Tenth Peregrine breather solution of the NLS equation.

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Tenth Peregrine breather solution of the NLS equation.

Pierre Gaillard

To cite this version:

Pierre Gaillard. Tenth Peregrine breather solution of the NLS equation.. 2012. �hal-00743859v2�

Tenth Peregrine breather solution of the NLS equation.

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

October 21, 2012

Abstract

We go on in this paper, in the study of the solutions of the focusing NLS equation. With a new representation given in a preceding paper, a very compact formulation without limit as a quotient of two determi- nants, we construct Peregrine breathers of orderN = 10. The explicit analytical expression of the Akhmediev’s solution is completely given.

1 Introduction

In 1972 Zakharov and Shabat solved the nonlinear Schr¨odinger equation (NLS) using the inverse scattering method. The case of periodic and al- most periodic algebro-geometric solutions to the focusing NLS equation were first constructed in 1976 by Its and Kotlyarov [14]. The first quasi-rational solutions of NLS equation were constructed in 1983 by Peregrine, nowadays called worldwide Peregrine breathers. In 1986 Eleonski, Akhmediev and Ku- lagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather[3].

Other families of higher order were constructed in a series of articles by Akhmediev et al. [1, 2] using Darboux transformations.

In 2010, it has been shown in [6] that rational solutions of the NLS equation can be written as a quotient of two Wronskians. With this formulation, it was possible to recover as a particular case, Akhmediev’s quasi-rational so- lution of the NLS equation.

Recently, it has been constructed in [12] a new representation of the solutions of the NLS equation in terms of a ratio of two wronskians determinants of even order 2N composed of elementary functions; the related solutions of NLS are called of order N. Quasi-rational solutions of the NLS equation were obtained by the passage to the limit when some parameter tended to 0.

Here we obtain a new representation of quasi-rational solutions of NLS in term of a quotient of two determinants different from preceding works which does not involve wronskians. As a consequence we obtain a more efficient method than the preceding one, to obtain families of multi-rogue wave so- lutions of the focusing NLS equation depending on a certain number of pa- rameters.

2 Expression of solutions of NLS equation in terms of Wronskian determinant and quasi- rational limit

2.1 Solutions of NLS equation in terms of Wronskian determinant

We recall results obtained in [12]. We consider the focusing NLS equation

ivt+vxx+ 2|v|²v = 0. (1)

From [11], the solution of the NLS equation can be written in the form v(x, t) = det(I+A3(x, t))

det(I+A1(x, t))exp(2it−iϕ). (2) In (2), the matrix Ar = (aνµ)1≤ν,µ≤2N (r = 3, 1) is defined by

a_νµ = (−1)^ǫν Y

λ6=µ

γλ+γν

γλ −γµ

exp(iκ_νx−2δ_νt+x_r,ν +e_ν). (3) κν, δν, γν are functions of the parameters λν, ν = 1, . . . ,2N satisfying the relations

0< λj <1, λN+j =−λj, 1≤j ≤N. (4)

They are given by the following equations, κν = 2p

1−λ²_ν, δν =κνλν, γν =

r1−λν

1 +λν

, (5)

and

κN+j =κj, δN+j =−δj, γN+j = 1/γj, j = 1. . . N. (6) The terms xr,ν (r= 3, 1) are defined by

xr,ν = (r−1) lnγ_ν −i

γν +i, 1≤j ≤2N. (7)

The parameters e_ν are defined by

e_j =ia_j −b_j, e_N+j =ia_j+b_j, 1≤j ≤N, (8) where aj and bj, for 1≤j ≤N are arbitrary real numbers.

The terms ǫν are defined by :

ǫν = 0, 1≤ν ≤N

ǫν = 1, N + 1≤ν ≤2N.

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,

φ_r,ν(y) = cos Θ_r,ν, N + 1 ≤ν≤2N. (9) Wr(y) =W(φr,1, . . . , φr,2N) is the Wronskian

Wr(y) = det[(∂_y^µ−1φr,ν)ν, µ∈[1,...,2N]]. (10) Then we get the following link between Fredholm and Wronskian determi- nants [9]

Proposition 2.1

det(I+Ar) = kr(0)×Wr(φr,1, . . . , φr,2N)(0), (11) where

k_r(y) = 2^2Nexp(iP2N ν=1Θr,ν) Q2N

ν=2

Qν−1

µ=1(γν −γµ).

In (11), the matrix Ar is defined by (3).

It can be deduced the following result : Proposition 2.2 The function v defined by

v(x, t) = W3(0)

W1(0)exp(2it−iϕ). (12)

is solution of the NLS equation (1)

ivt+vxx+ 2|v|²v = 0.

2.2 Quasi-rational solutions of NLS equation in terms of a limit of a ratio of wronskian determinants

In the following, we take the limit when the parametersλj →1 for 1≤j ≤N and λj → −1 for N+ 1 ≤j ≤2N.

For simplicity, we denote d_j the term ^√cj₂.

We consider the parameter λj written in the form

λj = 1−2ǫ²d²_j, 1≤j ≤N. (13) When ǫ goes to 0, we realize limited expansions at order p, for 1≤ j ≤ N, of the terms

κ_j = 4d_jǫ(1−ǫ²d²_j)^1/2,δ_j = 4d_jǫ(1−2ǫ²d²_j)(1−ǫ²d²_j)^1/2, γj =djǫ(1−ǫ²d²_j)^−1/2, xr,j = (r−1) ln^1+iǫd_1−iǫd^{j(1−ǫ2d2j)−1/2}

j(1−ǫ²d²_j)^−1/2,

κ_N+j = 4d_jǫ(1−ǫ²d²_j)^1/2, δ_N+j =−4d_jǫ(1−2ǫ²d²_j)(1−ǫ²d²_j)^1/2, γN+j = 1/(djǫ)(1−ǫ²d²_j)^1/2, xr,N+j = (r−1) ln^1−iǫd_1+iǫd^{j(1−ǫ2d2j)−1/2}

j(1−ǫ²d²_j)^−1/2. The parameters aj and bj, for 1≤N are chosen in the form

aj = ˜ajǫ^M−1, bj = ˜bjǫ^M−1, 1≤j ≤N, M = 2N. (14) We have the result given in [11] :

Theorem 2.1 With the parameters λ_j defined by (13), a_j and b_j chosen as in (14), for 1≤j ≤N, the functionv defined by

v(x, t) = exp(2it−iϕ) lim

ǫ→0

W3(0)

W₁(0), (15)

is a quasi-rational solution of the NLS equation (1) ivt+vxx+ 2|v|²v = 0, depending on 3N parameters dj, ˜aj, ˜bj, 1≤j ≤N.

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

We construct here solutions of the NLS equation which does not involve Wronskian determinant and a passage to the limit, but which is expressed as a quotient of two determinants.

For this we need 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 (5), (6) and (7).

The parameters eν are defined by (8). For simplicity of the reduction, we choose aj and bj in the form

aj = ˜a1j^2N−1ǫ^2N−1, bj = ˜b1j^2N−1ǫ^2N−1, 1≤j ≤N. (16) Below we use the following notations :

f_4j+1,k =γ_k^4j−1sinA_k, f_4j+2,k =γ_k^4jcosA_k, f_4j+3,k =−γ_k^4j+1sinA_k, f_4j+4,k =−γ_k^4j+2cosA_k, for 1≤k ≤N, and

f_4j+1,k =γ_k^2N−4j−2cosA_k, f_4j+2,k =−γ_k^2N−4j−3sinA_k, f4j+3,k =−γ_k^2N−4j−4cosAk, f4j+4,k =γ_k^2N−4j−5sinAk, 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.

g_4j+1,k =γ_k^4j−1sinB_k, g_4j+2,k =γ_k^4jcosB_k, g_4j+3,k =−γ_k^4j+1sinB_k, g_4j+4,k =−γ_k^4j+2cosB_k,

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, for N + 1≤k ≤2N.

Then it is clear that

q(x, t) := W3(0) W1(0) can be written as

q(x, t) = ∆3

∆₁ = det(f_j,k)_{j, k∈[1,2N]}

det(g_j,k)_{j, k∈[1,2N]}. (17) We recall that λj = 1−2jǫ². All the functionsfj,k and gj,k depend onǫ. We use the expansions

fj,k(x, t, ǫ) =

N−1

X

l=0

1

(2l)!fj,1[l]k^2lǫ^2l+O(ǫ^2N), fj,1[l] = ∂^2lfj,1

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

N−1

X

l=0

1

(2l)!fj,N+1[l]k^2lǫ^2l+O(ǫ^2N), fj,N+1[l] = ∂^2lfj,N+1

∂ǫ^2l (x, t,0), fj,N+1[0] = fj,N+1(x, t,0), 1≤j ≤2N, 1≤k ≤N, 1≤l≤N −1.

We have the same expansions for the functions gj,k. gj,k(x, t, ǫ) =

N−1

X

l=0

1

(2l)!gj,1[l]k^2lǫ^2l+O(ǫ^2N), gj,1[l] = ∂^2lgj,1

∂ǫ^2l (x, t,0), gj,1[0] = gj,1(x, t,0), 1≤j ≤2N, 1≤k ≤N, 1≤l ≤N −1,

gj,N+k(x, t, ǫ) =

N−1

X

l=0

1

(2l)!gj,N+1[l]k^2lǫ^2l+O(ǫ^2N), gj,N+1[l] = ∂^2lgj,N+1

∂ǫ^2l (x, t,0), gj,N+1[0] =gj,N+1(x, t,0), 1≤j ≤2N, 1≤k≤N, N + 1≤k ≤2N..

Combining the columns of the determinants appearing in q(x, t) successively to eliminate in each columnk (or N+k) of them the powers of ǫlower than 2(k−1), and factorizing and simplifying each common terms, q(x, t) can be replaced by Q(x, t)

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]

g_1,1[0] . . . g_1,1[N −1] g_1,N+1[0] . . . g_1,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]

(18)

Then we get the following result :

Proposition 3.1 The function v defined by

v(x, t) = exp(2it−iϕ)×Q(x, t) (19) is a quasi-rational solution of the NLS equation (1)

ivt+vxx+ 2|v|²v = 0, where Q(x, t) is defined in (18).

4 Quasi-rational solutions of order N

Wa have already constructed in [12] solutions for the casesN = 1 untilN = 9 , and this method gives the same results. We don’t reproduce it here. We only give solutions of (NLS) equation in the case N = 10.

Because of the length of the expressions of polynomials N and D in the solutions v of the NLS equation defined by

v(x, t) = N(x, t)

D(x, t)exp(2it−iϕ), we only give in the appendix.

If we choosea1 = 0, b1 = 0, we obtain the classical Akhmediev’s breather :

Figure 1: Solution of NLS, N=10, a1 = 0, b1 = 0.

5 Conclusion

The method described in the present paper provides a powerful tool to get explicitly solutions of the NLS equation.

As my knowledge, it is the first time that the Peregrine breather of order seven is presented.

It confirms the conjecture about the shape of the breather in the (x, t) co- ordinates, the maximum of amplitude equal to 2N + 1 and the degree of polynomials in x and t here equal toN(N + 1). This new formulation gives an infinite set of non singular solution of NLS equation. It opens a large way to future researches in this domain.

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Appendix

In the following, we choose all the parameters a and b equal to 0.

The solution of NLS equation takes the form v(x, t) = N(x, t)

D(x, t)exp(2it−iϕ).

The polynomials N and D are defined by

N(x, t) =−108420328134123403897198738214693026385651427406838874942554577384174484827680224051200000t⁸⁶x²

−2193282686500631525315272661590511295758112030951273637905997322912560068209926808398725120000000000000000t³²x²

−187934072913248324467472609403525609855405622141026966348743047366467178004741497741639680000000000000000t²⁸x⁶

−120601467392759729566333536622791770045432283392458329281712979523453964739333521093427200000000000000000t²⁸x⁴

+36229881635758328246481237959602655317263772520259159153204530475312521805135127511040000000000000000000t²⁸x²

−23370676175197207897649263183176728975352732649082158520993305618021843379969379532800000000000000000t²⁰x¹² +300628987289608740650577115438409389840692968676891521647123295445163696742400000000000000000000t¹²x⁴

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