The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation.

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The fifth order Peregrine breather and its

eight-parameters deformations solutions of the NLS equation.

Pierre Gaillard

To cite this version:

Pierre Gaillard. The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation.. 2013. �hal-00819359�

The fifth order Peregrine breather and its eight-parameters deformations

solutions of the NLS equation.

+Pierre Gaillard, ⁺ Universit´e de Bourgogne, Dijon, France : e-mail: Pierre.Gaillard@u-bourgogne.fr,

April 28, 2013

Abstract

We construct here new deformations of the Peregrine breather of order 5 with 8 real parameters. This gives new families of quasi- rational solutions of the NLS equation and thus one can describe in a more precise way the phenomena of appearance of multi rogue waves.

With this method, we construct new patterns of different types of rogue waves. We get at the same time, the triangular configurations as well as rings isolated. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings.

1 Introduction

One can use different approaches to modeling the evolution of deep wa- ter waves. In this study, we use of the nonlinear Schr¨odinger equation (NLS) [25, 30]. The story of the nonlinear Scr¨odinger equation has begun in 1972 where Zakharov and Shabat first solved it using the inverse scattering method. Its and Kotlyarov constructed in 1976 periodic and almost periodic solutions of the focusing NLS equation [19]. It is in 1979 that Ma found the first breather-type solution of the NLS equation [22]. Then Peregrine con- structed in 1983 the first quasi-rational solutions of NLS equation. Eleonski, Akhmediev and Kulagin obtained the first higher order analogue of the Pere- grine breather[3] in 1986. Akhmediev et al. [1, 2] , constructed other families

The notion of rogue waves first appeared in studies of ocean waves. Then it moved to other domains of physics; in nonlinear optics [27], Bose-Einstein condensate [7], atmosphere [28] and even finance [29]....

A lot of experiments about solutions of NLS equation have been realized these last years. In particular, the basic Peregrine soliton has been studied very recently in [9, 20]; furthermore, the NLS equation accurately describes physical rogue waves of relatively high order according to the work [8].

Actually, there is growing interest in studying higher order rational solu- tions. In 2010, rational solutions of the NLS equation have been written as a quotient of two Wronskians in [10]. In 2011, an other representation of the solutions of the NLS equation has been constructed in [13], also in terms of a ratio of two Wronskians determinants of order 2N.

In 2012, an other representation of the solutions of the focusing NLS equa- tion, as a ratio of two determinants has been given in [17] using generalized Darboux transform.

Ohta and Yang [24] have given a new approach where solutions of the focus- ing NLS equation by means of a determinant representation, obtained from Hirota bilinear method.

A the beginning of the year 2012, one obtained a representation in terms of determinants which does not involve limits [15].

These first two formulations given in [13, 15] did depend in fact only on two parameters; this was first pointed out by V.B. Matveev in 2012. Then we found for the order N (for determinants of order 2N), solutions depending on 2N −2 real parameters.

With this new method, we construct news deformations at order 5 with 8 real parameters.

The aim of this paper is to present new solutions depending this time on strictly more than two parameters, to get all the possible patterns for the solutions of NLS equation. As it will be shown in the following, we construct solutions depending on 8 parameters which give the Peregrine breather as particular case when all the parameters are equal to 0 : for this reason, we will call these solutions, 8 parameters deformations of the Peregrine of order 5.

The structure of the paper is the following. We first recall the expressions of solutions of the two dimensional focusing nonlinear Schr¨odinger equation [13] in terms of Wronskians. Then we explain how to construct quasi ratio- nal solutions of this equation. Then we give the new result concerning the

new solutions of the NLS equation and the notations used. We construct new quasi rational solutions depending a priori on 2N−2 parameters at the order N. After, one builds various drawings to illustrate the evolution of the solutions according to the parameters. One obtains at the same time trian- gular configurations and ring structures with a maximum of 15 peaks. The complete analytical expression of the solutions depending on 8 parameters is found; it takes more than 14000 pages. These deformations are completely new and gives by new patterns a best understanding of the NLS equation.

2 Determinant representation of solutions of NLS equation

2.1 Quasi-rational limit solutions of the NLS equation

We recall the results obtained in [13]. We consider the focusing NLS equation ivt+vxx+ 2|v|²v = 0. (1) In the following, we consider 2N parametersλν,ν = 1, . . . ,2N satisfying the relations

0< λj <1, λN+j =−λj, 1≤j ≤N. (2) We define the terms κν, δν,γν by the following equations,

κν = 2p

1−λ²_ν, δν =κνλν, γν =

r1−λν

1 +λν

, (3)

and

κN+j =κj, δN+j =−δj, γN+j = 1/γj, j = 1. . . N. (4) The terms xr,ν (r= 3, 1) are defined by

xr,ν = (r−1) lnγν −i

γν +i, 1≤j ≤2N. (5) The parameters eν are defined by

ej =iaj −bj, eN+j =iaj+bj, 1≤j ≤N, (6)

j j

We use the following notations :

Θr,ν =κνx/2 +iδνt−ixr,ν/2 +γνy−ieν, 1≤ν ≤2N. We consider the functions

φr,ν(y) = sin Θr,ν, 1≤ν ≤N,

φr,ν(y) = cos Θr,ν, N + 1 ≤ν≤2N. (7) Wr(y) =W(φr,1, . . . , φr,2N) is the Wronskian

Wr(y) = det[(∂_y^µ−1φr,ν)ν, µ∈[1,...,2N]]. (8) Then we get the following statement [14]

Theorem 2.1 The function v defined by v(x, t) = W3(0)

W₁(0)exp(2it−iϕ). (9) is solution of the NLS equation (1)

ivt+vxx+ 2|v|²v = 0.

To obtain quasi-rational solutions of the NLS equation, we take the limit when the parametersλj →1 for 1≤j ≤N andλj → −1 forN+1≤j ≤2N. For that, we consider the parameter λj written in the form

λj = 1−2j²ǫ², 1≤j ≤N. (10) When ǫ goes to 0, we obtain quasi-rational solutions of the NLS equation given by :

Theorem 2.2 The function v defined by v(x, t) = exp(2it−iϕ) lim

ǫ→0

W3(0)

W1(0), (11)

is a quasi-rational solution of the NLS equation (1) ivt+vxx+ 2|v|²v = 0.

2.2 Expression of solutions of NLS equation in terms of a ratio of two determinants

We construct the solutions of the NLS equation expressed as a quotient of two determinants which does not involve a passage to the limit.

We use the following notations :

Aν =κνx/2 +iδνt−ix3,ν/2−ieν/2, Bν =κνx/2 +iδνt−ix1,ν/2−ieν/2, for 1≤ν ≤2N, with κν, δν, xr,ν defined in (3), (4) and (5).

The parameters eν are defined by (6).

We consider the following functions :

f4j+1,k =γ_k^4j−1sinAk, f4j+2,k =γ_k^4jcosAk,

f4j+3,k =−γ_k^4j+1sinAk, f4j+4,k =−γ_k^4j+2cosAk, (12) for 1≤k ≤N, and

f_4j+1,k =γ_k^2N−4j−2cosAk, f_4j+2,k =−γ_k^2N−4j−3sinAk,

f4j+3,k =−γ_k^2N−4j−4cosAk, f4j+4,k =γ_k^2N−4j−5sinAk, (13) for N + 1≤k ≤2N.

We define the functions gj,k for 1 ≤j ≤2N, 1 ≤ k ≤2N in the same way, we replace only the term Ak byBk.

g4j+1,k =γ_k^4j−1sinBk, g4j+2,k =γ_k^4jcosBk,

g4j+3,k =−γ_k^4j+1sinBk, g4j+4,k =−γ_k^4j+2cosBk, (14) for 1≤k ≤N, and

g4j+1,k =γ_k^2N−4j−2cosBk, g4j+2,k =−γ_k^2N−4j−3sinBk,

g4j+3,k =−γ_k^2N−4j−4cosBk, g4j+4,k =γ_k^2N−4j−5sinBk, (15) for N + 1≤k ≤2N.

Then we get the following result :

v(x, t) = exp(2it−iϕ)×det((njk)_j,k∈[1,2N])

det((djk)_j,k∈[1,2N]) (16) is a quasi-rational solution of the NLS equation (1)

ivt+vxx+ 2|v|²v = 0, where

nj1 =fj,1(x, t,0), 1≤j ≤2N njk = ∂^2k−2fj,1

∂ǫ^2k−2 (x, t,0), 2≤k≤N, 1≤j ≤2N njN+1 =fj,N+1(x, t,0), 1≤j ≤2N njN+k= ∂^2k−2fj,N+1

∂ǫ^2k−2 (x, t,0), 2≤k≤N, 1≤j ≤2N dj1 =gj,1(x, t,0),1≤j ≤2N djk = ∂^2k−2gj,1

∂ǫ^2k−2 (x, t,0), 2≤k ≤N, 1≤j ≤2N djN+1 =g_j,N+1(x, t,0), 1≤j ≤2N d_jN+k= ∂^2k−2gj,N+1

∂ǫ^2k−2 (x, t,0), 2≤k ≤N, 1≤j ≤2N The functions f and g are defined in (12),(13), (14), (15).

We don’t give here the proof of this result. The ideas are the same as exposed in [15]. We will give the proof in a forthcoming paper.

The solutions of the NLS equation can also be written in the form : v(x, t) = exp(2it−iϕ)×Q(x, t)

where Q(x, t) is defined by :

Q(x, t) :=

f1,1[0] . . . f1,1[N −1] f1,N+1[0] . . . f1,N+1[N −1]

f2,1[0] . . . f2,1[N −1] f2,N+1[0] . . . f2,N+1[N −1]

... ... ... ... ... ...

f2N,1[0] . . . f2N,1[N −1] f2N,N+1[0] . . . f2N,N+1[N −1]

g1,1[0] . . . g1,1[N −1] g1,N+1[0] . . . g1,N+1[N −1]

g2,1[0] . . . g2,1[N −1] g2,N+1[0] . . . g2,N+1[N −1]

... ... ... ... ... ...

g2N,1[0] . . . g2N,1[N −1] g2N,N+1[0] . . . g2N,N+1[N −1]

(17)

3 Quasi-rational solutions of order 5 with eight parameters

We have already constructed in [13] solutions for the cases N = 1, 2, 3,, and in [15] for N = 4, 5, 6, with two parameters.

Because of the length of the expression v of the solution of NLS equation with eight parameters, we can’t give here. We only construct figures to show deformations of the fifth Peregrine breathers.

Conversely to the study with two parameters given in preceding works [13, 15], we get other type of symmetries in the plots in the (x, t) plane. We give some examples of this fact in the following.

3.1 Peregrine breather of order 5

If we choose ˜a1 = ˜b1 = ˜a2 = ˜b2 = ˜a3 = ˜b3 = ˜a4 = ˜b4 = 0, we obtain the classical Peregrine breather :

Figure 1

1 1 2 2 3 3 4 4

With other choices of parameters, we obtain different types of configura- tions : triangles, ring structures and configurations with multiple concentric rings with a maximum of 15 peaks.

3.2 Variation of parameters

Figure 2 a1 6= 0

Figure 2: Solution of NLS, N=5, ˜a1 = 10⁴.

1

Figure 3: Solution of NLS, N=5, ˜b₁ = 10⁴.

Figure 4 a2 6= 0

Figure 4: Solution of NLS, N=5, ˜a2 = 10⁵.

2

Figure 5: Solution of NLS, N=5, ˜b₂ = 10⁶.

Figure 6 a3 6= 0

Figure 6: Solution of NLS, N=5, ˜a₃ = 10⁶.

3

Figure 7: Solution of NLS, N=5, ˜b₄ = 10¹⁰.

Figure 8 a4 6= 0

Figure 8: Solution of NLS, N=5, ˜a₄ = 10¹⁰.

4

Figure 9: Solution of NLS, N=5, ˜b₄ = 10¹⁰.

4 Conclusion

In the present paper we construct for the first time to my knowledge, ex- plicitly solutions of the NLS equation of order N = 5 with 2N −2 = 8 real parameters.

. By different choices of these parameters, we obtained new patterns in the (x;t) plane; we recognized ring structure as already observed in the case of deformations depending on two parameters [13, 15]. We obtain triangular configurations; it was already reported in [21]. We get news concentric rings configurations.

We chose to present here the solutions of the NLS equation in the caseN = 5 with eight real parameters, only in order not to weigh down the text with . We postpone the presentation of the higher orders in another publication.

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Appendix

vN(x, t) = n(x, t)

d(x, t)exp(2it)

= (1− GN(2x,4t) +iHN(2x,4t) QN(2x,4t) )e^2it with

GN(X, T) =

12

X

k=0

gk(T)X^k,

HN(X, T) =

12

X

k=0

hk(T)X^k,

QN(X, T) =

12

X

k=0

qk(T)X^k. GN(X, T) = 7127040T²⁸ -

199557120T²⁷b1

+

92897280T²⁶a12

+

2694021120T²⁶b12

-

2415329280T²⁵a12b1

-

23348183040T²⁵b13

+

559104000T²⁴a14

+

30191616000T²⁴a12b12

+

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