RIMS-1777

Multi-rogue waves solutions :

from NLS to KP-I equation

By

P. DUBARD and V.B. MATVEEV

March 2013

RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES

KYOTO UNIVERSITY, Kyoto, Japan

Multi-rogue waves solutions : from NLS to KP-I

equation

P. Dubard and V.B. Matveev

IMB, Universite de Bourgogne, 9 av. Alain Savary, BP 47870, 21078 Dijon Cedex,

France

E-mail: matveev@u-bourgogne.fr

Abstract. The discovery of the multi-rogue waves solutions made in 2010 completely

changed the vision of the links of the theory of rogue waves and integrable systems

allowing to explain many phenomena which were not understood before. It’s enough to

mention the famous 3-sister waves observed in ocean, the creation of a regular approach

to study higher Peregrine breathers and the new understanding of 2 + 1 dimensional

rogue waves via the NLS-KP correspondence. This article continues the study of the

multi-rogue waves solutions of the NLS equation and their links with the KP-I equation

started in the series of articles [1, 2, 3, 4]. In particular, it contains the discussion of

the large parametric asymptotic of these solutions which was never studied before.

CONTENTS 2

Contents

1 Introduction 2

2 The main formulas: multi-rogue solutions of the NLS equation. 3

2.1 Multi-rogue waves solutions of the NLS equation : ϕj parametrization . . 3

2.2 Non-stationary Schrodinger equation and the KP-I equation . . . . . . . 4

2.3 Reduction to the nonlinear Schrodinger equation . . . . . . . . . . . . . . 5

3 α, β-parametrization and Pn-breathers 7

3.1 First and second order solutions . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Rank 3 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Rank 4 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Links with the KP-I equation and the related movies . . . . . . . . . . . 19

4 Concluding remarks 20

5 Aknowledgments 24

1. Introduction

In this article we discuss the multi-rogue waves (MRW) solutions of the focusing NLS

equation

iut + uxx + 2|u|2u = 0 (1)

and the related solutions of the KP-I equation

(4vt + 6vvx + vxxx)x = 3vyy (2)

which we construct below. Equation (1) is obviously invariant with respect to the scaling

transformations, phase transformations and Galilean transformations

u(x, t) → Bu(B2x,Bt) B > O (3)

u(x, t) → eiχu(x, t) χ ∈ R (4)

u(x, t) → u(x− V t) exp (iV x/2− iV 2t/4) V ∈ R. (5)

MRW solutions of the NLS equation are quasi rational solutions

u = e2iB2tR(x, t), R(x, t) =

N(x, t)

D(x, t), B > 0. (6)

Here N(x, t), D(x, t) are polynomials of x and t, and

degN(x, t) = degR(x, t) = n(n+ 1), such that

|u2| → B2, x2 + t2 → ∞.

The rational function R(x, t) obviously satisfies the 1D Gross-Pitaevskii (GP) equation

or more precisely the 1D GP equation with zero trapping potential

iRt + 2R(|R|2 −B2) +Rxx = 0, |R| = |u|. (7)

CONTENTS 3

Therefore, rational MRW solutions of the GP equation are trivially connected with

quasi-rational MRW solutions of the focusing NLS equation. MRW solutions of the

NLS equation and the related (see the explanation below) solutions of the KP-I equation

are labeled by an integer n which defines the degree of polynomials N and D. Below

we call this integer the rank of the solution. For given n these solutions of the NLS

equation depend on 2n + 3 free real parameters χ,B, V, ϕj , j = 1, . . . 2n, where χ, B

and V correspond to the phase freedom, scaling freedom and a freedom to perform a

Galilean transformation with velocity parameter V . Without loss of generality we can

always deal with the case χ = 0, B = 1, V = 0. But we will keep B which has a sense

of asymptotic magnitude of u(x, t).

2. The main formulas: multi-rogue solutions of the NLS equation.

2.1. Multi-rogue waves solutions of the NLS equation : ϕj parametrization

Let n be any positive integer. Following [5] we define two polynomials q2n and Φ of

degree 2n by

q2n(k) :=n∏

j=1

(

k2 − ω2mj+1 + 1

ω2mj+1 − 1B2

)

, ω := exp

(

iπ

2n+ 1

)

(8)

Φ(k) := i

2n∑

l=1

ϕl(ik)l, B > 0, ϕ ∈ R. (9)

We assume that the integers mj satisfy

0 ≤ mj ≤ 2n− 1, ml 6= 2n−mj. (10)

In particular this condition is satisfied when mj = j− 1 and below we use this choice of

mj. In [5] the last choice was replaced by mj = j, which is not valid.‡ It is clear that

for any k the function f defined by

f(k, x, t) :=exp(kx+ ik2t+ Φ(k))

q2n(k)(11)

is a solution of the non stationary linear Schrodinger equation with zero potential

−ift = fxx. (12)

The same is true for the functions f1, . . . , f2n defined by the formula

fj(x, t) := D2j−1k f(k, x, t) |k=B , Dk :=

k2

k2 + B2

∂

∂kj = 1 . . . n; (13)

fn+j(x, t) := D2j−1k f(k, x, t) |k=−B . (14)

Suppose that W1, W2 are Wronskian determinants composed from the functions fj and

f

W1 := W (f1, . . . , fn) ≡ detA, Alj = ∂l−1x fj, W2 := W (f1, . . . , fn, f).

Then the following proposition holds.

‡ See [4] for further comments concerning the condition (10).

CONTENTS 4

Theorem 1 The function

un(x, t) := −q2n(0)B1−2ne2iB2tW2 |k=0

W1

(15)

represents a 2n+ 1 parametric family of solutions of the NLS equation.

We call this solution multi-rogue waves solution of rank n or simply MRWn.§ We will

give a proof of theorem 1 slightly later.

2.2. Non-stationary Schrodinger equation and the KP-I equation

Consider the Lax system for the KP-I equation

−iψy = ψxx + v(x, y, t)ψ (16)

−4ψt = 4ψxxx + 6vψx + 3w(x, y, t)ψ. (17)

In particular the first equation of this system is the non stationary linear Schrodinger

equation with potential v(x, y) depending on the parameter t with the ”time” evolution

variable y.

Suppose v(x, y, t) is any solution of the KP-I equation (2) and f1, . . . , fn, f are

linearly independent solutions of (16) and (17). Then the following proposition [6, 7, 8]

holds.‖Theorem 2 The function

ψ :=W (f1, . . . , fn, f)

W (f1, . . . , fn)(18)

is a solution of (16) and (17) with potential

vn(x, y, t) := v(x, y, t) + 2∂2x logW (f1, . . . , fn) (19)

and the function vn(x, y, t) is a new solution of KP-I equation.

In particular, this is true when v(x, y, t) = 0, w = 0.

It is clear that all functions f , fj, j = 1, . . . 2n defined by (11) and (13) will satisfy

the Lax system (16) and (17) with v = 0, w = 0 if we denote t by y and ϕ3 by −t.Therefore we have the following result

Theorem 3 The function

v2n(x, y, t) := 2∂2x log W (f1, . . . , f2n), fj(x, y, t) := fj |t=y,ϕ3=−t (20)

where fj are defined in (13) represents a family of smooth real rational solutions of the

KP-I equation. This solution satisfies the relation∫

∞

−∞

v2n(x, y, t) dx = 0. (21)

These solutions are also rational functions of 2n− 1 parameters ϕ1, ϕ2, ϕ4, . . . , ϕ2n.

§ In a different and more complicated notations, with no use of Wronskian determinants, this solution

was first presented in [5] where it was also mentioned that for n = 1 it reproduces the so called Peregrine

breather or P1 breather [9].‖ [6] and [7] contain much more general statements directly applicable to the non-Abelian KP-I and

KP-II hierarchies and their different reductions.

CONTENTS 5

To understand why these solutions of the KP-I equation are real valued and non-singular

we have to remark that an important statement which we call NLS-KP correspondence

holds.

Theorem 4 The solution (20) can be also written as follows :

v2n = 2(|un|2 −B2), un(x, y, t) := un(x, t, ϕ1, . . . , ϕ2n) |t=y,ϕ3=−t . (22)

It turns out that v2n(x, y, t) satisfies the important inequality

v2n ≥ −2B2. (23)

A reasonable conjecture is that the maximum value of |un(x, t)| is described by the

formula

maxx,t,ϕ1,...,ϕ2n∈R

|un(x, t, ϕ1, . . . , ϕ2n)| = B(2n+ 1). (24)

The solution where the parameters ϕ1, . . . , ϕ2n are chosen is such a way that this

maximum is attained is denoted Pn(x, t) and called a Pn-breather. In the case of Pn-

breathers, this conjecture belongs to Akhmediev and was tested by him first for n ≤ 3

and next by Pierre Gaillard for n ≤ 10 but the conjecture in its full extent was first

introduced in our works where the whole family of solutions was first constructed. Of

course it is enough to prove this conjecture for B = 1 since the general result then

follows from the scaling invariance of the NLS equation. If this conjecture is true the

related maximal value of the solution of the KP equation described by (20) is given by

the formula

maxx,y,t∈R

v(x, y, t) = 8B2n(n+ 1), (25)

i.e. it is equal to the number of peaks of the generic solution of rank n of the NLS

equation times the magnitude of the KP-I image of the P1 breather.

2.3. Reduction to the nonlinear Schrodinger equation

By theorem 2 the function

ψ(x, t, k) :=W (f1, . . . , f2n, f)

W (f1, . . . , f2n)(26)

is a solution of

iψt + ψxx + vψ = 0 (27)

for the potential

v(x, t) := 2∂2x logW (f1, . . . , f2n) (28)

and so is

RC(x, t) := Cψ(x, t, 0). (29)

This is a general result that holds any time the functions fj are solutions of (12). We

can remark that RC is a rational solution of (27). The particular form of the fj and the

right choice for the constant C allow us to reduce (27) to (1).

CONTENTS 6

Proposition 5 If C = eiχq2n(0)B1−2n then v and R defined by (28) and (29) satisfy

v = 2(

|R|2 −B2)

¶Proof

Because of (9) we can define define the following three meromorphic differentials

dΩ :=(k2 +B2)2n+1 + (k2 −B2)2n+1

2k2(k2 −B2)2ndk, (30)

dΩ1 := ψ(x, t, k)ψ(x, t,−k) dΩ, (31)

dΩ2 := (k +B2

k) dΩ1. (32)

The polynomial q2n defined by (8) satisfy

2k2q2n(k)q2n(−k) = (k2 + B2)2n+1 + (k2 −B2)2n+1

hence dΩ and dΩ1 have poles at k = ±B and k = ∞ and dΩ2 have poles at k = 0,

k = ±B and k = ∞.

In the neighborhood of k = ±B we use the local parameter z = k − B/k. dΩ can

be expressed as

dΩ =

(

(z2 + 4B2)n

z2n+

z

2B

(

1 +z2

4B2

)−1

2

)

dz

2

and it admits the following expansion

dΩ =

(

α±

0

z2n+

α±

1

z2n−2+ . . .+

α±

n−1

z2+O(1)

)

dz. (33)

Given (26) we can check that the odd order derivatives of ψ with respect to z vanish up

to order 2n− 1 so ψ has an expansion of the form

ψ = β±

0 + β±

1 z2 + . . .+ β±

n−1z2n−2 +O(z2n). (34)

By (33) and (34) we obtain that the residues of dΩ1, and subsequently dΩ2, at k = ±Bvanish and the remaining residues must satisfy

res∞dΩ1 = 0, (35)

res0dΩ2 = −res∞dΩ2. (36)

¶ The proof of this proposition is the same as in [5]. The proof of the smoothness of the solution

(15) in [5] was too complicated. It follows directly from the structure of focusing NLS equation and

meromorphic nature of the discussed solution considered as a function of x. See [10] for a detailed

explanation.

CONTENTS 7

In the neighborhood of k = ∞, ψ admits the following expansion

ψ =

(

1 +ξ1(x, t)

k+ξ2(x, t)

k2+ . . .

)

ekx+ik2t+Φ(k) (37)

and (35) yields the reality of ξ1. We can easily check that

res0dΩ2 = |ψ(x, t, 0)|2|q2n(0)|2/B4n−2 = |R|2 (38)

and

−res∞ dΩ2 = ξ2 + ξ2 − ξ21 + (2n+ 1)B2. (39)

Substituting (37) into (27) we obtain the relations

v = −2∂xξ1

and

i∂tξ1 + 2∂xξ2 + ∂2xξ1 − 2∂xξ1ξ1 = 0. (40)

The real part of (40) combined with the reality of ξ1 yields

∂x(

ξ2 + ξ2 − v/2− ξ21)

= 0

or, equivalently,

ξ2 + ξ2 − ξ21 = v/2 +K(t).

This latest relation with (36), (38) and (39) gives us

|R|2 = v/2 +K(t) + (2n+ 1)B2.

A comparison of the behaviors of both sides when |x| → ∞ shows that K(t) = −2nB2

and gives the announced result.

R is now a rational solution of (7) and u(x, t) := R(x, t)e2iB2t is a solution of (1).

3. α, β-parametrization and Pn-breathers

3.1. First and second order solutions

In this subsection we set χ = 0, B = 1 and V = 0 but these parameters can easily

be restored performing the scaling transformation (3), the phase transformation (4)

and the Galilean transformation (5). Finally, phases ϕ1 and ϕ2 are simply space and

time translation parameters, so we can select their values in order to produce the most

compact expressions. Between the 2n + 2 parameters χ,B, ϕ1, . . . , ϕ2n, only 2n − 2

of them, namely ϕ3, . . . ϕ2n, have a direct influence on the shape of magnitude of the

solution.

In the case n = 1 it means we essentially get only one solution. By choosing ϕ1 = 0

and ϕ2 =√3/4 we get the Peregrine solution [9] :

P1(x, t) =

(

1− 41 + 4it

1 + 4x2 + 16t2

)

e2it ≡(

1− 41 + iT

1 +X2 + T 2

)

eiT/2, X := 2x, T := 4t.

CONTENTS 8

Its plot is presented in figure 1.

The case n = 2 is the first one where we get a family of solutions depending on 4

parameters including two nontrivial parameters ϕ3 and ϕ4. We already explained above

that ϕ3 has a connection with the KP-I equation. For convenience we choose ϕ1 = 3ϕ3

and ϕ2 = 2ϕ4+(3+√5) sin(π/5)/4 and we switch from the pair of parameters ϕ3, ϕ4

to the pair α, β defined by

α := 48ϕ3

β := 4(5 +√5) sin(π/5)− 96ϕ4.

The solutions read

u2(x, t, α, β) =

(

1− 12G(2x, 4t) + iH(2x, 4t)

Q(2x, 4t)

)

e2it (41)

where

G(X,T ) := X4 + 6(T 2 + 1)X2 + 4αX + 5T 4 + 18T 2 − 4βT − 3

H(X,T ) := TX4 + 2(T 3 − 3T + β)X2 + 4αTX + T 5 + 2T 3

− 2βT 2 − 15T + 2β

Q(X,T ) := (1 +X2 + T 2)3 − 4αX3 − 12(2T 2 − βT − 2)X2

+ 4(3α(T 2 + 1)X + 6T 4 − βT 3 + 24T 2 − 9βT

+ α2 + β2 + 2). (42)

When α = β = 0 this solution coincide with the P2-breather with maximum of

magnitude equal 5 obtained at the point x = t = 0. When α2 + β2 is small enough

the solution is obviously very close to the P2-breather. Formula (42) shows that the

solution u2(x, t, α, β) can be considered as a two-parametric quadratic deformation of

the P2-breather. When the two parameters are large enough we obtain the ”generic”

form of rank 2 multi-rogue waves : three rogue waves of similar height. These two

phenomenon are shown in figure 2. In between we can observe some transition states.

Some are represented in figure 3 or figure 4.

Formula (42) which was first discovered in [2, 3, 4] was very important, showing for

the first time that, contrary to the genuine Peregrine breathers, its higher versions are

not isolated.+ There exist a families of solutions with very similar properties (obtained

by the sufficiently small variation of parameters α and β in vicinity of their zero values)

having almost the same extreme rogue wave behaviour as the P2-breather. Formula (42)

also clearly answered the question posed by Eleonski and Kulagin about 28 years ago :

how to embed the P2 breather discovered in [11] into a larger family of quasi rational

solutions of the NLS equation.∗+ In [2, 3, 4] the parameters α, β were defined differently. They were proportional to those we use here

in order to shorten the writing of the formulas.∗ It is instructive to see the movies KP2a, KP2e providing at any fixed moment of time the plot of

the square of the absolute value of the related solution of the NLS equation. See the subsection ”Links

with KP-I equations and the related movies” below.

CONTENTS 9

When α2 + β2 tends to ∞, u2(x, t, α, β) tends to e2it. Thus, a simple wave, i.e.

rank 0 solution, can be interpreted as a large parametric limit of the rank 2 multi-rogue

wave solution. Below we will show that for higher ranks similar but more diversified

phenomena take place.

Figure 1. The Peregrine solution

Figure 2. Second order solution (41) for α = 0 and β = 0 on the left and α = 20 and

β = 20 on the right

CONTENTS 10

Figure 3. Second order solution (41) for α = 0 and β = 1 on the left and α = 0 and

β = 3 on the right

Figure 4. Second order solution (41) for α = 0 and β = 6 on the left and α = 5 and

β = 0 on the right

CONTENTS 11

3.2. Rank 3 solutions

The ϕ-parametrization described above makes it difficult to isolate the values of

parameters describing higher Peregrine breathers. Similarly to the previous section

here we introduce 4 ”essential” parameters α1, β1, α2, β2. We choose ϕj according to the

following linear system

ϕ1 = 3ϕ3 − 5ϕ5

ϕ2 = 2ϕ4 − 3ϕ6 +sin(π/7)

4(1−cos(π/7))

768ϕ3 = 26α1 − α2

1920ϕ4 = −40β1 + β2 + 96(3 sin(π/7) + 8 sin(2π/7) + 2 sin(3π/7))

3840ϕ5 = 10α1 − α2

7680ϕ6 = −20β1 + β2 + 32(4 sin(π/7) + 14 sin(2π/7) + sin(3π/7)).

Substituting these formulas in expression (15) for n = 3, we can after long

calculation, using Maple, convert the related solution of NLS equation in the following

explicit form :

u3(x, t, α1, β1, α2, β2) =

(

1− 24G3(2x, 4t) + iH3(2x, 4t)

Q3(2x, 4t)

)

e2it (43)

with

G3(X,T ) = X10 + 15(T 2 + 1)X8 +∑6

n=0 gn(T )Xn

H3(X,T ) = TX10 + 5(T 3 − 3T + β1)X8 +

∑6n=0 hn(T )X

n

Q3(X,T ) = (1 +X2 + T 2)6 − 20α1X9 − 60(2T 2 − β1T − 2)X8 + 4

∑7n=0 qn(T )X

n

where

g6 = 50T 4 − 60T 2 + 80β1T + 210

g5 = 120α1T2 − 18α2 + 300α1

g4 = 70T 6 − 150T 4 + 200β1T3 + 450T 2 + 30β2T − 450 + 150α2

1 − 50β21

g3 = 400α1T4 + (3000α1 − 60α2)T

2 − 800α1β1T − 600α1 − 60α2

g2 = 45T 8 + 420T 6 + 6750T 4 − (6000β1 − 180β2)T3 − (300α2

1 − 900β21 + 13500)T 2

+(3600β1 + 180β2)T − 675− 300α21 − 300β2

1

g1 = 280α1T6 + (150α2 − 2100α1)T

4 + 800α1β1T3 − (3600α1 − 540α2)T

2

+(120β2α1 + 1200α1β1 − 120α2β1)T − 200α1β21 − 900α1 − 90α2 − 200α3

1

g0 = 11T 10 + 495T 8 − 120β1T7 + 2190T 6 − (42β2 + 1200β1)T

5

+(350α21 + 150β2

1 − 7650)T 4 + (6600β1 − 420β2)T3

−(2100β21 + 2025− 120β2β1 − 120α2α1 + 900α2

1)T2 + (200α2

1β1 + 200β31 − 90β2)T

+675 + 150α21 + 6α2

2 + 150β21 + 6β2

2

CONTENTS 12

h6 = 10T 5 − 140T 3 + 40β1T2 − 150T + 60β1 − 5β2

h5 = 40α1T3 + (60α1 − 18α2)T + 40α1β1

h4 = 10T 7 − 210T 5 + 50β1T4 − 450T 3 + 15β2T

2 − (50β21 + 1350− 150α2

1)T

+150β1 − 15β2h3 = 80α1T

5 + (1000α1 − 20α2)T3 − 400α1β1T

2 − (1800α1 − 60α2)T

+200α1β1 + 20β2α1 − 20α2β1h2 = 5T 9 − 60T 7 + 1710T 5 + (45β2 − 2100β1)T

4 + (300β21 − 6300− 100α2

1)T3

+(1800β1 − 90β2)T2 + (4725 + 300α2

1 + 300β21)T − 135β2 − 100β3

1

−100α21β1 − 900β1

h1 = 40α1T7 + (30α2 − 1140α1)T

5 + 200α1β1T4 − (2400α1 − 60α2)T

3

+(60β2α1 − 60α2β1 + 600α1β1)T2 − (900α1 + 450α2 + 200α3

1 + 200α1β21)T

+60α2β1 − 60β2α1

h0 = T 11 + 25T 9 − 15β1T8 − 870T 7 + (40β1 − 7β2)T

6 + (70α21 − 9630 + 30β2

1)T5

+(5850β1 − 75β2)T4 + (40β2β1 + 40α2α1 − 2475− 900α2

1 − 1300β21)T

3

+(100α21β1 + 495β2 + 100β3

1)T2 + (6α2

2 + 4725− 240α2α1 − 240β2β1+750β2

1 + 6β22 + 750α2

1)T − 20α21β2 − 675β1 − 45β2 − 100α2

1β1 − 100β31

+40α2α1β1 + 20β21β2

q7 = 3α2 − 30α1

q6 = −60T 4 + 40β1T3 + 120T 2 − (15β2 − 60β1)T + 35β2

1 + 15α21 + 580

q5 = 30α1T4 − (27α2 − 90α1)T

2 + 120α1β1T − 27α2 + 540α1

q4 = 30β1T5 − 360T 4 + (15β2 + 600β1)T

3 + (3360 + 225α21 − 75β2

1)T2

+(135β2 − 1350β1)T + 225β21 − 30α2α1 + 525α2

1 − 30β2β1 + 840

q3 = 40α1T6 + (1950α1 − 15α2)T

4 − 400α1β1T3 + (90α2 + 4500α1)T

2

+(60β2α1 − 1800α1β1 − 60α2β1)T − 450α1 + 100α31 + 100α1β

21 − 135α2

q2 = 60T 8 + 3360T 6 − (1620β1 − 27β2)T5 + (225β2

1 − 75α21 + 19560)T 4

−(16200β1 − 270β2)T3 + (450α2

1 − 9120 + 4050β21)T

2

+(675β2 + 2700β1 − 300β31 − 300α2

1β1)T + 3036 + 9α22 − 180α2α1

+225β21 + 225α2

1 + 9β22 − 180β2β1

q1 = 15α1T8 + (15α2 − 90α1)T

6 + 120α1β1T5 + (405α2 − 5400α1)T

4

+(3000α1β1 − 60α2β1 + 60β2α1)T3 + (1485α2 − 300α1β

21 − 1350α1 − 300α3

1)T2

+(540β2α1 − 540α2β1)T + 300α31 − 120α1β1β2 − 60α2α

21 + 135α2 + 60α2β

21

+300α1β21 + 2025α1

q0 = 30T 10 − 5β1T9 + 930T 8 − (240β1 + 3β2)T

7 + (15β21 + 3820 + 35α2

1)T6

+(1710β1 − 153β2)T5 + (30β2β1 + 30α2α1 − 975β2

1 + 35940− 75α21)T

4

+(100β31 + 100α2

1β1 + 135β2 − 23400β1)T3

+(9β22 + 23286 + 9α2

2 − 360β2β1 − 360α2α1 + 4725α21 + 8325β2

1)T2

+(120α2α1β1 − 60α21β2 − 1500α2

1β1 + 60β21β2 − 7425β1 − 675β2 − 1500β3

1)T

+506 + 9β22 + 100β4

1 + 675α21 + 100α4

1 + 9α22 + 90β2β1 + 200α2

1β21

+675β21 + 90α2α1.

We can easily get the solutions of the first section by choosing the parameters

α1 = 48(ϕ3 − 5ϕ5),

CONTENTS 13

α2 = 480(ϕ3 − 13ϕ5),

β1 = 96(4ϕ6 − ϕ4) + 8(sin(π/7) + 2 sin(2π/7) + sin(3π/7)),

β2 = 1920(8ϕ6 − ϕ4) + 32(sin(π/7)− 4 sin(2π/7) + 4 sin(3π/7))

and performing the translation x, t x− ϕ1, t− ϕ2.

It is easy to see that P3 can be obtained as u3(x, t, 0, 0, 0, 0) and in particular

|P3(0, 0)| = 7. Let us indicate the particular values of ϕj corresponding to the P3

breather corresponding to particular solution of the system (3.2) with αj = βj = 0, j =

1, 2 :

ϕ1 = ϕ3 = ϕ5 = 0,

ϕ4 = (3 sin(π/7) + 8 sin(2π/7) + 2 sin(3π/7))/20,

ϕ6 = (4 sin(π/7) + 14 sin(2π/7) + sin(3π/7)/240,

ϕ2 = 2ϕ4 − 3ϕ6 +sin(π/7)

4(1− cos(π/7)).

One of the advantage of this representation of the solution is that we can analyze

its limit behavior when one or several parameters tend to infinity and x and t remain

bounded.

• If α2 and β2 remain finite then u3(x, t, α1, β1, α2, β2) tends to e2it when α21 + β2

1

tends to ∞.

• If α1 and β1 remain finite then u3(x, t, α1, β1, α2, β2) tends to P1(x, t) when α22 + β2

2

tends to ∞.

• If α1, β1, α2 and β2 all tend to ∞ in the following way

β1 ∼ bα1, α2 ∼ cαr1, β2 ∼ dαr

1

then the limit of u3(x, t, α1, β1, α2, β2) depends on r according to the following table

r limit

< 2 e2it

> 2 u1(x, t)

2 u1(x− x1, t− t1)

where x1 and t1 are defined by

x1 = 10(1−b2)c+20bd3(c2+d2)

t1 = 10(1−b2)d−20bc3(c2+d2)

Thus we have shown that u3 contain the solutions of rank 0 and 1 as appropriate large

parametric limits. Plots for several choices of parameters are shown in figure 5 and 6.

CONTENTS 14

Figure 5. Third order solution (43) for α1 = β1 = α2 = β2 = 0 on the left and

α1 = β1 = α2 = β2 = 50 on the right

Figure 6. Third order solution (43) for α1 = α2 = 0 and β1 = β2 = 50 on the left

and α1 = β1 = 0 and α2 = β2 = 5000 on the right

CONTENTS 15

3.3. Rank 4 solutions

Here we present without details the formulas providing the 6-parametric family of multi-

rogue wave solutions similar to the one of the previous section :

u4(x, t, α1, β1, α2, β2, α3, β3) =

(

1− 40G4(2x, 4t) + iH4(2x, 4t)

Q4(2x, 4t)

)

e2it (44)

with

G4(X,T ) = X18 + 27(T 2 + 1)X16 − 24α1X15 +

∑14n=0 gn(T )X

n

H4(X,T ) = TX18 + 9(T 3 − 3T + β1)X16 − 24α1TX

15 +∑14

n=0 hn(T )Xn

Q4(X,T ) = (1 +X2 + T 2)10 − 60α1X17 − 180(2T 2 − β1T − 2)X16 + 4

∑15n=0 qn(T )X

n.

The interested reader can find the explicit formulas for the coefficients in appendix.

These solutions are polynomials of order 6 with respect to α1 and β1, of order 4 with

respect to α2, β2 and quadratic with respect to α3, β3. The P4-breather is obtained from

it by setting αj = βj = 0, ∀j. It is easy to see that |P4(0, 0)| = 9. As above we can

investigate the limit of this solution when one or several parameter tends to infinity and

x and t remain bounded.

• If α2, β2, α3 and β3 remain finite then u4(x, t, α1, β1, α2, β2, α3, β3) tends to P1(x, t)

when α21 + β2

1 tends to ∞.

• If α1, β1, α3 and β3 remain finite then u4(x, t, α1, β1, α2, β2, α3, β3) tends to e2it

when α22 + β2

2 tends to ∞.

• If α1, β1, α2 and β2 remain finite then u4(x, t, α1, β1, α2, β2, α3, β3) tends to

u2(x, t, α1, β1) when α23 + β2

3 tends to ∞.

• If α3 and β3 remain finite and α1, β1, α2 and β2 all tend to ∞ in the following way

β1 ∼ bα1, α2 ∼ cαr1, β2 ∼ dαr

1

then the limit of u4(x, t, α1, β1, α2, β2, α3, β3) depends on r according to the follow-

ing table

r limit

< 3/2 u1(x, t)

> 3/2 e2it

3/2 u1(x− x2, t− t2)

where x2 and t2 are defined by

x2 = 3((1−3b2)(d2−c2)+2(b2−3)bcd)50(b2+1)3

t2 = 3((3−b2)b(d2−c2)+2(1−3b2)cd)50(b2+1)3

.

CONTENTS 16

• If α2 and β2 remain finite and α1, β1, α3 and β3 all tend to ∞ in the following way

β1 ∼ bα1, α3 ∼ eαs1, β3 ∼ fαs

1

then the limit of u4(x, t, α1, β1, α2, β2, α3, β3) depends on s according to the follow-

ing table

s limit

< 2 u1(x, t)

> 2 e2it

2 u1(x− x3, t− t3)

where x3 and t3 are defined by

x3 = (2bf+(1−b2)e)35(b2+1)2

t3 = (2be−(1−b2)f)35(b2+1)2

.

• If α1 and β1 remain finite and α2, β2, α3 and β3 all tend to ∞ in such a way that

β2 ∼ dα2, α3 ∼ eαp2, β2 ∼ fαp

2,

then, the limit of u4(x, t, α1, β1, α2, β2, α3, β3) depends on p according to the follow-

ing table

p limit

< 2 e2it

> 2 u2(x, t, α1, β1)

2 u2(x, t, α1 − α0, β1 − β0)

where α0 and β0 are defined by

α0 = 21(2df+(1−d2)e)10(e2+f2)

β0 = 21(2de−(1−d2)f)10(e2+f2)

.

• If α1, β1, α2, β2, α3 and β3 all tend to ∞ in the following way

β1 ∼ bα1, α2 ∼ cαr1, β2 ∼ dαr

1, α3 ∼ eαs1, β3 ∼ fαs

1.

then the limit of u4(x, t, α1, β1, α2, β2, α3, β3) depends on r and s according to the

following table

CONTENTS 17

r s limit

> 3/2 any e2it

any > 2 e2it

< 3/2 < 2 u1(x, t)

3/2 < 2 u1(x− x2, t− t2)

< 3/2 2 u1(x− x3, t− t3)

3/2 2 u1(x− x2 − x3, t− t2 − t3)

where x2, t2, x3 and t3 are defined as above.

It seems that un contain all solutions of order 0 to n− 2 as appropriately chosen large

parametric limits. Of course for higher ranks further work has to be produced to get

the similar results. In figure 7 to 9 we present some plots of fourth order solutions.

Figure 7. Fourth order solution (44) for α1 = β1 = α2 = β2 = α3 = β3 = 0

CONTENTS 18

Figure 8. Fourth order solution (44) for α1 = β1 = α2 = β2 = α3 = β3 = 20 on the

left and α1 = β1 = 0 and α2 = β2 = α3 = β3 = 1500 on the right

Figure 9. Fourth order solution (44) for α1 = β1 = α2 = β2 = 0 and α3 = β3 = 105

on the left and α1 = β1 = 20, α2 = β2 = 0 and α3 = β3 = 105 on the right

CONTENTS 19

3.4. Links with the KP-I equation and the related movies

Assume now that B = 1 and ϕj, j 6= 3 are selected in such a way that when t = 0

un(x, y, ϕ1, ϕ2, 0, ϕ4, . . . , ϕ2n) = Pn(x, y).

Then the related solution of the KP-I equation attains its absolute maximum at the

point x = t = y = 0 and

v(0, 0, 0) = 2[(2n+ 1)2 − 1] = 8n(n+ 1).

For instance for n = 2 this corresponds to

ϕ1 = 0, ϕ2 = (7 + 5√5)sin π/5

6, ϕ4 = (5 +

√5)sin π/5

24. (45)

Therefore similarly to the P2-breather the solution of the KP-I equation vp2(x, y, t)

generated by the selection of the phases above is rigid. It attains at the point

x = y = t = 0 the absolute maximum v(0,0,0)=48. This maximum is 3 times greater

then the height of the KP-I image of the P1 breather.

Quite similarly, for the case n = 3, from the formulas (44) we see that selecting the

phases as

ϕ1 = ϕ5 = 0, ϕ3 = −t,ϕ4 = 3 sin(π/7) + 8 sin(2π/7) + 2 sin(3π/7)/20,

ϕ6 = (4 sin(π/7) + 14 sin(2π/7) + sin(3π/7)/240,

ϕ2 = 2ϕ4 − 3ϕ6 +sin(π/7)

4(1− cos(π/7))(46)

we obtain via the same formula (22) the smooth rational solution of the KP-I equation

which we denote vp3(x, y, t) such that vp3(0, 0, 0) = 96. The related maximum is 6 times

higher then the KP-I image of the P1-breather.

These solutions of the KP-I equation can also be obtained from the α, β

parametrization of the solutions of the NLS equation. Investigating the relation be-

tween the parameters ϕj and the parameters αj, βj shows that performing the changes

of variables t y, x x + 3t, α1 −48t and α2 −480t + γ where γ is a free

real parameter is equivalent to the transformation presented above. It gives us a fam-

ily of solutions depending on one parameter β1 in the case n = 2 and three parameters

β1, β2, γ in the case n = 3 that differ from the families constructed above by translations

in x, y, t. The time evolution of some of these solutions can be observed on the 5 movies

available at http://www.kurims.kyoto-u.ac.jp/ kirillov/MATVEEV.

The first movie KP2a shows the triangular configuration of 3 peaks corresponding

to the choice of parameters β1 = 20 which at the moment of their appearance have

CONTENTS 20

the average height 16. At the large negative times it is close to an isosceles triangle

of growing size when t → −∞. Close to t = 0 the triangle loses its form and rotates

while propagating so that the peaks do not produce the confluence but rather the peak,

initially coinciding with the left vertex of the back side of the triangle, after some

rotation acquires the maximal height 25 at the moment when the distances between the

peaks are minimal. After this the peaks diverge forming asymptotically at large times

an isosceles triangle of growing size.

For the second movie (KP2e) with β1 = 0 the scenario is different. The initial

triangle (again almost isosceles for large negative times) does not rotate and contracts

progressively so that first happens the confluence of two peaks situated behind a first

one going ahead. Next, the so formed higher peak approaches the slower and smaller

one, so that at the moment of their confluence they form one peak of the height 48,

surrounded by four small peaks.] After the full confluence the mirror image of the

previous configuration appears forming asymptotically the isosceles triangle of growing

size .

The next three movies represent the evolution of a triangular array containing 6

peaks forming for large negative times the almost isosceles triangle. The first of them

(KP3a) with β1 = 20, β2 = 20, γ = 0 behaves as follows. First for large negative times

we have the almost isosceles triangular array containing 6 peaks. When t → 0, t < 0

the triangle rotates and contracts arriving to the maximal height 35 at some moment

of time with no confluence between peaks after which it starts to diverge again forming

at large positive times the configuration close to an isosceles triangle. For the second

movie KP3b with β1 = 0, β2 = 20, γ = 0 the maximum of amplitude in the ”collision

area” is a bit higher but still there is no total confluence of 6 peaks. Finally, in a third

movie (KP3e) with β1 = 0, β2 = 0, γ = 0 we see the formation of the extremal rogue

wave of the height 96 reaching at the moment t = 0 absolute maximum of the solution

located at the point (x, y) = (0, 0).

It is worthwhile to mention that, taking into account the NLS-KP correspondence

(22) the presented movies also show the infinite numbers of space time plots for the

square of magnitude of the NLS multi rogue wave solutions of the ranks 2 and 3 in

particular illustrating a variety of important non symmetric configurations including 2

peaks configurations.

4. Concluding remarks

1. Our works [1, 2, 3, 4] were the first to explain the concept of the multiple-rogue

waves solutions and their links with higher order Peregrine breathers thus, in particular,

answering a question posed by Eleonski and Kulagin almost 30 years ago : how to

incorporate the second order Peregrine breather discovered in [11] to the larger family

of rational solutions. The work [5] was written in 1986 exactly for this purpose but it

] this already gives an idea of what is an extremal rogue wave solution of the KP-I equation.

CONTENTS 21

was not properly understood until the appearance of our works written in 2010-2011.††The explanation of the related formula (15) for the multi-rogue waves solutions takes

less than one page and contains all necessary definitions. This formula is of about the

same length with respect to other recently appeared works and it was the first which

allowed to understand that generic multi-rogue wave solutions (unknown before) can

be considered as a simple rational (with respect to free parameters) deformations of

the higher Peregrine breathers or Pn breathers as we call them here. In addition this

formulation has the advantage of providing the natural construction of the multi-rogue

waves smooth rational solutions of the KP-I equation. By a way these solutions are quite

different from the so called multi-lumps smooth rational solutions of the KP-I equation

found in 1977 in [12]. Already in our first work [1] we mentioned that a number of other

approaches namely Darboux transformation formulas for the NLS equation (known since

1882, see for instance [8, 13, 14]) and also a passage to appropriate infinite periods limit

in multi-periodic solutions (first obtained in [10, 15] via degeneration of finite gap multi-

periodic solutions) should provide another description of the same multi-rogue wave

solutions. This prediction was fully realized for DT approach by Guo, Ling and Liu

[16] and for the other mentioned approach it was done (to some extent) by P. Gaillard

[17, 18, 19, 20]. In these works another formulas for the multi-rogue waves solutions were

obtained producing for all tested ranks the higher Peregrine breathers just setting all

the parameters to be zeros. In particular, in 2011-2012 Pierre Gaillard computed all the

Pn breathers of the rank n ≤ 10. Before that, only the genuine Peregrine breather, the

P2-breather found in 1985 and the P3-breather discovered in 2009 [21, 22] were explicitly

described.

2. In addition, Hirota direct method was successfully applied to obtain also the

description of the multiple rogue waves solutions by Ohta and Yang [23]. The multi-

rogue wave solutions in [23] were expressed as a ratio of two tau-functions polynomial

with respect to complex parameters aj, aj. In their work the authors considered the

complex conjugate of the NLS equation and Gross-Pitaevskii equation used here. For

their form of the GP equation they obtained a beautiful explicit formula (3.20) in which

dominator and denominator of the solution are polynomials of the complex parameters

am, am, a0 = 1, a2j = 0, ∀j ≥ 1. For particular values of their parameters Ohta and Yang

first detected, for the rank 3 solutions, not only the circular arrays found slightly before

[18, 19, 24] but also the configurations of the six peaks forming a triangular array close to

an equilateral triangle with a slightly curved base (concave or convex) depending on the

choice of parameters (see the figure 2 of [23]). This was an indication that the solutions

discussed in [17, 18, 24, 25] were not generic contrary to our works [1, 2, 3, 4] and the

works [16] and [23]. Our films (see the reference on the related URL above) show that

the evolution of these triangular configurations is responsible for formation of extreme

rogue waves events described by KP-I equations and provide for ranks 2 and 3 an infinite

††That’s true that every of the formulas describing the multi-rogue wave solutions including our works

[1, 2, 3, 4] and those which appeared later [16, 23] , [19] have there own advantages and disadvantages

and quite a different analytic and combinatorial structures.

CONTENTS 22

number of such triangular configurations from the point of view of NLS equation. Some

triangular static configurations were also found slightly later with respect to [23] by

Guo, Ling and Liu [16] using their approach based on Darboux transformation. In [16]

similar triangular structure was also first found for the Hirota equation.

3. One further comment concerns two more points in the work by Ohta and Yang.

In [23] the values of parameters leading to the higher Peregrines breather were pointed

out only for ranks n = 1, 2, 3. The connection between the real parameters αj and βjused above and the complex valued parameters aj used in [23] for j = 2, 3, 4 is given by

the formulas

12a3 = α1 − 1− iβ1, 240a5 = α2 − 1− iβ2, 10080a7 = α3 − 1− iβ3.

In general the passage from the variables αj, βj to the variables aj of [23] is given by

the formula

a2j+1 =αj − 1− iβj2(2j + 1)!

. (47)

One of us (Ph. Dubard) conjectured that as all Pn-breathers with the maximum of

magnitude located at the point x = −1/2, t = 0 in the approach of [23] correspond to

the choice of aj given by (47) with αj = βj = 0 i.e.

a2j+1 = − 1

2(2j + 1)!, j ≤ n, aj = 0 ∀j ≥ 2n+ 2. (48)

This conjecture was confirmed by Ph.Dubard for the ranks n ≤ 7 although the general

proof is still missing .

Ohta and Yang also detailed their formula (3.20) expressing the matrix elements of

the related determinants by means of Schur polynomials with the arguments containing

some rational numbers rk, sk defined by means of simple generating functions (see the

formulas (4-9) of [23]), satisfying the condition r2k+1 = s2k+1 = 0. The short calculation

proves that r2k, s2k are simply expressed by means of Bernoulli numbers B2n. The laters

are defined by the formula

t

et − 1=

∞∑

n=0

Bn tn

n!. (49)

It follows from the definition above that B2n+1 = 0, ∀n ≥ 1.

Now it is easy to prove the formulas:

r2n =(22n − 1)

2n (2n)!B2n, s2n = − 22n − 2

2n (2n)!B2n, r2n + s2n =

B2n

n(2n!). (50)

Since all Bernoulli numbers are rational this proves that it is also true for sk and rk. Also

taking into account that Bernoulli numbers are very well studied this gives a simplest

way to compute r2j , s2j.

4. Another important comment concerns the works of Pierre Gaillard [17, 18, 19, 20]

and several others deposed on Arxiv hal. The key idea of these works suggested by

one of us (VM) in 2010 was to obtain the multi-rogue waves solutions considering an

CONTENTS 23

appropriate passage to the limit in the formulas describing the multi-phase trigonometric

modulations of the plane wave solutions [10, 15]. In fact in all of these works by

P. Gaillard despite the apparent presence of 2n or even more free parameters (for the

rank n) the related solutions were always 2 parametrical as we will briefly explain below.

The reason to think so was the absence in his plots of the triangular arrays similar to

those appearing already for n = 3 in [16] and [23]. Therefore we asked the author to

print out the related analytical expressions for ranks 3, 4 and 5 including explicitly all

his parameters aj, bj . At that time we already have got a formulas equivalent to α, β

parametrisation of this article for n=3 and 4. The comparison of Gaillard solutions with

ours shown that they are:

1. Simply equivalent to solutions of this work with α1, β1 = 0 for n = 3 and to the

solutions of this work for n = 4 with α1 = α2 = β1 = β2 = 0.

2. As we explained to Gaillard his solutions in fact are equivalent to those obtained

from his own formulas by setting for instance aj = bj = 0 ∀j > 1 . One of us (VM)

showed that this can be proved in intrinsic terms: first of all for any rank n it can be

checked that all the terms in numerators and denominators of his formulas are just the

quadratic polynomials of the same linear combinations of aj or bj, and such a structure

can not correspond to generic rational solution for the rank greater then 2. For instance,

for one of the works of PG for rank 4 these linear combinations found by VM are:

a = (a1 + 26a2 + 36a3 − 212a4), b = (b1 + 26b2 + 36b3 − 212b4).

All coefficients depending on the parameters in this case are proportional to a2 + b2

or to a and b. This was also checked for higher ranks but of course the coefficients in

the aforementioned linear combinations are depending on the rank but qualitatively the

fact rest the same. All the formally multi-parametric solutions of the works by Gaillard

of the period 2011-2012 were dealing with two parametric quadratic deformations of

the higher Peregrine breathers. The work [19] was written already on the base of

this understanding which allowed to compactify drastically his previous calculations.

Unfortunately he never explained clearly all these points. Therefore the history of

coming to modern understanding explained above, which was asking a big amount of

work and intelligent study of monstrous formulas obtained as a Maple output having

volumes of hundreds of pages already for n=5, was never explained to the reader before.

Quite recently in [20] he succeeded to produce, by taking a new kind of long periods

limit of [10] a full family of the rank 3 solutions. Initially he supposed that he found

a new solution. In fact it is the same solution which he learned from us already in

March 2012, corresponding to the formula (43). Exact correspondence between these

two solutions computed by Ph. Dubard reads as follows :

a1 = −α1

2, b1 = −β1

2, a2 = 60α1 − 6α2, b2 =

83

2β1 − 6β2.

Of course this latest result of PG means that in general the method of passage to the

limit in [10, 15] produces also the generic solutions under appropriate choice of the

limiting procedure but further work in this direction for higher ranks should be done.

CONTENTS 24

5. Recently on arXiv appeared a preprint by He, Fokas et al [26] which can be

considered as a natural continuation of [16]. Some of the conclusions of our work here

are also relevant to this article, although we considered a more diversified class of large

parametric limits of the multi-rogue wave solutions.

6. The writing of MRW solutions to the NLS and KP-I equation by means of

explicit polynomials of several variables (independent variables and parameters) allows

not only to make a punctual numerical evaluation of these solutions and producing

stationary plots but now we can also study the large parametric asymptotic of the

solutions and also see some symmetries which were not visible from initial determinant

representations. The simplest symmetries concern the Pn-breathers. Namely it can be

seen that for all the examples tested in our works and those by P.Gaillard, i.e. for

n ≤ 10, the denominator D(x, t) in (6) is always an even function of both x and t, the

real part of N(x,t) is an even function of x and t and the imaginary part of N(x, t) is an

even function of x and an odd function of t. Moreover some symmetries can be observed

even for the deformations of Pn breathers i.e for generic MRW solutions due to the α, β

representations of the solutions. For instance for n = 2 it follows from (41) that

u2(x, t, α, β) = u2(−x, t,−α, β) = u2(x,−t, α,−β)

and the same kind of symmetries appear in (43) and (44). These symmetries in the

structure of the MRW solutions of the NLS equation give some symmetry to the

associated solutions of the KP-I equation. If we denote by v2 and v3 the solutions

associated to u2 and u3 we can easily see that they have the following properties :

v2(x, y,−t, β) = v2(−x, y, t, β),

v2(x,−y, t, β) = v2(x, y, t,−β),v3(x, y,−t, β1, γ, β2) = v3(−x, y, t, β1,−γ, β2),v3(x,−y, t, β1, γ, β2) = v3(x, y, t,−β1, γ,−β2).

7. We can easily construct multi-rogue waves solutions for the Johnson-I aka CKP-I

equation or for recently discovered ECKP or Elliptic cylindrical KP-I equation [27, 28]

using their links with KP-I equation. For CKP-I equation some plots of the multi-rogue

wave solutions with fixed time values can be found in [4] .

5. Aknowledgments

One of the authors (VM) wishes to thank Professor Nakajima for the invitation to

visit RIMS and the kind hospitality during his stay (February 27 -March 19 2013 )

where this work was completed and also for opportunity to participate in the conference

”Bethe Ansatz , Quantum Groups and Beyond” (March 7-9 2013 ) at RIMS dedicated

to retirement of Professor Kirillov, where this work was reported. I also wish to thank

Anatol N Kirillov for many illuminating discussions and a wonderful time we spent

together. Both of the authors were partially supported by ANR via the program ANR-

09-BLAN-0117-01.

CONTENTS 25

Appendix

g14 = 180T 4 − 360T 2 + 360β1T + 900

g13 = 420α1 − 42α2

g12 = 588T 6 − 5460T 4 + 3360β1T3 − 1260T 2 + (5880β1 − 378β2)T

+18900 + 630α21 + 630β2

1

g11 = 2520α1T4 + (22680α1 − 1764α2)T

2 + 38808α1 − 4284α2 + 72α3

g10 = 1134T 8 − 17640T 6 + 10584β1T5 + 18900T 4 − (84β2 + 16800β1)T

3

+(11340α21 − 37800 + 11340β2

1)T2 + (4620β2 − 168β3 + 37128β1)T

+43260α21 − 3192α2α1 + 1176β2β1 − 15540β2

1 − 107730

g9 = 11760α1T6 + (140700α1 − 8190α2)T

4 + (600α3 + 546000α1 − 42420α2)T2

+(2520α2β1 − 142800α1β1 + 17640β2α1)T + 9800α31 + 600α3 − 29400α1β

21

−342300α1 − 24150α2

g8 = 1386T 10 − 20790T 8 + 15120β1T7 + 235620T 6 + (7938β2 − 219240β1)T

5

+(47250β21 + 170100 + 47250α2

1)T4 + (149940β2 − 866880β1 − 2520β3)T

3

+(472500α21 − 132300β2

1 − 5698350− 15120β2β1)T2 + (1464120β1 − 189000α2

1β1

+63000β31 + 6930β2 − 2520β3)T + 1638α2

2 + 17640β2β1 − 129150β21

+126β22 − 481950− 7560α2α1 − 353430α2

1 − 720α1α3

g7 = 22680α1T8 + (317520α1 − 10584α2)T

6 + (47880α2 + 3119760α1 − 2160α3)T4

+(90720β2α1 − 2923200α1β1 − 10080α2β1)T3 + (453600α1β

21 − 151200α3

1

−15840α3 + 425880α2 − 9621360α1)T2 + (5760α3β1 + 211680β2α1

−131040α2β1 + 2600640α1β1 − 12096α2β2)T + 12600α2 + 9360α3

−352800α31 + 20160α2

1α2 − 352800α1β21 + 20160α2β

21 − 1317960α1

g6 = 1092T 12 − 1848T 10 + 9240β1T9 + 696780T 8 + (19656β2 − 564480β1)T

7

+(88200α21 + 88200β2

1 + 5609520)T 6 + (518616β2 − 3024β3 − 9603216β1)T5

+(105840α2α1 − 105840β2β1 − 1499400α21 − 34265700 + 2381400β2

1)T4

+(36960β3 + 18392640β1 + 352800α21β1 − 535080β2 − 117600β3

1)T3

+(31752β22 − 2910600α2

1 − 20160β1β3 + 423360α2α1 + 19845000− 10584α22

−141120β2β1 − 1428840β21)T

2 + (352800α21β1 − 7285320β1 − 588000β3

1

−1464120β2 − 35280β3)T + 112896α2α1 + 1176α22 − 452760α2

1 + 3360α1α3

−336α2α3 − 117600α41 + 117600β4

1 + 265776β2β1 + 793800β21 + 8232β2

2

−336β2β3 − 1190700

g5 = 22176α1T10 + (378α2 + 328860α1)T

8 + (758520α2 − 7056α3 − 2850624α1)T6

+(21168β2α1 − 1058400α1β1 − 190512α2β1)T5 + (2249100α2 + 529200α1β

21

+15120α3 − 30217320α1 − 176400α31)T

4 + (84672α2β2 + 19192320α1β1 − 70560β2α1

−1764000α2β1 − 20160α1β3 − 20160α3β1)T3 + (211680α2

1α2 − 5292000α1β21

−45360α3 − 3880800α31 − 3969000α2 − 3810240α1 + 211680α2β

21)T

2 + (105840α2β2

−3024α3β2 + 1764000α1β1 − 50400α1β3 − 818496β2α1 + 705600α1β31 + 3024α2β3

+994896α2β1 + 705600α31β1)T − 7699860α1 + 39690α2 + 45360α3 − 987840α3

1

+317520α21α2 + 5040α2

1α3 − 14112α1α22 − 21168α2β1β2 + 352800α1β1β2

−987840α1β21 + 7056α1β

22 − 35280α2β

21 + 5040α3β

21

CONTENTS 26

g4 = 540T 14 + 16380T 12 + 903420T 10 + (16170β2 − 634200β1)T9 + (85050β2

1 + 10791900

+85050α21)T

8 + (3600β3 + 68040β2 − 15372000β1)T7 + (70560α2α1 + 5027400β2

1

−61519500− 4145400α21)T

6 + (25200β3 + 1587600α21β1 + 52038000β1 − 529200β3

1

−3501540β2)T5 + (529200α2α1 + 79380α2

2 + 17860500− 50400α1α3 − 26460β22

−18566100α21 − 16978500β2

1 + 2293200β2β1)T4 + (1764000β3

1 + 2910600β2

−22226400β1 + 8820000α21β1 − 352800β2

1β2 − 352800α21β2 − 75600β3)T

3

+(370440α22 − 52920β2

2 − 100800α1α3 + 14200200β21 − 2469600β2β1 − 2822400α2α1

+100800β1β3 + 20197800α21 + 65488500)T 2 + (1411200α1α2β1 − 25200α2

1β3

+105840α1α2β2 + 75600β3 + 705600β21β2 − 1653750β2 − 32016600β1 − 4762800α2

1β1

−105840α22β1 − 4762800β3

1 − 705600α21β2 − 25200β2

1β3)T + 17860500− 940800α2α1

+14700α22 − 1365630α2

1 + 11760α1α3 − 8400α2α3 + 120α23 − 588000α4

1 + 58800α31α2

−176400α1α2β21 + 176400α2

1β1β2 − 58800β31β2 − 58800β2β1 + 67620β2

2 + 6660570β21

+120β23 − 13440β1β3 − 8400β2β3 + 588000β4

1

g3 = 10920α1T12 + (157080α1 + 8316α2)T

10 + (795060α2 − 20661480α1 − 2520α3)T8

+(11995200α1β1 − 30240β2α1 − 211680α2β1)T7 + (2640120α2 − 1764000α1β

21

+588000α31 − 23520α3 − 97290480α1)T

6 + (90175680α1β1 − 84672α2β2 + 40320α1β3

−635040α2β1 − 776160β2α1)T5 + (2910600α2 − 378000α3 + 54243000α1 − 26460000α1β

21

−2940000α31)T

4 + (2352000α1β31 − 10080α3β2 − 31281600α1β1 + 94080β2α1

+336000α3β1 + 2352000α31β1 + 23520α2β1 + 33600α1β3 + 10080α2β3 − 776160α2β2)T

3

+(211680α2β1β2 − 11289600α31 − 50400α3β

21 + 2116800α1β1β2 + 756000α3

+1764000α21α2 − 352800α2β

21 − 211680α1β

22 − 11289600α1β

21 − 6218100α2

−109809000α1 − 50400α21α3)T

2 + (2352000α31β1 + 2352000α1β

31 + 211680α2β2

+100800α1β3 + 10080α2β3 − 201600α3β1 + 1270080β2α1 − 10080α3β2 + 16228800α1β1

+1199520α2β1)T + 17728200α1 + 1455300α2 + 37800α3 + 2587200α31 − 211680α2

1α2

+16800α21α3 + 23520α1α

22 − 6720α1α2α3 + 7056α3

2 − 70560α2β1β2 − 6720α1β2β3

−329280α1β1β2 + 2587200α1β21 + 94080α1β

22 + 117600α2β

21 + 7056α2β

22 + 16800α3β

21

g2 = 153T 16 + 13320T 14 − 2520β1T13 + 550620T 12 + (3276β2 − 332640β1)T

11 + (41580β21

−11680200 + 41580α21)T

10 + (11357640β1 + 3960β3 − 477540β2)T9 + (4063500α2

1

−52920α2α1 + 143640β2β1 − 3723300β21 − 173133450)T 8 + (256757760β1 + 352800β3

1

−1058400α21β1 + 102240β3 − 8031240β2)T

7 + (14376600α21 + 253222200 + 3386880β2β1

+3528α22 − 20160β1β3 − 125261640β2

1 + 45864β22 − 564480α2α1)T

6 + (25048800β31

+559440β3 − 9261000β2 − 197232840β1 − 423360β21β2 − 3175200α2

1β1 − 423360α21β2)T

5

+(5040β2β3 + 5040α2α3 + 55654200β21 − 1764000β4

1 + 405720β22 − 1693440α2α1

+1764000α41 − 335160α2

2 − 89302500− 403200β1β3 − 50400α1α3 − 458640β2β1

−1146600α21)T

4 + (70560α22β1 − 4586400β3

1 + 2116800β21β2 + 50400α2

1β3 + 60328800β1

+50400β21β3 − 1360800β3 − 3528000α2

1β2 − 211680α1α2β2 − 4586400α21β1 + 9128700β2

−141120β1β22 + 5644800α1α2β1)T

3 + (826560β1β3 − 1058400α1α2β21 + 675360α1α3

+1781640α22 − 80640β2β3 + 1058400α2

1β1β2 + 352800α31α2 + 720α2

3 − 14535360β2β1

+53193420α21 + 1464120β2

2 + 720β23 + 52664220β2

1 − 12418560α2α1 − 80640α2α3

+464373000− 352800β31β2)T

2 + (50400α21β3 − 21168β3

2 − 21168α22β2 + 3810240α1α2β1

−705600α21β2 − 17992800α2

1β1 − 493920α22β1 + 113400β3 + 211680α1α2β2

−201600α1α3β1 − 17992800β31 + 20160α2α3β1 − 137327400β1 − 5159700β2 − 282240β1β

22

+20160β1β2β3 + 3104640β21β2 − 151200β2

1β3)T + 22325625− 511560α2α1 + 88200α22

+16149420α21 − 80640α1α3 − 5040α2α3 + 720α2

3 + 1764000α41 − 352800α3

1α2 + 35280α21α

22

−352800α1α2β21 − 352800α2

1β1β2 + 35280β21β

22 − 352800β3

1β2 − 511560β2β1 + 88200β22

+16149420β21 + 720β2

3 − 80640β1β3 − 5040β2β3 + 1764000β41 + 3528000α2

1β21

+35280α21β

22 + 35280α2

2β21

CONTENTS 27

g1 = 2160α1T14 + (27300α1 + 3822α2)T

12 + (3510192α1 + 1848α3 − 6468α2)T10 + (64680β2α1

+9240α2β1 − 2629200α1β1)T9 + (90088740α1 − 5448870α2 + 83160α3 + 340200α1β

21

−113400α31)T

8 + (12096α2β2 + 14400α1β3 + 151200β2α1 − 20160α3β1 − 85760640α1β1

+2610720α2β1)T7 + (367920α3 + 23637600α1β

21 + 8114400α3

1 − 352800α2β21

−24307920α1 − 352800α21α2 − 25525080α2)T

6 + (42406560α1β1 + 7056α2β3

−16962624β2α1 + 359856α2β2 + 18409104α2β1 + 30240α1β3 − 7056α3β2 − 201600α3β1

−2116800α31β1 − 2116800α1β

31)T

5 + (25200α3β21 − 105840α1β

22 + 22954050α2

−211680α1α22 − 19051200α3

1 − 41145300α1 − 1285200α3 + 25200α21α3 + 5115600α2

1α2

+10231200α1β1β2 − 105840α2β1β2 − 5115600α2β21 − 19051200α1β

21)T

4 + (70560α2β3

−1411200α21α2β1 + 1108800α3β1 + 470400α3

1β2 + 2352000α31β1 + 470400α2β

31

−20532960α2β1 − 70560α3β2 + 20180160β2α1 + 2352000α1β31 + 15523200α1β1

−1008000α1β3 − 1411200α1β21β2 − 211680α2β2)T

3 + (28224000α1β21 + 211680α2β1β2

−13829760α1β1β2 + 604800α1β1β3 − 20160α2β1β3 − 7761600α21α2 − 340200α3

+20160α3β1β2 + 211680α1α22 + 28224000α3

1 + 233377200α1 + 5953500α2 − 21168α2β22

+252000α21α3 + 6068160α2β

21 − 352800α3β

21 − 21168α3

2)T2 + (33600α3

1β3

+2822400α1β21β2 − 317520α2β2 − 2063880β2α1 − 470400α3

1β2 − 100800α21α3β1

−10348800α31β1 − 10348800α1β

31 − 15120α3β2 − 79909200α1β1 − 4815720α2β1

+151200α1β3 − 1176000α2β31 + 15120α2β3 + 2116800α2

1α2β1 − 100800α1β21β3

+33600α3β31)T + 23417100α1 + 3770550α2 + 113400α3 + 12001080α3

1 + 929040α21α2

−28560α21α3 + 44688α1α

22 − 13440α1α2α3 + 480α1α

23 + 14112α3

2 − 1008α22α3

−2016α2β2β3 + 7056α2β1β2 − 13440α1β2β3 − 53760α1β1β3 − 305760α1β1β2

+12001080α1β21 + 37632α1β

22 + 480α1β

23 + 1234800α2β

21 + 14112α2β

22 + 25200α3β

21

+1008α3β22 + 117600α2β

41 + 1176000α5

1 − 117600α41α2 + 2352000α3

1β21 − 235200α3

1β1β2

+1176000α1β41 − 235200α1β

31β2

CONTENTS 28

g0 = 19T 18 + 3315T 16 − 816β1T15 + 129420T 14 − (66360β1 + 882β2)T

13 + (8190β21

+8190α21 + 328860)T 12 − (678048β1 + 312β3 + 43260β2)T

11 + (11088β2β1 − 170940α21

+4862970 + 22176α2α1 + 198660β21)T

10 + (46200α21β1 − 31883880β1 + 1218210β2

−15400β31 − 21720β3)T

9 + (8694α22 + 5760β1β3 + 7182β2

2 + 5040α1α3 − 17640α2α1

+18473490β21 − 902790α2

1 − 201540150− 677880β2β1)T8 + (90720β2

1β2 − 3645600β31

+90720α21β2 − 1539720β2 + 152767440β1 − 91440β3 − 352800α2

1β1)T7 + (4704β2β3

−184558500 + 7074816α2α1 − 39954600α21 − 51361800β2

1 + 33600β1β3 + 6604416β2β1

−323400β22 + 4704α2α3 − 252840α2

2 + 235200β41 − 235200α4

1)T6 + (21168α1α2β2

−5040β21β3 + 55658610β2 + 63504α2

2β1 + 8925840β31 + 1975680α2

1β2 − 650160β3

+8925840α21β1 − 5040α2

1β3 + 84672β1β22 − 4233600α1α2β1 − 2257920β2

1β2

+67367160β1)T5 + (198413250β2

1 − 57765120α2α1 + 206263050α21 + 600α2

3 + 588000α41

−52080α2α3 + 1142400β1β3 + 600β23 − 70642320β2β1 + 529200α1α2β

21 − 588000β4

1

+1167600α1α3 + 1458607500− 176400α31α2 + 1484700α2

2 − 52080β2β3 − 529200α21β1β2

+1537620β22 + 176400β3

1β2)T4 + (38478720α1α2β1 − 806400α1α3β1 + 420000α2

1β3

−149469600β31 − 6656160α2

1β2 − 1075334400β1 − 36647100β2 − 823200α22β1 + 7056α2

2β2

+7056β32 − 149469600α2

1β1 − 893760β1β22 − 386400β2

1β3 + 13440α2α3β1 + 20160α1α3β2

−20160α1α2β3 + 2759400β3 + 31822560β21β2 + 13440β1β2β3 − 70560α1α2β2)T

3

+(423360α2α1 + 793800α22 + 145291860α2

1 − 695520α1α3 − 120960α2α3 + 2160α23

+15993600α41 − 2704800α3

1α2 − 33600α31α3 + 105840α2

1α22 + 100800α1α3β

21 − 8820000α1α2β

21

−100800α21β1β3 + 352800α2

1β1β2 + 105840β21β

22 + 33600β3

1β3 − 5762400β31β2 + 17886960β2β1

+1746360β22 + 321515460β2

1 + 2160β23 − 1149120β1β3 − 120960β2β3 + 40454400β4

1

+56448000α21β

21 + 105840α2

1β22 + 105840α2

2β21 + 209860875)T 2 + (129360β2

1β3 − 432768β1β22

−480β1β23 + 1008β2

2β3 − 480α23β1 − 1008α2

2β3 − 35280α22β2 − 214032α2

2β1 − 25200α21β3

−2822400α21β2 − 41459880α2

1β1 − 352800α41β2 + 352800β4

1β2 − 218736α1α2β2

−1176000α1α2β1 + 2016α2α3β2 + 33600α2α3β1 + 154560α1α3β1 + 33600β1β2β3

+705600α1α2β31 + 705600α3

1α2β1 − 53184600β1 − 4961250β2 − 113400β3 − 35280β32 − 4704000β5

1

−41459880β31 − 9408000α2

1β31 − 4704000α4

1β1 − 3998400β21β2)T − 4465125 + 1076040α2α1

−57330α22 + 1847790α2

1 + 40320α1α3 − 5040α2α3 − 360α23 + 1764000α4

1 + 364560α31α2

+11760α21α

22 − 3360α2

1α2α3 + 7056α1α32 − 3360α2α3β

21 + 7056α2

2β1β2 + 7056α1α2β22

+364560α1α2β21 − 3360α2

1β2β3 + 364560α21β1β2 + 7056β1β

32 + 11760β2

1β22 + 364560β3

1β2

+1076040β2β1 − 57330β22 + 1847790β2

1 − 360β23 + 40320β1β3 − 5040β2β3 + 1764000β4

1

−3360β21β2β3 + 3528000α2

1β21 + 11760α2

1β22 + 11760α2

2β21 + 196000α6

1 + 588000α41β

21

+588000α21β

41 + 196000β6

1

h14 = 36T 5 − 600T 3 + 180β1T2 − 540T + 240β1 − 21β2

h13 = (420α1 − 42α2)T

h12 = 84T 7 − 2940T 5 + 840β1T4 + 1260T 3 − (2940β1 + 189β2)T

2 + (630β21 + 630α2

1 − 13860)T

+4676β1 − 595β2 + 14β3

h11 = 504α1T5 + (4200α1 − 588α2)T

3 + (72α3 − 756α2 − 6552α1)T + 7560α1β1 − 588α2β1 − 84β2α1

h10 = 126T 9 − 6552T 7 + 1764β1T6 + 25956T 5 − (21β2 + 29400β1)T

4 + (3780β21 − 52920 + 3780α2

1)T3

+(42β2 − 16716β1 − 84β3)T2 + (20580α2

1 + 1176β2β1 − 198450− 7980β21 − 3192α2α1)T

+3780α21β1 − 4641β2 + 84β3 − 1260β3

1 + 10416β1

h9 = 1680α1T7 + (4620α1 − 1638α2)T

5 + (200α3 − 4900α2 − 145600α1)T3 + (1260α2β1 + 8820β2α1

+46200α1β1)T2 + (9800α3

1 + 45570α2 − 577500α1 − 600α3 − 29400α1β21)T + 200α3β1

+168α2β2 + 54880α1β1 − 11620α2β1 − 280α1β3 + 5180β2α1

CONTENTS 29

h8 = 126T 11 − 7770T 9 + 1890β1T8 + 71820T 7 + (1323β2 − 84420β1)T

6 + (9450α21 + 9450β2

1

−532980)T 5 + (29295β2 − 630β3 + 312480β1)T4 + (220500α2

1 − 233100β21 − 614250

−5040β2β1)T3 + (31500β3

1 + 1260β3 − 94500α21β1 − 73395β2 − 202860β1)T

2 + (2976750

+412650β21 − 22680β2β1 − 7560α2α1 − 164430α2

1 + 1638α22 + 126β2

2 − 720α1α3)T + 6300β21β2

−682290β1 − 67095β2 + 1890β3 + 6300α21β2 − 31500α2

1β1 − 90300β31

h7 = 2520α1T9 − (1512α2 + 15120α1)T

7 + (32760α2 + 573552α1 − 432α3)T5 + (22680β2α1

−1033200α1β1 − 2520α2β1)T4 + (346920α2 + 151200α1β

21 − 6240α3 − 50400α3

1 − 6076560α1)T3

+(2880α3β1 + 15120β2α1 − 6048α2β2 + 2005920α1β1 − 136080α2β1)T2 + (20160α2

1α2

+20160α2β21 + 6480α3 + 2320920α1 − 50400α3

1 − 52920α2 − 50400α1β21)T − 50400α3

1β1

−50400α1β31 + 168α2β3 + 1512α2β1 − 512400α1β1 + 480α3β1 − 77952β2α1 − 1680α1β3

−168α3β2 + 2520α2β2

h6 = 84T 13 − 4872T 11 + 924β1T10 + 71820T 9 + (2457β2 − 110880β1)T

8 + (297360 + 12600β21

+12600α21)T

7 + (54684β2 − 1294776β1 − 504β3)T6 + (370440β2

1 − 21168β2β1 − 546840α21

+21168α2α1 − 7276500)T 5 + (7114800β1 + 88200α21β1 + 21000β3 − 29400β3

1 − 986370β2)T4

+(43659000− 3528α22 − 6720β1β3 + 88200α2

1 + 1522920β21 + 10584β2

2 + 94080β2β1)T3

−(17966340β1 + 176400α21β1 + 1117200β3

1 + 7560β3 + 291060β2)T2 + (117600β4

1 + 22344α22

−336β2β3 + 3251640α21 + 3360α1α3 − 392784β2β1 + 15288β2

2 + 4121880β21 + 6720β1β3 − 336α2α3

−733824α2α1 + 14685300− 117600α41)T − 4480560β1 + 11025β2 + 32760β3 − 664440β3

1

+129360β21β2 + 840β2

1β3 − 7056β1β22 − 3528α2

2β1 + 840α21β3 − 11760α2

1β2 − 664440α21β1

−3528α1α2β2 + 141120α1α2β1

h5 = 2016α1T11 + (42α2 − 37380α1)T

9 + (125496α2 − 1465632α1 − 1008α3)T7 + (3528β2α1

−317520α1β1 − 31752α2β1)T6 + (21168α3 − 418068α2 + 105840α1β

21 − 11716488α1 − 35280α3

1)T5

+(11501280α1β1 − 5040α3β1 − 52920β2α1 − 5040α1β3 + 21168α2β2 − 264600α2β1)T4

+(23919840α1 + 70560α21α2 − 3880800α1β

21 + 70560α2β

21 − 1528800α3

1 + 45360α3 − 3545640α2)T3

+(352800α31β1 + 1512α2β3 + 352800α1β

31 − 1512α3β2 + 10584α2β2 + 1132488α2β1 − 12947760α1β1

−5040α1β3 − 56448β2α1)T2 + (5040α3β

21 + 6271020α1 − 105840α2

1α2 + 105840α2β21

−21168α2β1β2 + 992250α2 + 136080α3 + 2540160α31 + 5040α2

1α3 − 211680α1β1β2 + 7056α1β22

+2540160α1β21 − 14112α1α

22)T − 588000α1β

31 − 1512α2β3 − 23520α2β

31 − 23520α3

1β2 − 15120α3β1

−402192β2α1 − 74088α2β1 − 4021920α1β1 − 588000α31β1 − 31752α2β2 + 30240α1β3

+1512α3β2 + 70560α21α2β1 + 70560α1β

21β2

h4 = 36T 15 − 1260T 13 + 13860T 11 + (1617β2 − 70980β1)T10 + (9450α2

1 − 585900 + 9450β21)T

9

+(450β3 − 40005β2 − 1115100β1)T8 + (642600β2

1 − 32829300 + 10080α2α1 − 970200α21)T

7

+(264600α21β1 − 1295070β2 − 4200β3 − 88200β3

1 + 39660600β1)T6 + (107559900 + 343980α2

1

+599760β2β1 − 5292β22 − 35280α2α1 − 10080α1α3 − 12744900β2

1 + 15876α22)T

5 + (5799150β2

−441000α21β1 + 1323000β3

1 − 88200α21β2 − 88200β2

1β2 − 88464600β1 − 144900β3)T4 + (67200β1β3

+77395500 + 17640β22 + 33600α1α3 + 88200α2

2 + 32075400α21 − 3998400α2α1 − 4821600β2β1

+39013800β21)T

3 + (52920α1α2β2 − 7673400β31 − 14729400α2

1β1 − 352800α21β2 − 189000β3

−12600β21β3 − 33075β2 − 12600α2

1β3 + 1058400β21β2 + 1411200α1α2β1 − 52920α2

2β1

−23152500β1)T2 + (176400α2

1β1β2 + 58800α31α2 − 58800β3

1β2 + 8953770β21 + 87360β1β3

+53581500− 91140α22 + 120α2

3 + 588000β41 + 120β2

3 − 176400α1α2β21 + 173460β2

2 − 1470000β2β1

−4717230α21 + 213360α1α3 − 8400α2α3 − 588000α4

1 − 8400β2β3 − 3763200α2α1)T − 1680α1α2β3

−52920α1α2β2 + 564480α1α2β1 + 1680α1α3β2 − 33600α1α3β1 − 8599500β1 + 429975β2 + 47250β3

+1764β32 − 1087800β3

1 + 323400β21β2 − 4200β2

1β3 − 35280β1β22 + 1764α2

2β2 + 17640α22β1

+29400α21β3 − 241080α2

1β2 − 1087800α21β1

CONTENTS 30

h3 = 840α1T13 + (756α2 − 29400α1)T

11 + (72380α2 − 3911320α1 − 280α3)T9 + (1751400α1β1

−3780β2α1 − 26460α2β1)T8 + (7195440α1 − 1501080α2 + 3360α3 + 84000α3

1 − 252000α1β21)T

7

+(388080α2β1 + 682080α1β1 − 58800β2α1 + 6720α1β3 − 14112α2β2)T6 + (108291960α1

−3175200α1β21 − 6650280α2 − 2234400α3

1 − 95760α3)T5 + (117600α3β1 + 2005080α2β1

−101194800α1β1 + 1317120β2α1 + 588000α1β31 − 2520α3β2 − 123480α2β2 − 58800α1β3

+588000α31β1 + 2520α2β3)T

4 + (70560α2β1β2 − 16800α3β21 + 588000α2

1α2 − 117600α2β21

−70560α1β22 + 22108800α1β

21 + 352800α3 − 13009500α2 + 705600α1β1β2 − 16800α2

1α3

−6115200α31 − 100459800α1)T

3 + (5040α3β2 + 4339440α2β1 + 740880α2β2 − 1176000α1β31

+35632800α1β1 + 4304160β2α1 − 100800α3β1 − 1176000α31β1 − 5040α2β3 − 252000α1β3)T

2

+(134400α1β1β3 − 6720α1β2β3 + 3307500α2 − 6720α1α2α3 + 117600α21α3 − 3622080α1β1β2

−264600α3 + 23520α1α22 − 211680α2β1β2 + 11054400α1β

21 + 235200α1β

22 + 11054400α3

1

+7056α2β22 + 106104600α1 − 117600α2β

21 + 7056α3

2 − 16800α3β21 − 3739680α2

1α2)T

+588000α21α2β1 + 39200α2β

31 − 313600α3

1β2 − 25200α1β3 − 1724800α31β1 − 1724800α1β

31

−7560α2β3 − 1261260β2α1 + 114660α2β1 − 13318200α1β1 + 50400α3β1 − 52920α2β2

−16800α21α3β1 − 16800α1β

21β3 + 235200α1β

21β2 + 7560α3β2 + 5600α3β

31 + 5600α3

1β3

h2 = 9T 17 + 120T 15 − 180β1T14 − 18900T 13 + (273β2 − 19320β1)T

12 + (3780α21 + 3780β2

1

−4014360)T 11 + (396β3 + 2693124β1 − 66150β2)T10 + (371700α2

1 + 15960β2β1 − 5880α2α1

−30778650− 611100β21)T

9 + (38727360β1 + 44100β31 + 2340β3 − 143325β2 − 132300α2

1β1)T8

+(181440β2β1 − 18650520β21 + 504α2

2 + 60480α2α1 + 208542600− 7572600α21 − 2880β1β3

+6552β22)T

7 + (1940400α21β1 − 98280β3 − 338608620β1 + 4292400β3

1 − 70560α21β2 − 70560β2

1β2

+15267420β2)T6 + (352800α4

1 + 17640β22 + 1008β2β3 + 175641480β2

1 − 338688α2α1

−37802520α21 + 1008α2α3 − 88200α2

2 − 446512500− 7994448β2β1 − 10080α1α3 − 352800β41

−60480β1β3)T5 + (12600α2

1β3 + 12600β21β3 + 1587600β2

1β2 − 529200α21β2 + 2116800α1α2β1

+340540200β1 − 52920α1α2β2 + 7673400α21β1 + 47528775β2 − 1474200β3 + 17640α2

2β1

−35280β1β22 − 34662600β3

1)T4 + (880320β1β3 + 2352000β4

1 + 426720α1α3 − 18533760α2α1

−117600β31β2 − 352800α1α2β

21 + 240α2

3 − 35468160β2β1 + 440559000 + 558600β22 − 47040β2β3

−47040α2α3 + 1511160α22 + 117600α3

1α2 + 352800α21β1β2 − 32190060β2

1 + 240β23 − 2352000α4

1

+32901540α21)T

3 + (510300β3 − 126000β21β3 + 8608320β2

1β2 + 10080β1β2β3 − 25200α21β3

+6138720α1α2β1 + 2469600α21β2 + 10080α2α3β1 − 595350β2 − 458640α2

2β1 − 165507300β1

−100800α1α3β1 + 317520α1α2β2 − 11466000β31 − 11466000α2

1β1 − 141120β1β22 − 10584β3

2

−10584α22β2)T

2 + (35280β21β

22 − 221760β1β3 − 511560β2

2 + 1764000β41 − 1058400α3

1α2

−120558375− 720β23 + 35280α2

1β22 + 35280β2β3 + 1464120β2β1 + 784980α2

1 − 1058400α21β1β2

+35280α21α

22 + 35280α2α3 + 784980β2

1 + 1464120α2α1 − 1058400α1α2β21 + 1764000α4

1 − 720α23

−221760α1α3 + 35280α22β

21 − 511560α2

2 − 1058400β31β2 + 3528000α2

1β21)T − 10080α1α2β3

−102312α1α2β2 + 235200α1α2β1 − 1008α2α3β2 − 6720α2α3β1 + 10080α1α3β2 + 23520α1α3β1

−6720β1β2β3 + 117600α1α2β31 + 117600α3

1α2β1 + 14288400β1 + 4465125β2 + 170100β3 + 17640β32

+1855140β31 + 552720β2

1β2 + 36120β21β3 − 30576β1β

22 + 240β1β

23 − 504β2

2β3 + 240α23β1

+504α22β3 + 17640α2

2β2 + 71736α22β1 + 12600α2

1β3 + 317520α21β2 + 1855140α2

1β1

−58800α41β2 + 58800β4

1β2

CONTENTS 31

h1 = 144α1T15 + (294α2 − 7980α1)T

13 + (202272α1 + 168α3 − 16212α2)T11 + (924α2β1 + 6468β2α1

−293160α1β1)T10 + (37800α1β

21 + 4200α3 − 878430α2 + 6058500α1 − 12600α3

1)T9 + (1800α1β3

+1512α2β2 + 349020α2β1 − 7721280α1β1 − 2520α3β1 − 175140β2α1)T8 + (3074400α1β

21

−50400α2β21 − 50400α2

1α2 + 3535560α2 + 1663200α31 − 146160α3 − 153372240α1)T

7

+(6720α3β1 − 352800α1β31 − 352800α3

1β1 − 1176α3β2 − 28560α1β3 + 1176α2β3 − 389256α2β1

+161923440α1β1 − 5531904β2α1 + 45864α2β2)T6 + (2610720α1β1β2 − 42336α1α

22 − 17216640α3

1

+57616650α2 − 599760α2β21 − 21168α2β1β2 − 1617840α3 + 1296540α1 + 5040α3β

21 + 5040α2

1α3

−45440640α1β21 + 2010960α2

1α2 − 21168α1β22)T

5 + (4116000α1β31 + 12600α2β3 − 352800α1β

21β2

−12600α3β2 + 117600α2β31 + 30164400β2α1 − 352800α2

1α2β1 + 117600α31β2 − 44893800α2β1

+4116000α31β1 + 5644800α1β1 + 982800α3β1 − 582120α2β2 − 705600α1β3)T

4 + (35750400α31

+211680α2β1β2 + 261424800α1 − 7056α32 + 352800α1α

22 − 7996800α2

1α2 + 8599500α2

−22014720α1β1β2 + 117600α21α3 + 6720α3β1β2 + 141120α1β

22 + 336000α1β1β3 + 35750400α1β

21

−6720α2β1β3 − 7056α2β22 + 14017920α2β

21 − 218400α3β

21 − 415800α3)T

3 + (2469600α21α2β1

+16800α3β31 − 12230400α1β

31 − 87582600α1β1 − 1999200α2β

31 + 83160α3β2 − 12230400α3

1β1

−50400α1β21β3 − 50400α2

1α3β1 − 235200α31β2 − 6112260β2α1 − 502740α2β1 − 83160α2β3

+75600α1β3 + 16800α31β3 + 4233600α1β

21β2 − 158760α2β2)T

2 + (15677550α2 + 793800α3

+6356280α31 + 3751440α2

1α2 + 72240α21α3 − 378672α1α

22 − 13440α1α2α3 + 480α1α

23 + 56448α3

2

−1008α22α3 − 2016α2β2β3 − 40320α3β1β2 + 40320α2β1β3 − 416304α2β1β2 − 13440α1β2β3

−53760α1β1β3 + 1952160α1β1β2 + 6356280α1β21 + 37632α1β

22 + 480α1β

23 + 1799280α2β

21

+56448α2β22 + 126000α3β

21 + 1008α3β

22 + 117600α2β

41 + 1176000α5

1 − 117600α41α2 + 2352000α3

1β21

−235200α31β1β2 + 1176000α1β

41 − 235200α1β

31β2 − 14685300α1)T − 70560α1β1β

22 + 70560α2β

21β2

+3360α21α3β2 + 70560α1α

22β1 + 7056α1α

22β2 − 70560α2

1α2β2 − 3360α21α2β3 − 7056α2β1β

22

+3360α3β21β2 − 3360α2β

21β3 + 113400α1β3 + 7560α2β3 + 2407860β2α1 − 2407860α2β1

−113400α3β1 − 16800α21α3β1 + 16800α1β

21β3 + 282240α1β

21β2 − 7560α3β2 − 16800α3β

31

+16800α31β3 + 7056α1β

32 − 7056α3

2β1 − 282240α21α2β1 − 282240α2β

31 + 282240α3

1β2

CONTENTS 32

h0 = T 19 + 93T 17 − 51β1T16 − 8604T 15 − (1020β1 + 63β2)T

14 + (630β21 + 630α2

1 − 701820)T 13

+(244216β1 − 26β3 − 119β2)T12 + (2016α2α1 + 1008β2β1 − 4810050− 49980α2

1 − 13020β21)T

11

+(4620α21β1 − 1540β3

1 − 765996β1 + 330897β2 − 1284β3)T10 + (966α2

2 + 560α1α3 + 640β1β3

+1888810β21 + 798β2

2 − 122360β2β1 + 167090α21 − 54290250− 75880α2α1)T

9 + (99645210β1

−476700β31 + 11340β2

1β2 − 2529135β2 + 48690β3 + 11340α21β2 − 182700α2

1β1)T8 + (672α2α3

−5030760α21 − 48337800β2

1 − 33600α41 + 603911700 + 1178688α2α1 − 65352α2

2 − 9600β1β3

+672β2β3 − 13440α1α3 + 2778048β2β1 − 73416β22 + 33600β4

1)T7 + (14112β1β

22 − 658560β2

1β2

+28874475β2 − 22680β3 − 840α21β3 + 3528α1α2β2 + 10584α2

2β1 + 188160α21β2 − 840β2

1β3

−846720α1α2β1 + 2075640α21β1 + 9131640β3

1 − 699258420β1)T6 + (35280β3

1β2 + 501321450β21

+228554130α21 + 394800α1α3 + 1006068β2

2 + 120β23 − 105840α2

1β1β2 − 35280α31α2 − 22512α2α3

−588000β41 − 22512β2β3 − 35995008α2α1 − 44074128β2β1 + 105840α1α2β

21 + 329280β1β3

+588000α41 + 120α2

3 + 868476α22 + 1294290900)T 5 + (1764α2

2β2 + 18892440β21β2 + 147000α2

1β3

+614250β3 − 382200α22β1 − 5074440α2

1β2 − 204418200β31 + 1764β3

2 − 162082200α21β1

+23966880α1α2β1 − 907578000β1 + 3360α2α3β1 − 34894125β2 + 3360β1β2β3 − 268800α1α3β1

−5040α1α2β3 + 5040α1α3β2 − 52920α1α2β2 − 435120β1β22 − 121800β2

1β3)T4 + (6162240α2α1

+770280α22 + 96229140α2

1 − 278880α1α3 − 47040α2α3 + 240α23 + 12387200α4

1 − 1136800α31α2

−11200α31α3 + 35280α2

1α22 + 33600α1α3β

21 − 5056800α1α2β

21 − 33600α2

1β1β3 + 823200α21β1β2

+35280β21β

22 + 11200β3

1β3 − 3096800β31β2 + 16922640β2β1 + 241080β2

2 + 248815140β21 + 240β2

3

−26880β1β3 − 47040β2β3 + 44060800β41 + 56448000α2

1β21 + 35280α2

1β22 + 35280α2

2β21 + 93767625)T 3

+(31355100β1 − 19547325β2 − 1304100β3 − 38808β32 − 4704000β5

1 − 22493940β31 − 9408000α2

1β31

−4704000α41β1 − 6938400β2

1β2 − 187320β21β3 + 630336β1β

22 − 240β1β

23 + 504β2

2β3 − 240α23β1

−504α22β3 − 38808α2

2β2 − 318696α22β1 − 264600α2

1β3 − 3104640α21β2 − 22493940α2

1β1 − 176400α41β2

+176400β41β2 + 20160α1α2β3 + 949032α1α2β2 − 3833760α1α2β1 + 1008α2α3β2 + 16800α2α3β1

−20160α1α3β2 + 77280α1α3β1 + 16800β1β2β3 + 352800α1α2β31 + 352800α3

1α2β1)T2 + (504000α1α3

+35280α2α3 − 1800α23 − 1999200α4

1 + 1540560α31α2 + 67200α3

1α3 − 199920α21α

22 − 3360α2

1α2α3

+7056α1α32 − 3360α2α3β

21 + 7056α2

2β1β2 + 67200α1α3β21 + 7056α1α2β

22 + 1540560α1α2β

21

−3360α21β2β3 + 67200α2

1β1β3 + 1540560α21β1β2 + 7056β1β

32 − 199920β2

1β22 + 67200β3

1β3

+1540560β31β2 + 7285320β2β1 − 22050β2

2 − 31509450β21 − 1800β2

3 + 504000β1β3 + 35280β2β3

−1999200β41 − 3360β2

1β2β3 − 564480α1α2β1β2 − 3998400α21β

21 + 82320α2

1β22 + 82320α2

2β21 + 196000α6

1

+588000α41β

21 + 588000α2

1β41 + 196000β6

1 − 66976875 + 7285320α2α1 − 22050α22 − 31509450α2

1)T

+5040α1α2β3 + 3528α1α2β2 − 1317120α1α2β1 − 1008α2α3β2 − 6720α2α3β1 − 5040α1α3β2

−77280α1α3β1 − 6720β1β2β3 + 35280α21α

22β1 − 23520α3

1α2β2 − 11200α31α3β1 − 35280α2

1β1β22

−196000α1α2β31 − 196000α3

1α2β1 − 11200α1α3β31 + 8831025β1 + 1091475β2 + 28350β3 + 1764β3

2

+313600β51 + 3442740β3

1 + 627200α21β

31 + 313600α4

1β1 − 373380β21β2 − 39480β2

1β3 − 30576β1β22

+240β1β23 − 504β2

2β3 + 240α23β1 + 504α2

2β3 + 1764α22β2 − 34104α2

2β1 + 37800α21β3 + 943740α2

1β2

+3442740α21β1 + 98000α4

1β2 − 11760α22β

31 − 5600β4

1β3 + 5600α41β3 − 98000β4

1β2 + 11760β31β

22

+70560α1α2β21β2

q15 = −60α1T2 + 21α2 − 270α1

q14 = −540T 4 + 300β1T3 + 360T 2 + (600β1 − 105β2)T + 4020 + 225α2

1 + 225β21

q13 = (1050α1 − 105α2)T2 − 1540α1 + 245α2 − 10α3

q12 = −1260T 6 + 840β1T5 + 1260T 4 + (2100β1 − 315β2)T

3 + (28140 + 1575α21 + 1575β2

1)T2 + (70β3

−875β2 + 2380β1)T + 106260 + 5775α21 − 420α2α1 − 840β2β1 + 9975β2

1

q11 = 420α1T6 + (13650α1 − 735α2+)T 4 + (180α3 + 34020α1 − 1890α2)T

2 + (21000α1β1 − 420β2α1

−2940α2β1)T + 131670α1 − 700α31 − 12915α2 + 180α3 + 2100α1β

21

q10 = −1260T 8 + 1260β1T7 + 15120T 6 − (21β2 + 1680β1)T

5 + (4725α21 + 178920 + 4725β2

1)T4

−(53060β1 + 4130β2 + 140β3)T3 + (750960 + 2940β2β1 + 51450α2

1 + 55650β21 − 7980α2α1)T

2

+(58275β2 + 18900α21β1 − 269640β1 − 1260β3 − 6300β3

1)T + 411012− 27160α2α1 + 399α22

+245105α21 + 320α1α3 − 16520β2β1 + 77245β2

1 + 280β1β3 + 483β22

CONTENTS 33

q9 = 1050α1T8 + (43050α1 − 1365α2)T

6 + (250α3 − 35525α2 + 112000α1)T4 + (14700β2α1 + 273000α1β1

+2100α2β1)T3 + (24500α3

1 − 1500α3 − 62475α2 + 2084250α1 − 73500α1β21)T

2 + (1000α3β1

−1400α1β3 + 840α2β2 + 84700β2α1 + 700α2β1 − 901600α1β1)T − 645750α1 − 121275α2

+2250α3 + 87500α31 − 6300α2

1α2 − 6300α2β21 − 10500α1β

21

q8 = 1050β1T9 + 63000T 8 + (945β2 − 18900β1)T

7 + (7875β21 + 7875α2

1 − 142800)T 6 + (602280β1

−630β3 + 60795β2)T5 + (5846400− 6300β2β1 + 590625α2

1 − 291375β21)T

4 + (688275β2

−157500α21β1 + 52500β3

1 − 6300β3 − 5978700β1)T3 + (1283625β2

1 − 1800α1α3 − 18900α2α1

−233100β2β1 + 315β22 + 4095α2

2 + 1478925α21 − 7867440)T 2 + (94500β3

1 − 15750β3 + 527625β2

+31500β21β2 + 31500α2

1β2 + 3102750β1 − 787500α21β1)T + 4358760− 274680α2α1 + 18795α2

2

−32025α21 + 2400α1α3 − 420α2α3 + 21000α4

1 − 21000β41 − 420β2β3 − 186480β2β1 + 15015β2

2

+339675β21 + 4200β1β3

q7 = 1260α1T10 + (66150α1 − 945α2)T

8 + (2930760α1 − 6300α2 − 360α3+)T 6 + (22680β2α1

−1335600α1β1 − 2520α2β1)T5 + (9708300α1 + 853650α2 + 189000α1β

21 − 19800α3 − 63000α3

1)T4

+(327600β2α1 + 4800α3β1 − 10080α2β2 − 394800α2β1 − 10096800α1β1)T3 + (50400α2β

21 − 55800α3

+50400α21α2 + 875700α2 − 11333700α1 − 126000α3

1 + 2898000α1β21)T

2 + (26400α3β1 + 3150000α1β1

−252000α31β1 − 252000α1β

31 + 1021440β2α1 + 840α2β3 − 328440α2β1 − 8400α1β3 − 37800α2β2

−840α3β2)T + 330750α1 − 23625α2 + 27000α3 − 218400α31 − 130200α2

1α2 − 600α21α3 + 10080α1α

22

+2520α2β1β2 − 176400α1β1β2 − 218400α1β21 + 7560α1β

22 + 46200α2β

21 − 600α3β

21

q6 = 1260T 12 + 420β1T11 + 118440T 10 + (1365β2 − 37800β1+)T 9 + (7875α2

1 + 4109700 + 7875β21)T

8

+(114660β2 − 360β3 − 2890440β1)T7 + (17640α2α1 + 661500β2

1 − 220500α21 + 40681200

−17640β2β1)T6 + (88200α2

1β1 + 930510β2 − 29400β31 − 49674240β1 + 17640β3)T

5 + (352800α2α1

+21307650β21 − 235200β2β1 + 13230β2

2 − 3417750α21 − 4410α2

2 − 8400β1β3 − 38698380)T 4

+(882000α21β1 − 3822000β3

1 + 88200β3 + 26151300β1 + 2866500β2)T3 + (214620β2

2 + 55860α22

+294000β41 − 2628360β2β1 − 67200β1β3 − 294000α4

1 − 840β2β3 − 2454900α21 − 867300β2

1

+282240α2α1 + 32744040− 840α2α3 + 8400α1α3)T2 + (294000α2

1β2 − 17640α22β1 + 4200α2

1β3

+294000β21β2 + 4200β2

1β3 − 35280β1β22 + 205800β3

1 − 189000β3 + 205800α21β1 − 826875β2

−1323000β1 − 17640α1α2β2)T + 32744220− 736960α2α1 + 112210α22 + 4418575α2

1 + 60200α1α3

−7840α2α3 + 100α23 − 29400α3

1α2 + 88200α1α2β21 − 88200α2

1β1β2 + 29400β31β2 − 560560β2β1

+138670β22 + 4462675β2

1 + 100β23 + 47600β1β3 − 7840β2β3

q5 = 840α1T12 + (55230α1 + 21α2)T

10 + (1225980α1 + 91035α2 − 630α3)T8 + (2520β2α1 − 327600α1β1

−22680α2β1)T7 + (88200α1β

21 − 18513180α1 − 2520α3 + 2403450α2 − 29400α3

1)T6 + (21168α2β2

−1040760α2β1 + 17640β2α1 + 18557280α1β1 − 5040α1β3 − 5040α3β1)T5 + (88200α2

1α2 + 9856350α2

+3439800α1 + 88200α2β21 − 3087000α3

1 + 56700α3 − 4851000α1β21)T

4 + (1940400α1β1 − 75600α1β3

+2257920β2α1 − 7990920α2β1 + 588000α1β31 + 370440α2β2 + 2520α2β3 + 588000α3

1β1 − 100800α3β1

−2520α3β2)T3 + (1675800α2β

21 − 1940400α1β1β2 + 17640α1β

22 − 35280α1α

22 − 705600α3

1

+12600α3β21 − 567000α3 + 6350400α1β

21 + 34728750α1 − 264600α2

1α2 − 694575α2 + 12600α21α3

−52920α2β1β2)T2 + (1764000α1β1 + 2504880β2α1 + 264600α2β2 − 117600α3

1β2 + 226800α3β1

−588000α31β1 − 1005480α2β1 + 352800α1β

21β2 − 117600α2β

31 + 22680α2β3 − 22680α3β2

+352800α21α2β1 − 352800α1β3 − 588000α1β

31)T + 6232800α3

1 − 1517040α21α2 + 46200α2

1α3

+135240α1α22 − 3360α1α2α3 − 1764α3

2 + 5040α3β1β2 − 5040α2β1β3 − 17640α2β1β2 − 3360α1β2β3

+84000α1β1β3 − 1622880α1β1β2 + 6232800α1β21 + 152880α1β

22 + 105840α2β

21 − 1764α2β

22

−37800α3β21 + 49612500α1 − 99225α2 − 141750α3

CONTENTS 34

q4 = 1260T 14 + 114660T 12 + (735β2 − 35700β1)T11 + (4725β2

1 + 4960620 + 4725α21)T

10 + (250β3

−3174500β1 + 31675β2)T9 + (38448900 + 6300α2α1 − 417375α2

1 + 779625β21)T

8 + (9000β3 − 63000β31

+189000α21β1 − 24885000β1 − 513450β2)T

7 + (617400β2β1 − 8400α1α3 + 13230α22 + 175627620

+5843250β21 + 205800α2α1 − 10885350α2

1 − 4410β22)T

6 + (4851000α21β1 + 6945750β2 − 88200α2

1β2

−88200β21β2 − 94500β3 − 199773000β1 − 441000β3

1)T5 + (90515250β2

1 − 6174000α2α1 + 463050α22

−66150β22 + 126000β1β3 − 3087000β2β1 − 22325940− 126000α1α3 − 7827750α2

1)T4 + (18301500β1

+441000β3 − 18669000β31 + 3528000α1α2β1 − 2352000α2

1β2 − 88200α22β1 + 1176000β2

1β2

+16611000α21β1 − 21000α2

1β3 + 88200α1α2β2 + 18466875β2 − 21000β21β3)T

3 + (300α23 + 147000α3

1α2

−33600β1β3 + 300β23 + 1470000β4

1 − 21000α2α3 − 474600α1α3 + 1888950α22 + 962850β2

2 − 1470000α41

+71114925α21 − 16023000β2β1 − 5880000α2α1 + 36496425β2

1 + 441000α21β1β2 − 441000α1α2β

21

+312558660− 21000β2β3 − 147000β31β2)T

2 + (4086600α21β2 − 617400α2

2β1 − 105000α21β3 + 992250β3

+8400α1α3β2 + 1646400α1α2β1 − 8400α1α2β3 + 8820β32 − 529200β1β

22 + 88200α1α2β2

−18963000α21β1 + 8820α2

2β2 − 144868500β1 + 5457375β2 + 5733000β21β2 + 168000α1α3β1

+63000β21β3 − 18963000β3

1)T + 27907020 + 1837500α2α1 + 169050α22 + 53695425α2

1 − 222600α1α3

−21000α2α3 + 300α23 + 6566000α4

1 − 1225000α31α2 + 14000α3

1α3 + 44100α21α

22 − 42000α1α3β

21

+147000α1α2β21 + 42000α2

1β1β3 − 1911000α21β1β2 + 44100β2

1β22 − 14000β3

1β3 − 539000β31β2

−1029000β2β1 + 36750β22 + 20399925β2

1 + 300β23 − 159600β1β3 − 21000β2β3 + 2254000β4

1

+8820000α21β

21 + 44100α2

1β22 + 44100α2

2β21

q3 = 300α1T14 + (315α2 + 24150α1)T

12 + (69790α2 − 140α3 − 1233260α1)T10 + (1113000α1β1 − 2100β2α1

−14700α2β1)T9 + (1926225α2 − 157500α1β

21 − 6300α3 + 52500α3

1 − 52236450α1)T8 + (4800α1β3

−10080α2β2 + 44839200α1β1 − 428400α2β1 − 142800β2α1)T7 + (19624500α2 − 263497500α1

−315000α3 − 9702000α1β21 − 294000α3

1)T6 + (2520α2β3 − 4110120α2β1 + 588000α3

1β1 + 588000α1β31

+151200α3β1 + 75600α1β3 − 1975680β2α1 − 2520α3β2 + 291589200α1β1 − 476280α2β2)T5

+(226343250α1 + 116038125α2 − 147000α2β21 − 21000α3β

21 − 21000α2

1α3 − 78204000α1β21

−1827000α3 − 7644000α31 + 882000α1β1β2 + 88200α2β1β2 + 735000α2

1α2 − 88200α1β22)T

4

+(5880000α31β1 + 5880000α1β

31 − 1587600α2β2 + 49039200β2α1 + 25200α2β3 − 78145200α2β1

+1512000α3β1 − 756000α1β3 − 121716000α1β1 − 25200α3β2)T3 + (17640α3

2 + 98196000α1β21

−16800α1α2α3 + 176400α2β1β2 − 9349200α21α2 + 577489500α1 − 378000α3β

21 + 25578000α2β

21

+850500α3 + 98196000α31 + 17640α2β

22 + 672000α1β1β3 − 16800α1β2β3 + 50604750α2 + 58800α1α

22

+294000α21α3 − 34927200α1β1β2 − 117600α1β

22)T

2 + (8232000α1β21β2 + 882000α1β3 − 32144000α1β

31

−252000α3β1 + 63000α2β3 − 1323000α2β2 − 63000α3β2 − 4189500β2α1 − 84000α21α3β1

+2940000α21α2β1 + 784000α3

1β2 + 28000α3β31 − 19183500α2β1 + 28000α3

1β3 − 4508000α2β31

−84000α1β21β3 − 32144000α3

1β1 − 227115000α1β1)T + 294000α2β41 + 2940000α5

1 − 294000α41α2

+5880000α31β

21 − 588000α3

1β1β2 + 2940000α1β41 − 588000α1β

31β2 + 992250α1 − 6449625α2

−283500α3 + 21457100α31 + 1636600α2

1α2 − 144200α21α3 + 374360α1α

22 − 5600α1α2α3 − 400α1α

23

−11760α32 + 840α2

2α3 + 1680α2β2β3 + 16800α3β1β2 − 16800α2β1β3 + 346920α2β1β2 − 5600α1β2β3

−123200α1β1β3 + 842800α1β1β2 + 21457100α1β21 + 27440α1β

22 − 400α1β

23 + 793800α2β

21

−11760α2β22 − 21000α3β

21 − 840α3β

22

CONTENTS 35

q2 = 540T 16 − 60β1T15 + 56160T 14 + (105β2 − 16800β1)T

13 + (1105440 + 1575β21 + 1575α2

1)T12

+(172620β1 + 180β3 − 22050β2)T11 + (7980β2β1 − 196350β2

1 + 353850α21 − 24877440

−2940α2α1)T10 + (15300β3 − 1598625β2 + 24500β3

1 − 73500α21β1 + 58300200β1)T

9 + (9883125α21

−1800β1β3 + 315α22 + 642600β2β1 − 27910575β2

1 + 4095β22 − 138600α2α1 − 56062440)T 8 + (433800β3

+424676700β1 − 2142000α21β1 − 50400β2

1β2 + 4914000β31 − 20418300β2 − 50400α2

1β2)T7 + (294000α4

1

−73500α22 − 294000β4

1 + 5803560β2β1 − 134400β1β3 + 102561900α21 − 8400α1α3 + 2426051040

+840α2α3 + 840β2β3 − 5221440α2α1 − 242300100β21 + 191100β2

2)T6 + (12600β2

1β3 − 24078600α21β1

−35280β1β22 + 17640α2

2β1 + 12600α21β3 + 529200β2

1β2 + 1701000β3 − 14387625β2 + 46481400β31

+2822400α1α2β1 − 2293200α21β2 − 52920α1α2β2 − 2182950000β1)T

5 + (2940000α41 − 159600β1β3

+533400α1α3 + 782006925α21 − 441000α1α2β

21 − 58800α2α3 + 300α2

3 + 3795356160 + 300β23

−2940000β41 + 3344250β2

2 − 70795200α2α1 − 147000β31β2 + 1888950α2

2 + 1273942425β21

−63739200β2β1 + 441000α21β1β2 + 147000α3

1α2 − 58800β2β3)T4 + (16800β1β2β3 − 2712811500β1

−17640α22β2 + 36691200β2

1β2 − 8820000α21β2 + 5386500β3 − 940800β1β

22 − 42000β2

1β3

−168000α1α3β1 − 176400α1α2β2 − 764400α22β1 + 45511200α1α2β1 − 504798000α2

1β1 + 16800α2α3β1

−179597250β2 + 126000α21β3 − 17640β3

2 − 504798000β31)T

3 + (88200α21α

22 + 35412300α2α1

+644940450β21 + 88200α2

2β21 + 98916300β2β1 − 554400α1α3 + 88200β2β3 + 110250000β4

1

−2646000α31α2 − 2822400β1β3 − 1278900α2

2 − 1800β23 + 882000α2

1β1β2 + 88200α21β

22 − 480000960

+3483900β22 + 39690000α4

1 + 88200β21β

22 + 160722450α2

1 + 149940000α21β

21 + 88200α2α3

−6174000β31β2 − 1800α2

3 − 9702000α1α2β21)T

2 + (67473000β1 + 42170625β2 + 1984500β3 + 88200β32

−11760000β51 − 71868300β3

1 − 23520000α21β

31 − 11760000α4

1β1 − 9584400β21β2 + 684600β2

1β3

−2269680β1β22 + 1200β1β

23 − 2520β2

2β3 + 1200α23β1 + 2520α2

2β3 + 88200α22β2 + 358680α2

2β1

+567000α21β3 − 176400α2

1β2 − 71868300α21β1 − 294000α4

1β2 + 294000β41β2 − 50400α1α2β3

−2628360α1α2β2 − 9408000α1α2β1 − 5040α2α3β2 − 33600α2α3β1 + 50400α1α3β2 + 117600α1α3β1

−33600β1β2β3 + 588000α1α2β31 + 588000α3

1α2β1)T + 55814060− 9261000α2α1 + 165375α22

+16570575α21 − 491400α1α3 + 2700α2

3 + 4704000α41 − 617400α3

1α2 − 84000α31α3 + 323400α2

1α22

+8400α21α2α3 − 17640α1α

32 + 8400α2α3β

21 − 17640α2

2β1β2 − 84000α1α3β21 − 17640α1α2β

22

−617400α1α2β21 + 8400α2

1β2β3 − 84000α21β1β3 − 617400α2

1β1β2 − 17640β1β32 + 323400β2

1β22

−84000β31β3 − 617400β3

1β2 − 9261000β2β1 + 165375β22 + 16570575β2

1 + 2700β23 − 491400β1β3

+4704000β41 + 8400β2

1β2β3 + 705600α1α2β1β2 + 9408000α21β

21 − 29400α2

1β22 − 29400α2

2β21

+490000α61 + 1470000α4

1β21 + 1470000α2

1β41 + 490000β6

1

CONTENTS 36

q1 = 45α1T16 + (105α2 + 4350α1)T

14 + (5845α2 + 70α3 + 395080α1)T12 + (420α2β1 + 2940β2α1

−147000α1β1)T11 + (22308930α1 + 8820α3 + 18900α1β

21 − 6300α3

1 − 326655α2)T10 + (1000α1β3

+840α2β2 + 118300β2α1 + 252700α2β1 − 1400α3β1 − 15097600α1β1)T9 + (264275550α1 − 31500α2β

21

+3433500α1β21 + 261450α3 − 31500α2

1α2 + 787500α31 − 13426875α2)T

8 + (840α2β3 − 252000α31β1

−67200α3β1 + 83160α2β2 − 2808960β2α1 − 189428400α1β1 + 27600α1β3 + 9600360α2β1 − 840α3β2

−252000α1β31)T

7 + (2646000α1β1β2 + 6820800α31 + 735000α2

1α2 + 42100800α1β21 − 17640α1β

22

−1911000α2β21 + 4200α2

1α3 + 812873250α1 + 4200α3β21 + 1071000α3 − 20782125α2 − 17640α2β1β2

−35280α1α22)T

6 + (687960α2β2 − 2940000α31β1 − 370440α2β1 + 24801840β2α1 + 117600α2β

31

−352800α21α2β1 − 806400α1β3 + 478800α3β1 + 117600α3

1β2 − 352800α1β21β2 − 2940000α1β

31

−42840α3β2 − 838958400α1β1 + 42840α2β3)T5 + (10466400α2β

21 + 282240000α1β

21 − 8820α3

2

+588000α1β1β3 + 70560000α31 − 264600α1α

22 − 146853000α1 − 176400α1β

22 + 10064250α3

−85829625α2 − 1764000α21α2 − 12230400α1β1β2 − 8820α2β

22 + 315000α2

1α3 − 88200α2β1β2

−8400α2β1β3 + 8400α3β1β2 − 273000α3β21)T

4 + (28665000α1β1 + 37800α3β2 − 84000α21α3β1

−43904000α1β31 + 64077300α2β1 + 9261000α2β2 + 2142000α1β3 − 43904000α3

1β1 − 22182300β2α1

−84000α1β21β3 + 28000α3

1β3 + 28000α3β31 − 6552000α3β1 + 1960000α3

1β2 + 4704000α1β21β2

−3332000α2β31 − 588000α2

1α2β1 − 37800α2β3)T3 + (94759875α2 + 6520500α3 − 40557300α3

1

+16434600α21α2 + 1188600α2

1α3 − 5180280α1α22 − 33600α1α2α3 + 1200α1α

23 + 141120α3

2 − 2520α22α3

−5040α2β2β3 − 100800α3β1β2 + 100800α2β1β3 − 7391160α2β1β2 − 33600α1β2β3 − 1142400α1β1β3

+18992400α1β1β2 − 40557300α1β21 + 2210880α1β

22 + 1200α1β

23 − 2557800α2β

21 + 141120α2β

22

+2331000α3β21 + 2520α3β

22 + 294000α2β

41 + 2940000α5

1 − 294000α41α2 + 5880000α3

1β21 − 588000α3

1β1β2

+2940000α1β41 − 588000α1β

31β2 − 290729250α1)T

2 + (3763200α31β2 − 3763200α2β

31 − 3763200α2

1α2β1

+3763200α1β21β2 − 420000α3β

31 + 420000α3

1β3 − 35280α32β1 + 35280α1β

32 + 39028500β2α1

−39028500α2β1 − 189000α3β2 + 189000α2β3 − 420000α21α3β1 + 420000α1β

21β3 − 1764000α1β1β

22

+1764000α2β21β2 − 16800α2β

21β3 − 35280α2β1β

22 + 16800α3β

21β2 − 1764000α2

1α2β2 − 16800α21α2β3

+16800α21α3β2 + 1764000α1α

22β1 + 35280α1α

22β2 + 2079000α1β3 − 2079000α3β1)T + 1568000α5

1

−490000α41α2 − 28000α4

1α3 + 58800α31α

22 + 3136000α3

1β21 − 980000α3

1β1β2 − 56000α31β1β3

−58800α31β

22 + 352800α2

1α2β1β2 − 176400α1α22β

21 + 1568000α1β

41 − 980000α1β

31β2 − 56000α1β

31β3

+176400α1β21β

22 + 28000α3β

41 + 44155125α1 + 5457375α2 + 141750α3 + 17213700α3

1 − 1866900α21α2

−197400α21α3 − 152880α1α

22 − 33600α1α2α3 + 1200α1α

23 + 8820α3

2 − 2520α22α3 − 5040α2β2β3

+25200α3β1β2 − 25200α2β1β3 + 17640α2β1β2 − 33600α1β2β3 − 386400α1β1β3 − 6585600α1β1β2

+17213700α1β21 − 170520α1β

22 + 1200α1β

23 + 4718700α2β

21 + 8820α2β

22 + 189000α3β

21 + 2520α3β

22

+490000α2β41 − 117600α2β

31β2

CONTENTS 37

q0 = 90T 18 − 15β1T17 + 11070T 16 − (3180β1 + 21β2)T

15 + (225β21 + 225α2

1 + 552120)T 14 − (3115β2

+10β3 + 201040β1)T13 + (15698760 + 840α2α1 + 22575β2

1 + 420β2β1 + 5775α21)T

12 + (2100α21β1

−700β31 − 32025β2 − 1740β3 − 9056460β1)T

11 + (24745α21 + 2439605β2

1 + 35980α2α1 + 483α22

+173346012 + 280α1α3 − 29260β2β1 + 399β22 + 320β1β3)T

10 + (6300α21β2 + 52500α2

1β1 + 6300β21β2

−833175β2 − 353500β31 − 58950β3 − 112130550β1)T

9 + (1912680α2α1 + 420β2β3 − 6542025α21

+1660282260 + 420α2α3 + 400680β2β1 − 21000α41 − 26985β2

2 + 33696075β21 − 15645α2

2 + 21000β41

+8400α1α3 + 13800β1β3)T8 + (487200α2

1β2 + 7560α22β1 + 53006625β2 − 218400β2

1β2 − 2315470500β1

+2520α1α2β2 − 600α21β3 − 5069400β3

1 − 600β21β3 − 1233000β3 + 1986600α2

1β1 − 705600α1α2β1

+10080β1β22)T

7 + (325113775α21 − 5320α2α3 + 665000α1α3 − 294000α4

1 + 1567190275β21

−24162880α2α1 + 394450α22 − 29400α3

1α2 + 100α23 + 294000β4

1 + 728000β1β3 − 5320β2β3 + 262150β22

+1086513720 + 88200α1α2β21 + 29400β3

1β2 + 100β23 − 88200α2

1β1β2 − 61824280β2β1)T6 + (1764β3

2

−570271800β31 − 336000α1α3β1 + 5040α1α3β2 + 24890040β2

1β2 + 2099160α21β2 − 358591800α2

1β1

−5040α1α2β3 + 3360α2α3β1 − 223440β1β22 + 17640α1α2β2 + 163800α2

1β3 + 1764α22β2

+22790880α1α2β1 − 241080α22β1 + 3360β1β2β3 − 691929000β1 + 120822975β2 − 172200β2

1β3

−7701750β3)T5 + (7078050α2

2 + 510159825α21 + 5430600α1α3 − 226800α2α3 + 2700α2

3 + 27244000α41

−2597000α31α2 − 14000α3

1α3 + 44100α21α

22 + 42000α1α3β

21 − 6321000α1α2β

21 − 42000α2

1β1β3

−735000α21β1β2 + 44100β2

1β22 + 14000β3

1β3 − 4459000β31β2 − 172298700β2β1 + 1389150β2

2

+613133325β21 + 2700β2

3 + 8139600β1β3 − 226800β2β3 + 113876000β41 + 141120000α2

1β21 + 44100α2

1β22

+44100α22β

21 + 3510704520− 141649200α2α1)T

4 + (1984500β3 − 64680β32 − 11760000β5

1

−318553900β31 − 23520000α2

1β31 − 11760000α4

1β1 + 68756800β21β2 − 3168200β2

1β3 + 815360β1β22

−400β1β23 + 840β2

2β3 − 400α23β1 − 840α2

2β3 − 64680α22β2 − 4294360α2

2β1 + 735000α21β3

−27165600α21β2 − 318553900α2

1β1 − 294000α41β2 + 294000β4

1β2 − 33600α1α2β3 + 5109720α1α2β2

+95922400α1α2β1 + 1680α2α3β2 + 95200α2α3β1 + 33600α1α3β2 − 3903200α1α3β1 + 95200β1β2β3

+588000α1α2β31 + 588000α3

1α2β1 − 2343694500β1 − 118573875β2)T3 + (9040500α2α1 + 2767275α2

2

+323594775α21 − 352800α1α3 − 315000α2α3 + 9900α2

3 + 30282000α41 − 3204600α3

1α2 + 168000α31α3

−499800α21α

22 − 8400α2

1α2α3 + 17640α1α32 − 8400α2α3β

21 + 17640α2

2β1β2 + 840000α1α3β21

+17640α1α2β22 − 19668600α1α2β

21 − 8400α2

1β2β3 − 168000α21β1β3 + 5027400α2

1β1β2 + 17640β1β32

−499800β21β

22 + 504000β3

1β3 − 11436600β31β2 + 43438500β2β1 + 4354875β2

2 + 723140775β21 + 9900β2

3

−1108800β1β3 − 315000β2β3 + 82026000β41 − 8400β2

1β2β3 − 2822400α1α2β1β2 + 112308000α21β

21

+911400α21β

22 + 911400α2

2β21 + 490000α6

1 + 1470000α41β

21 + 1470000α2

1β41 + 490000β6

1 + 725582810)T 2

+(88200β21β3 − 599760β1β

22 − 3600β1β

23 + 7560β2

2β3 − 3600α23β1 − 7560α2

2β3 − 132300α22β2

−546840α22β1 − 63000α2

1β3 − 7629300α21β2 − 102797100α2

1β1 − 686000α41β2 − 58800α2

2β31 − 28000β4

1β3

+28000α41β3 + 686000β4

1β2 + 58800β31β

22 + 352800α1α2β

21β2 + 25200α1α2β3 − 52920α1α2β2

−3528000α1α2β1 + 15120α2α3β2 + 100800α2α3β1 − 25200α1α3β2 + 151200α1α3β1 + 100800β1β2β3

+176400α21α

22β1 − 117600α3

1α2β2 − 56000α31α3β1 − 176400α2

1β1β22 + 1372000α1α2β

31 + 1372000α3

1α2β1

−56000α1α3β31 − 190015875β1 − 32248125β2 − 992250β3 − 132300β3

2 − 10192000β51 − 102797100β3

1

−20384000α21β

31 − 10192000α4

1β1 − 11157300β21β2)T + 490000β6

1 + 5581406 + 3528000α2α1

+474075α22 + 14564025α2

1 + 88200α1α3 + 31500α2α3 + 900α23 + 5517400α4

1 + 891800α31α2 − 2800α3

1α3

+98980α21α

22 − 2800α2

1α2α3 + 400α21α

23 + 5880α1α

32 − 1680α1α

22α3 + 1764α4

2 − 2800α2α3β21

+1680α22β1β3 + 5880α2

2β1β2 + 1680α1α3β22 − 2800α1α3β

21 + 5880α1α2β

22

+891800α1α2β21 − 2800α2

1β2β3 − 2800α21β1β3 + 891800α2

1β1β2 + 5880β1β32 + 400β2

1β23 + 98980β2

1β22

−2800β31β3 + 891800β3

1β2 + 3528000β2β1 + 474075β22 + 14564025β2

1 + 900β23 + 88200β1β3 + 31500β2β3

+5517400β41 + 1764β4

2 − 2800β21β2β3 − 1680β1β

22β3 + 23520α1α2β1β2 + 11034800α2

1β21 + 87220α2

1β22

+400α21β

23 + 87220α2

2β21 + 3528α2

2β22 + 400α2

3β21 − 3360α1α2β2β3 − 3360α2α3β1β2 + 490000α6

1

+1470000α41β

21 + 1470000α2

1β41 .

References

[1] Dubard P, Gaillard P, Klein C and Matveev VB 2010 Eur. Phys. J. ST 185 pp 247–258

[2] Dubard P and Matveev VB 2011 Nat. Haz. Earth Syst. Sci. 11 pp 667–672

CONTENTS 38

[3] Matveev VB and Dubard P 2010 Proceedings of the international conference FNP-2010 Novgorod-

St-Petersburg (ed Institut of Applied physics (IAP) of Russian Academy of sciences) pp 100–101

[4] Dubard P 2010 Ph.D thesis Multirogue solutions to the focusing NLS equation tel-00625446

[5] Eleonski VZ, Krichever I and Kulagin N 1986 Sov. Dokl. sect. Math.Phys. 287 pp 606–610

[6] Matveev VB 1979 Lett. Math. Phys. 3 pp 213–216

[7] Matveev VB 2000 Darboux Transformations, Covariance Theorems and Integrable Systems

L.D. Faddeev’s Seminar on Mathematical Physics (American Mathematics Society Translations

vol 201) ed MA Semenov-Tyan-Shanskij pp 179–209

[8] Matveev VB, Salle MA and Rybin AV 1988 Inverse Problems 4 pp 175–183

[9] Peregrine DH 1983 J. Austral. Math. Soc. B 25 pp 16–43

[10] Its AR, Rybin AV and Salle MA 1988 Theor. Math. Phys. 74 pp 29–45

[11] Akhmediev N, Eleonski VZ and Kulagin N 1985 Sov. Phys.-JETP 62 pp 894–899

[12] Bordag L, Its AR, Manakov SV, Matveev VB and Zakharov VE 1977 Phys. Lett. A 63 pp 205–206

[13] Salle MA 1982 Teor. Mat. Fiz. 53 pp 227-237

[14] Matveev VB and Salle MA 1991 Darboux transformations and solitons (Spinger series in Nonlinear

Dynamics,Springer Verlag)

[15] Belokolos ED, Bobenko AI, Its AR, Eleonski VZ and Matveev VB 1994 Algebro-geometric Approach

to Nonlinear Integrable Equations (Springer series in nonlinear dynamics, Springer Verlag)

[16] Guo B, Ling L and Liu QP 2012 Phys. Rev. E 85 pp 1–9

[17] Gaillard P 2011 J. Phys. A: Math. Gen. 44 pp 1–15

[18] Gaillard P 2012 Jour. Of Math. : Adv. And Appl., Sci. Adv. 13 pp 71–153

[19] Gaillard P 2013 J. Math. Phys. 54 pp 013504–013537

[20] Gaillard P 2013 hal-00783882

[21] Akhmediev N, Ankiewicz A and Soto-Crespo JM 2009 Phys. Rev. E 80 026601

[22] Akhmediev N, Ankiewicz A and Clarkson PA 2010 J. Phys. A: Math. Gen. 43 pp 1–9

[23] Ohta Y and Yang J 2012 Proc.R.Soc. A 468 pp 1716–1740

[24] Ankiewicz A, Kedziora D and Akhmediev N 2011 Phys. Lett. A 375 pp 2782–2785

[25] Kedziora D, Ankiewicz A and Akhmediev N 2011 Phys. Rev. E 84 056611

[26] He JS, Zhang HR, Wang LH Porsezian K and Fokas AS 2012 arXiv: 1209.3742v3 [nlin.SI]

[27] Klein C, Matveev VB and Smirnov AO 2007 Theor. Math. Phys. 152 pp 1132–1145

[28] Khusnutdinova KR, Klein CF, Matveev VB and Smirnov AO 2013 Chaos 23 031126