ABSTRACT

In this paper, we use the representation of the solutions of the focusing nonlinear Schrödinger equation we have constructed recently, in terms of wronskians; when we perform a special passage to the limit, we get quasi-rational solutions expressed as a ratio of two determinants. We have already construct breathers of orders N = 4, 5, 6 in preceding works; we give here the breather of order seven.

1. Introduction

From fundamental work of Zakharov and Shabat in 1968 [1,2], a lot of research has been carried out on the nonlinear Schrödinger equation (NLS). The case of periodic and almost periodic algebro-geometric solutions to the focusing NLS equation were first constructed in 1976 by Its and Kotlyarov [3]. The first quasi-rational solutions of NLS equation were construted in 1983 by Peregrine [4]; they are nowadays called worldwide Peregrine breathers. In 1986, Eleonski, Akhmediev and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [5,6]. Other families of higher order were constructed in a series of articles by Akhmediev et al. [7,8] using Darboux transformations.

In 2010, it has been shown in [9] that rational solutions of NLs equation can be writen as a quotient of two wronskians.

In this paper, we use a result [10] giving 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. When we perform the passage to the limit when some parameter tends to 0, we got families of multi-rogue wave solutions of the focusing NLS equation depending on a certain number of parameters. It allows to recognize the famous Peregrine breather [4] and also higher order Peregrine’s breathers constructed by Akhmediev [7,11].

Recently, another representation of the solutions of the focusing NLS equation, as a ratio of two determinants has been given in [12] using generalized Darboux transform.

A new approach has been done in [13] which gives a determinant representation of solutions of the focusing NLS equation, obtained from Hirota bilinear method, derived by reduction of the Gram determinant representation for Davey-Stewartson system.

We have already given breathers of order N = 1 to N = 6 in [14]. Here, we construct the breather of order N = 7 which shows the efficiency of this method.

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 briefly recall results obtained in [10,14]. We consider the focusing NLS equation

(1)

From [14], the solution of the NLS equation can be written in the form

(2)

In (2), the matrix is defined by

(3)

The terms and are functions of the parameters satisfying the relations

They are given by the following equations,

and

.

The terms are defined by

The coefficients are defined by :

(4)

We consider the following functions

(5)

We use the following notations:

.

is the wronskian

(6)

We consider the matrix defined by

Then we get the following link between Fredohlm and Wronskian determinants [14]

Theorem 2.1

(7)

where

It can be deduced the following result:

Theorem 2.2 The function v defined by

(8)

is solution of the NLS Equation (1)

2.2. Quasi-Rational Solutions of NLS Equation

In the following, we take the limit when the parameters for and for .

For simplicity, we denote the term.

We consider the parameter written in the form

(9)

When goes to 0, we realize limited expansions at order p, for, of the terms

We have the central result formulated in [14] :

Theorem 2.3 The function v defined by

(10)

is a quasi-rational solution of the NLS Equation (1)

Proof: Let be the complex number

, ,. We use the following functions:

(11)

for, and

(12)

for.

We define the functions for, in the same way, where the term in is replaced by.

Then it is clear that

(13)

All the functions and and their derivatives depend on and can all be prolonged by continuity when.

For simplicity we denote the term

, and the term.

Then we use the expansions

We have the same expansions for the functions.

The components j of the columns 1 and N + 1 are respectively equal by definition to for, for of, and for, for of.

At the first step of the reduction, we replace the columns by and by for, for; we do the same changes for. Each component j of the column of can be rewritten as

and the column replaced by

for. For, we have the same reductions, each component j of the column of can be rewritten as

and the column replaced by

for.

We can factorize in D₃ and D₁ in each column k and the term for, and so simplify these common terms in numerator and denominator.

If we restrict the developments at order 1 in columns 2 and, we get respectively for the component j of D₂, for the component j of of D₃, and for the component j of, for the component j of of D₁. This algorithm can be continued until the columns C_N, C_2N of D₃ and, of D₁.

Then taking the limit when tends to 0, can be replaced by

(14)

Each element of these determinants is a polynomial in x and t. So the solution of the NLS equation takes the form with a rational function in x and t, and which ends the proof.

3. Seventh-Order Breather Solution of NLS Equation

3.1. Fiber-Optics Case

To get solutions of NLS equation written in the context of fiber optics

(15)

from these of (1), we can make the following changes of variables

(16)

Equation (15) plays a fundamental role in optics and is the object of active research as recent work [8] attests it where the solutions of the two-breathers are studied.

3.2. Case of the Initial Conditions

In the case of order N = 7, we make an expansion at order 13. Taking the limit when with d_j = j, 1 ≤ j ≤ N, the solution of NLS Equation (15) takes the form

Because of the length of the complete analytical expression, we only give it in the appendix.

We give here the expression of the solution in the form

in the case t = 0:

Remark 3.1 The expressions of and can be easily verified from the recursive formulae given in [11].

3.3. Plot in the (x, t) Coordinates

Please see Figure 1.

4. Conclusion

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

To the best of my knowledge, it is the first time that the breather of order seven solution of the NLS equation is presented.

It confirms the conjecture about the shape of the rogue wave in the coordinates, the maximum of amplitude equal to 2N + 1 = 15 and the degree of polynomials in x and t here equal to 56 as already formulated in [7]. This new formulation gives the possibility, by introduction of parameters in the arguments of preceding functions defined in the text, to create an infinite set of non singular solutions of NLS equation. It will be the next step of the work which will open a large way to future researches in this domain.

REFERENCES

Appendix

Rather than to give the analytical expression in the form, to shorten the formulation one prefers to give that inspired by Akhmediev et al. in [11].

The solution of NLS equation takes the form, with N = 7

with