J. Hietarinta, F. W. Nijhoff, and J. Satsuma, eds., Symmetries and Integrability

of Difference Equations, Special Issue dedicated to the SIDE IV Meeting, Tokyo,

Japan, 2000, J. Phys. A: Math. Gen. 34 (2001) 10337.

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[22] R. Hirota and J. Satsuma, J. Phys. Soc. Jpn. 40 (1976) 891.

[23] E. G. B. Hohler and K. Olaussen, Int. J. Mod. Phys. A11 (1996) 1831.

[24] M. Ito, A REDUCE program for finding symmetries of nonlinear evolution equa-

tions with uniform rank, Comp. Phys. Comm. 42 (1986) 351-357.

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[26] D. Levi and O. Ragnisco, Lett. Nuovo Cimento 22 (1978)691.

[27] D. Levi and O. Ragnisco, eds., SIDE III—Symmetries and Integrability of Differ-

ence Equations, Meeting SIDE III, Sabaudia, Italy, 1998, CRM Proceedings and

Lecture Notes 25 (American Mathematical Society, Providence, Rhode Island,

2000).

[28] D. Levi, L. Vinet, and P. Winternitz, eds., Symmetries and Integrability of Differ-

ence Equations, Meeting SIDE I, Esterel, Quebec, Canada, 1994, CRM Proceed-

ings and Lecture Notes 9 (American Mathematical Society, Providence, Rhode

Island, 1996).

[29] D. Levi and P. Winternitz, J. Math. Phys. 34 (1993) 3713.

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[31] S. Manakov, Sov. Phys. JETP 40 (1975) 269.

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25 (1992) L883.

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[35] A. B. Shabat and R. I. Yamilov, Phys. Lett. A 130 (1988) 271.

[36] A. B. Shabat and R. I. Yamilov, Leningrad Math. J. 2 (1991) 377.

[37] A. B. Shabat and R. I. Yamilov, Phys. Lett. A 227 (1997) 15.

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[39] T.R. Taha, Maths. Comput. in Simul. 35 (1993) 509.

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107

Appendix A

DATA FILES

Samples of existing data files are given in the following pages. One may create

a new data file, though the format must be similar to those shown here. Specifically,

a new data file must include all lines not ‘commented out’ by (* *). We recommend

that existing data files be copied and modified.

Two data files are included here, one for the Kac-van Moerbeke (KvM) lattice,

and the other for the Ablowitz-Ladik (AL) lattice. Softcopy forms of the data files

are available from the same location as the DDEDensityFlux.m code itself.

108

(? Data file for KvM lattice, d volter.m ?)

u[1][n_]’[t]:= u[1][n][t]*(u[1][n+1][t]-u[1][n-1][t]);

noeqs = 1;

name = "Volterra (Kac-Van Moerbeke) Equation";

parameters = ;

weightpars = ;

formrho = 0;

(* FORCING OPTIONS *)

(* If forcesinglescale=True, only scale1 with w(d/dt)=1 is used to *)

(* construct the form of rho. The density is not split into pieces. *)

(* If forcemultiplescale=True, rho is constructed based on scale1, *)

(* but split into pieces according to scale0 with w(d/dt) = 0. *)

(* NOTE: If both are false, the code uses scale0 when appropriate. *)

forcesinglescale = False;

forcemultiplescale = False;

(* If forceshiftsinrho=True, the code will generate the form of rho *)

(* based on u_n, u_n+1, ..., u_n+maximumshift. *)

109

(* To compute the maximumshift there are two options (see below). *)

(* If forceshiftsinrho=False, no shifts on the dependent variable *)

(* u_n will be used to generate the form of rho. *)

forceshiftsinrho = False;

(* If forcemaximumshiftrhsdde=True, then the maximumshift is based *)

(* on the maximum of the shifts occuring in the rhs of the DDEs. *)

(* If forcemaximumshiftpowers=True, then maximumshift = p-1 where *)

(* p = (rhorank/weightu[1]), that is the highest power in rho. *)

(* Rho will be generated from the list u_n, u_n+1,...,u_n+p-1.*)

(* If u_n^p is the highest-power term in rho, then e.g. the term *)

(* u_n u_n+1 u_n+2 ... u_n+p-1 will occur in the form of rho. *)

(* For example, if p=4 then maximumshift = p-1 =3. The terms in rho *)

(* are based on u_n, u_n+1, u_n+2. So, rho has terms like *)

(* u_n^4, u_n^2 u_n+3^2, ..., u_n u_n+1 u_n+2 u_n+3. *)

(* NOTE: forcemaximumshiftrhsdde and forcemaximumshiftpowers must *)

(* opposite Boolean values. *)

forcemaximumshiftrhsdde = False;

forcemaximumshiftpowers = False;

(* If forcediscreteeuler=True, then the linear system for the c[i] *)

(* is computed via the Discrete Variational Derivative (Euler) *)

110

(* Operator. If forceshifting=True, then the original shifting *)

(* algorithm is used to compute the linear system for the c[i] and *)

(* the fluxes J_n. *)

forcediscreteeuler = True;

forceshifting = False;

(* If forcestripparameters=True, then power of the nonzero *)

(* parameters, are removed in the factored form of the linear *)

(* system for the c[i]. *)

(* Example: aa, bb^3, aa*bb, ... are removed during simplification, *)

(* but not factors like (aa^2-bb), (aa+bb)^4, etc. *)

forcestripparameters = True;

(* If forceextrasimplifications=True, then while generating the *)

(* linear system, equations of type c[i] == 0 are automatically *)

(* applied to the rest of the system for the c[i]. *)

forceextrasimplifications = True;

(* d_volter.m *)

111

(? Data file for AL lattice, d ablnls.m ?)

u[1][n_]’[t]:= aa*(u[1][n+1][t]-2*u[1][n][t]+u[1][n-1][t]) +

u[1][n][t]*u[2][n][t]*(u[1][n+1][t]+u[1][n-1][t]);

u[2][n_]’[t]:= -aa*(u[2][n+1][t]-2*u[2][n][t]+u[2][n-1][t]) -

u[1][n][t]*u[2][n][t]*(u[2][n+1][t]+u[2][n-1][t]);

noeqs = 2;

name = "NLS Equation (Ablowitz-Ladik Discretization)";

parameters = ;

weightpars = aa;

(* weightu[1] = weightu[2]; *)

formrho = 0;

(* FORCING OPTIONS *)

(* If forcesinglescale=True, only scale1 with w(d/dt)=1 is used to *)

(* construct the form of rho. The density is not split into pieces. *)

(* If forcemultiplescale=True, rho is constructed based on scale1, *)

(* but split into pieces according to scale0 with w(d/dt) = 0. *)

(* NOTE: If both are false, the code uses scale0 when appropriate. *)

112

forcesinglescale = False;

forcemultiplescale = False;

(* If forceshiftsinrho=True, the code will generate the form of rho *)

(* based on u_n, u_n+1, ..., u_n+maximumshift. *)

(* To compute the maximumshift there are two options (see below). *)

(* If forceshiftsinrho=False, no shifts on the dependent variable *)

(* u_n will be used to generate the form of rho. *)

forceshiftsinrho = False;

(* If forcemaximumshiftrhsdde=True, then the maximumshift is based *)

(* on the maximum of the shifts occuring in the rhs of the DDEs. *)

(* If forcemaximumshiftpowers=True, then maximumshift = p-1 where *)

(* p = (rhorank/weightu[1]), that is the highest power in rho. *)

(* Rho will be generated from the list u_n, u_n+1,...,u_n+p-1.*)

(* If u_n^p is the highest-power term in rho, then e.g. the term *)

(* u_n u_n+1 u_n+2 ... u_n+p-1 will occur in the form of rho. *)

(* For example, if p=4 then maximumshift = p-1 =3. The terms in rho *)

(* are based on u_n, u_n+1, u_n+2. So, rho has terms like *)

(* u_n^4, u_n^2 u_n+3^2, ..., u_n u_n+1 u_n+2 u_n+3. *)

(* NOTE: forcemaximumshiftrhsdde and forcemaximumshiftpowers must *)

(* opposite Boolean values. *)

113

forcemaximumshiftrhsdde = False;

forcemaximumshiftpowers = False;

(* If forcediscreteeuler=True, then the linear system for the c[i] *)

(* is computed via the Euler Operator. *)

(* If forceshifting=True, then the original shifting algorithm is *)

(* used to compute the linear system for the c[i] and the fluxe J_n *)

forcediscreteeuler = True;

forceshifting = False;

(* If forcestripparameters=True, then power of the nonzero *)

(* parameters are removed in the factored form of the linear system *)

(* for the c[i]. *)

(* Example: aa, bb^3, aa*bb, ... are removed during simplification, *)

(* but not factors like (aa^2-bb), (aa+bb)^4, etc. *)

forcestripparameters = True;

(* If forceextrasimplifications=True, then while generating the *)

(* linear system, equations of type c[i] == 0 are automatically *)

(* applied to the rest of the system for the c[i]. *)

forceextrasimplifications = True;

(* d_ablnls.m *)

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