Solving Nonlinear Wave Equations and Lattice with Mathematica

7/29/2019 Solving Nonlinear Wave Equations and Lattice with Mathematica

1/26

Solving Nonlinear Wave Equations and Lattices

with Mathematica

Willy Hereman

Department of Mathematical and Computer Sciences

Colorado School of Mines

Golden, Colorado, USAhttp://www.mines.edu/fs home/whereman/

[email protected]

Departement Toegepaste Wiskunde

Universiteit StellenboschMatieland, Zuid-Afrika

Woensdag, 21 Februarie 2001, 16:00

Collaborators: Unal Goktas (WRI), Doug Baldwin (CSM)

Ryan Martino, Joel Miller, Linda Hong (REUNSF 99)

Steve Formenac, Andrew Menz (REUNSF 00)

Research supported in part by NSF under Grant CCR-9901929

1


2/26

OUTLINE

Purpose & Motivation

Typical Examples Algorithm for Tanh Solutions Algorithm for Sech Solutions Extension: Tanh Solutions for Differential-difference Equations (DDEs) Solving/Analyzing Systems of Algebraic Equations with Parameters Implementation Issues Demo Mathematica Software Package Future Work

2


3/26

Purpose & Motivation

Develop various symbolic algorithms to compute exact solutionsof nonlinear (systems) of partial differential equations (PDEs) and

differential-difference equations (DDEs, lattices).

Fully automate the tanh and sech methods to compute closedform solitary wave solutions.

Solutions of tanh or sech type model solitary waves in fluid dy-namics, plasmas, electrical circuits, optical fibers, bio-genetics, etc.

Class of nonlinear PDEs and DDEs solvable with the tanh/sechmethod includes famous evolution and wave equations.

Typical examples: Korteweg-de Vries, Fisher and Boussinesq PDEs,

Toda and Volterra lattices (DDEs).

Research aspect: Design a high-quality application package forthe computation of exact solitary wave solutions of large classes of

nonlinear evolution and wave equations.

Educational aspect: Software as course ware for courses in non-linear PDEs, theory of nonlinear waves, integrability, dynamical sys-

tems, and modeling with symbolic software.

Users: scientists working on nonlinear wave phenomena in fluid dy-namics, nonlinear networks, elastic media, chemical kinetics, mate-

rial science, bio-sciences, plasma physics, and nonlinear optics.

3


4/26

Typical Examples of Single PDEs and Systems of PDEs

The Korteweg-de Vries (KdV) equation:

ut + uux + u3x = 0.Solitary wave solution:

u(x, t) =8c31 c2

6c1 2c

21

tanh2 [c1x + c2t + ] ,

or, equivalently,

u(x, t) = 4c31 + c2

6c1 +

2c21

sech

2

[c1x + c2t + ] .

The modified Korteweg-de Vries (mKdV) equation:ut + u

2ux + u3x = 0.

Solitary wave solution:

u(x, t) = 6

c1 sechc1x c

31t +

.

The Fisher equation:ut uxx u (1 u) = 0.


u(x, t) =

1

4 1

2 tanh+

1

4 tanh2

,

with

= 12

6x 5

12t + .

4


5/26

The generalized Kuramoto-Sivashinski equation:ut + uux + uxx + u3x + u4x = 0.

Solitary wave solutions

(ignoring symmetry u u, x x, ) :For = 4 :

u(x, t) = 9 2c2 15 tanh (1 + tanh tanh2),with = x

2+ c2t + .

For = 0 :

u(x, t) = 219

11c2 135

19

1119

tanh+16519

1119

tanh3,

with = 12

1119 x + c2t + .

For = 1247

:

u(x, t) =45 4418c2

47

47

45

47

47

tanh

4547

47

tanh2 1547

47

tanh3,

with = 1247

x + c2t + .

For = 16/

73 :

u(x, t) =2(30 5329c2)

7373

75

7373tanh

6073

73

tanh2 1573

73

tanh3,

with = 1273

x + c2t + .

5


6/26

Three-dimensional modified Korteweg-de Vries equation:ut + 6u

2ux + uxyz = 0.


u(x,y,z,t) = c2c3 sech [c1x + c2y + c3z c1c2c3t + ] .

The Boussinesq (wave) equation:utt u2x + 3uu2x + 3ux2 + u4x = 0,

or written as a first-order system (v auxiliary variable):

ut + vx = 0,vt + ux 3uux u3x = 0.


u(x, t) =c21 c22 + 8c41

3c21 4c21 tanh2 [c1x + c2t + ]

v(x, t) = b0 + 4c1c2 tanh2 [c1x + c2t + ] .

The Broer-Kaup system:uty + 2(uux)y + 2vxx uxxy = 0,

vt + 2(uv)x + vxx = 0.


u(x, t) = c3

2c1+ c1 tanh [c1x + c2y + c3t + ]

v(x, t) = c1c2 c1c2 tanh2 [c1x + c2y + c3t + ]

6


7/26

Typical Examples of DDEs (lattices)

The Toda lattice:

un = (1 + un) (un1 2un + un+1) .Solitary wave solution:

un(t) = a0 sinh(c1) tanh[c1n sinh(c1) t + ] .

The Volterra lattice:un = un(vn

vn

1)

vn = vn(un+1 un).Solitary wave solution:

un(t) = c2 coth(c1) + c2 tanh [c1n + c2t + ]vn(t) = c2 coth(c1) c2 tanh [c1n + c2t + ] .

The Relativistic Toda lattice:un = (1 + un)(vn vn1)vn = vn(un+1 un + vn+1 vn1).


un(t) = c2 coth(c1) 1

+ c2 tanh [c1n + c2t + ]

vn(t) =c2 coth(c1)

c2

tanh [c1n + c2t + ] .

7


8/26

Algorithm for Tanh Solutions for PDE system

Given is a system of PDEs of order n

(u(x), u(x), u(x),

u(n)(x)) = 0.

Dependent variable u has components ui (or u,v,w,...)

Independent variable x has components xi (or x,y,z,t)

Step 1:

Seek solution of the form ui(x) = Ui(T), withT = tanh[c1x + c2y + c3z + c4t] = tanh .

Observe cosh2

sinh2

= 1, ( tanh) = 1 tanh2

or T = 1 T2.

Repeatedly apply the operator rulexi

ci(1 T2) ddT

This produces a coupled system of Legendre equations of type

P(T, Ui, U

i, . . . , U

(n)

i) = 0

for Ui(T).

Example: For Boussinesq systemut + vx = 0

vt + ux 3uux u3x = 0,we obtain after cancelling common factors 1 T2

c2U + c1V = 0

c2V + c1U 3c1U U

+c312(1 3T2)U + 6T(1 T2)U (1 T2)2U

= 0

8


9/26

Step 2:

Seek polynomial solutionsUi(T) =

Mi

j=0

aijTj

Balance the highest power terms in T to determine Mi.

Example: Powers for Boussinesq systemM1 1 = M2 1, 2M1 1 = M1 + 1

gives M1 = M2 = 2.

Hence, U1(T) = a10 + a11T+ a12T2

, U2(T) = a20 + a21T+ a22T2

.

Step 3:

Determine the algebraic system for the unknown coefficients aij bybalancing the coefficients of the various powers of T.

Example: Boussinesq system

a11 c1 (3a12 + 2 c21) = 0a12 c1 (a12 + 4 c

21) = 0

a21 c1 + a11 c2 = 0

a22 c1 + a12 c2 = 0

a11 c1 3a10 a11 c1 + 2a11 c31 + a21 c2 = 03a211 c1 + 2 a12 c1 6a10 a12 c1 + 16 a12 c31 + 2a22 c2 = 0.

9


10/26

Step 4:

Solve the nonlinear algebraic system with parameters.Reject complex solutions? Test the solutions.

Example: Solution for Boussinesq casea10 =

c21 c22 + 8c413c21

a11 = 0

a12 = 4c21a20 = free

a21 = 0a22 = 4c1c2.

Step 5:

Return to the original variables.Test the final solution in the original equations

Example: Solitary wave solution for Boussinesq system:

u(x, t) =c21 c22 + 8c41

3c21 4c21 tanh2 [c1x + c2t + ]

v(x, t) = a20 + 4c1c2 tanh2 [c1x + c2t + ] .

10


11/26

Algorithm for Sech Solutions for PDE system

Given is a system of PDEs of order n

(u(x), u(x), u(x),

u(n)(x)) = 0.

Dependent variable u has components ui (or u,v,w,...)

Independent variable x has components xi (or x,y,z,t)

Step 1:

Seek solution of the form ui(x) = Ui(S), withS = sech[c1x + c2y + c3z + c4t] = tanh .

Observe ( sech ) = tanh sech or S = T S = 1 S2 S.Also, cosh2 sinh2 = 1, hence, T2 + S2 = 1 and dTdS = ST.

Repeatedly apply the operator rulexi

ci

1 S2SddS

This produces a coupled system of Legendre type equations of type

P(S, Ui, Ui , . . . , U

(m)i ) +

1 S2 Q(S, Ui, Ui , . . . , U (n)i ) = 0

for Ui(S).

For every equation one must have Pi = 0 or Qi = 0. Only oddderivatives produce the extra factor

1 S2.

Conclusion: The total number of derivatives in each term in thegiven system should be either even or odd. No mismatch is allowed.

Example: For the 3D mKdV equationut + 6u

2ux + uxyz = 0.

11


12/26

we obtain after cancelling a common factor 1 S2 Sc4U

+6c1U2U+c1c2c3[(16S2)U+3S(12S2)U+S2(1S2)U] = 0

Step 2: Seek polynomial solutions

Ui(S) =Mi

j=0aijS

j

Balance the highest power terms in S to determine Mi.

Example: Powers for the 3D mKdV case

3M1 1 = M1 + 1gives M1 = 1. Hence, U(S) = a10 + a11S.

Step 3:

Determine the algebraic system for the unknown coefficients aij bybalancing the coefficients of the various powers of S.

Example: System for 3D mKdV casea11c1 (a

211 c2 c3) = 0

a11 (6a210 c1 + c1 c2 c3 + c4) = 0

a10 a211 c1 == 0

12


13/26

Step 4:


Example: Solution for 3D mKdV casea10 = 0

a11 = c1 c3c4 = c1 c2 c3

Step 5:


Example: Solitary wave solution for the 3D mKdV equationu(x,y,z,t) = c2 c3 sech(c1 x + c2 y + c3 z c1c2c3 t).

13


14/26

Extension: Tanh Solutions for DDE system

Given is a system of differential-difference equations (DDEs) of order n

(..., un1, un, un+1,..., un,..., u

(m)n ,...) = 0.

Dependent variable un has components ui,n (or un, vn, wn,...)

Independent variable x has components xi (or n, t).

No derivatives on shifted variables are allowed!

Step 1:

Seek solution of the form ui,n(x) = Ui,n(T(n)), withT(n) = tanh[c1n + c2t + ] = tanh .

Note that the argument T depends on n. Complicates matters. Repeatedly apply the operator rule on ui,n

t

c2(1 T2) ddT

This produces a coupled system of Legendre equations of type

P(T, Ui,n, Ui,n, . . .) = 0

for Ui,n(T).

Example: Toda latticeun = (1 + un) (un1

2un + un+1) .

transforms into

c22(1 T2)2T Un (1 T2)Un

+1 + c2(1 T2)Un

[Un1 2Un + Un+1] = 0

14


15/26

Step 2:

Seek polynomial solutionsUi,n(T(n)) =

Mi

j=0

aijT(n)j

For Un+p, p = 0, there is a phase shift:

Ui,np (T(n p) =Mi

j=0ai,j [T(n + p)]

j =Mi

j=0ai,j

T(n) tanh(pc1)

1 T(n) tanh(pc1)

Balance the highest power terms in T(n) to determine Mi.

Example: Powers for Toda lattice2M1 1 = M1 + 1

gives M1 = 1.

Hence,

Un(T(n)) = a10 + a11T(n)

Un1(T(n 1)) = a10 + a11T(n 1) = a10 + a11T(n)

tanh(c1)

1 T(n) tanh(c1)Un+1(T(n + 1)) = a10 + a11T(n + 1) = a10 + a11

T(n) + tanh(c1)

1 + T(n) tanh(c1).

Step 3:

Determine the algebraic system for the unknown coefficients aij bybalancing the coefficients of the various powers of T(n).

Example: Algebraic system for Toda latticec22 tanh2(c1) a11c2 tanh2(c1) = 0, c2 a11 = 0

15


16/26

Step 4:


Example: Solution of algebraic system for Toda latticea10 = free, a11 = c2 = sinh(c1)

Step 5:


Example: Solitary wave solution for Toda lattice:un(t) = a0 sinh(c1) tanh[c1n sinh(c1) t + ] .

16


17/26

Example: System of DDEs: Relativistic Toda lattice

un = (1 + un)(vn vn1)vn = vn(un+1

un + vn+1

vn1).

Step 1: Change of variables

un(x, t) = Un(T(n)), vn(x, t) = Vn(T(n)),

with

T(n) = tanh[c1n + c2t + ] = tanh .

gives

c2(1 T2)Un (1 + Un)(Vn Vn1) = 0c2(1 T2)Vn Vn(Un+1 Un + Vn+1 Vn1) = 0.

Step 2: Seek polynomial solutions

Un(T(n)) =M1

j=0

a1jT(n)j

Vn(T(n)) =M2

j=0a2jT(n)

j.

Balance the highest power terms in T(n) to determine M1, and M2 :

M1 + 1 = M1 + M2, M2 + 1 = M1 + M2

gives M1 = M2 = 1.

Hence,Un = a10 + a11T(n), Vn = a20 + a21T(n).

17


18/26

Step 3: Algebraic system for aij :

a11 c2 + a21 tanh(c1) + a10 a21 tanh(c1) = 0a11 tanh(c1) ( a21 + c2) = 0

a21 c2 + a11 a20 tanh(c1) + 2 a20 a21 tanh(c1) = 0tanh(c1) (a11 a21 + 2 a

221 a11 a20 tanh(c1)) = 0

a21 tanh2(c1) (c2 a11) = 0

Step 4: Solution of the algebraic system

a10 = c2 coth(c1) 1

, a11 = c2, a20 =c2 coth(c1)

, a21 = c2

.

Step 5: Solitary wave solution in original variables:

un(t) = c2 coth(c1) 1

+ c2 tanh [c1n + c2t + ]

vn(t) =c2 coth(c1)

c2

tanh [c1n + c2t + ] .

18


19/26

Solving/Analyzing Systems of Algebraic Equations with Parameters

Class of fifth-order evolution equations with parameters:

ut + 2u2ux + uxu2x + uu3x + u5x = 0.

Well-Known Special cases

Lax case: = 310

, = 2, = 10. Two solutions:

u(x, t) = 4c21 6c21 tanh2c1x 56c51t +

and

u(x, t) = a0 2c21 tanh2c1x 2(15a20c1 40a0c31 + 28c51)t +

where a0 is arbitrary.

Sawada-Kotera case: = 15, = 1, = 5. Two solutions:

u(x, t) = 8c21 12c21 tanh2c1x 16c51t +

and

u(x, t) = a0 6c21 tanh2c1x (5a20c1 40a0c31 + 76c51)t +

where a0 is arbitrary.

Kaup-Kupershmidt case: = 15

, = 52

, = 10. Two solutions:

u(x, t) = c21 3

2c21 tanh

2c1x c51t +

and

u(x, t) = 8c21 12c2 tanh2c1x 176c51t +

,

no free constants!

Ito case: = 29

, = 2, = 3. One solution:

u(x, t) = 20c21 30c21 tanh2c1x 96c51t +

.

19


20/26

What about the General case?

Q1: Can we retrieve the special solutions?

Q2: What are the condition(s) on the parameters , , for solutions

of tanh-type to exist?

Tanh solutions:

u(x, t) = a0 + a1 tanh [c1x + c2t + ] + a2 tanh2 [c1x + c2t + ] .

Nonlinear algebraic system must be analyzed, solved (or reduced!):

a1(2a22 + 6a2c

21 + 2a2c

21 + 24c

41) = 0

a1(2a21 + 6

2a0a2 + 6a0c21 18a2c21 12a2c21 120c41) = 0

2a22 + 12a2c21 + 6a2c

21 + 360c

41 = 0

22a21a2 + 22a0a

22 + 3a

21c

21 + a

21c

21 + 12a0a2c

21

8a22c21 8a22c21 480a2c41 = 0

a1(2a20c1 2a0c31 + 2a2c31 + 16c51 + c2) = 0

2a0a21c1 +

2a20a2c1 a21c31 a21c31 8a0a2c31 + 2a22c31+136a2c

51 + a2c2 = 0

Unknowns: a0, a1, a2.

Parameters: c1, c2, , , .

Solve and Reduce cannot be used on the whole system!

20


21/26

Strategy to Solve/Reduce Nonlinear Systems

Assumptions:

All ci = 0 Parameters (,,,...) are nonzero. Otherwise the highest powers

Mi may change.

All aj Mi = 0. Coefficients of highest power in Ui are present. Solve for aij, then ci, then find conditions on parameters.

Strategy followed by hand:

Solve all linear equations in aij first (cost: branching). Start withthe ones without parameters. Capture constraints in the process.

Solve linear equations in ci if they are free of aij. Solve linear equations in parameters if they free of aij, ci. Solve quasi-linear equations for aij, ci, parameters.

Solve quadratic equations for aij, ci, parameters. Eliminate cubic terms for aij, ci, parameters, without solving. Show remaining equations, if any.

Alternatives:

Use (adapted) Grobner Basis Techniques.

Use combinatorics on coefficients aij = 0 or aij = 0.

21


22/26

Actual Solution: Two major cases:

CASE 1: a1 = 0, two subcases

Subcase 1-a:

a2 = 3

2a0

c2 = c31(24c

21 a0)

where a0 is one of the two roots of the quadratic equation:

2a20 8a0c21 4a0c21 + 160c41 = 0.

Subcase 1-b: If = 10 1, thena2 = 6

c21

c2 = 1

(22a20c1 8a0c31 + 12c51 + 16c51)where a0 is arbitrary.

CASE 2: a1 = 0, then

=1

392(39 + 38+ 82)

and

a2 = 168(3 + 2)

c21

provided is one of the roots of(1042 + 886+ 1487)(5203 + 21582 1103 8871) = 0

22


23/26

Subcase 2-a: If2 = 1104(886+ 1487), then

= 2+ 526

a0 = 49c21(9983 + 4378)26(8 + 3)(3 + 2)2

a1 = 336c21

(3 + 2)

a2 = 168c21

(3 + 2)

c2 = 364 c51 (3851 + 1634)

6715 + 2946.

Subcase 2-b: If3 = 1520(8871 + 1103 21582), then

=39 + 38+ 82

392

a0 = 28 c21 (6483 + 5529+ 10662)(3 + 2)(23 + 6)(81 + 26)

a21 =28224 c41 (4 1)(26 17)

(3 + 2)2(23 + 6)(81 + 26)2

a2 = 168c21

(3 + 2)

c2 = 8 c51 (1792261977 + 1161063881+ 188900114

2)

959833473 + 632954969+ 1051767862.

23


24/26

Implementation Issues Software Demo Future Work

Demonstration of Mathematica package for tanh/sech methods. Long term goal: Develop PDESolve for closed form solutions of

nonlinear PDEs and DDEs.

Implement various methods: Lie symmetry (similarity) methods. Look at other types of explicit solutions involving

hyperbolic functions sinh, cosh, tanh,...

other special functions.

complex exponentials combined with sech or tanh.

Example: Set of ODEs from quantum field theoryuxx = u + u3 + auv2vxx = bv + cv

3 + av(u2 1).Try solutions (c2 = 0 for ODEs)

ui(x, t) =Mi

j=0aij tanh

j[c1x + c2t + ] +Nij=0

bij sech2j+1[c1x + c2t + ].

or

ui(x, t) =Mi

j=0(aij + bij sech[c1x + c2t + ]) tanh

j[c1x + c2t + ].

Solitary wave solutions:

u = tanh[

a2 c2(a

c)

x + ]

v = 1 a

a c sech[ a

2 c2(a c)x + ],

provided b =

a2c2(ac).

24


25/26

Example: Nonlinear Schrodinger equation (focusing/defocusing):i ut + uxx |u|2u = 0.

Bright soliton solution (+ sign):

u(x, t) =k

2exp[i (

c

2x + (k2 c

2

4)t)] sech[k(x ct x0)]

Dark soliton solution ( sign):

u(x, t) =1

2exp[i(Kx (2k2 + 3K2 2Kc + c

2

2)t)]

k tanh[k(x ct x0)] i(K

c

2)

.

Example: Nonlinear sine-Gordon equation (light cone coordinates):uxt = sin u.

Setting = ux, = cos(u) 1, gives

xt = 02 + 2 + 2t = 0.

Solitary wave solution (kink):

= ux = 1c sech[1c(x ct) + ],

= cos(u)

1 = 1

2 sech2[

1

c(x

ct) + ],

in final form:

u(x, t) = 4 arctanexp( 1c(x ct) + )

.

25


26/26

Example: Coupled nonlinear Schrodinger equations:i ut = uxx + u(|u|2 + h|v|2)i vt = vxx + v(|v|2 + h|u|2)

Seek particular solutions

u(x, t) = a tanh(x) exp(iAt)

v(x, t) = b sech(x) exp(iBt).

Seek solutions u(x, t) = U(F()), where derivatives of F() arepolynomial in F.

Now,

F() = 1 F2() F = tanh().Other choices are possible.

Add the constraining differential equations to the system of PDEsdirectly.

Why are tanh and sech solutions so prevalent? Other applications:

Computation of conservation laws, symmetries, first integrals, etc.

leading to linear parameterized systems for unknowns coefficients(see InvariantsSymmetries by Goktas and Hereman).

26

Date post:	04-Apr-2018
Category:	Documents
Upload:	amir
View:	222 times
Download:	0 times

Solving Nonlinear Wave Equations and Lattice with Mathematica

Documents