_{1}

^{*}

In this paper, we use the representation of the solutions of the focusing nonlinear Schrodinger 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.

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 [

In 2010, it has been shown in [

In this paper, we use a result [

Recently, another representation of the solutions of the focusing NLS equation, as a ratio of two determinants has been given in [

A new approach has been done in [

We have already given breathers of order N = 1 to N = 6 in [

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

From [

In (2), the matrix is defined by

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 :

We consider the following functions

We use the following notations:

.

is the wronskian

We consider the matrix defined by

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

where

It can be deduced the following result:

Theorem 2.2 The function v defined by

is solution of the NLS Equation (1)

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

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

We have the central result formulated in [

Theorem 2.3 The function v defined by

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

Proof: Let be the complex number

, ,. We use the following functions:

for, and

for.

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

Then it is clear that

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 D_{3} and D_{1} 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 D_{2}, for the component j of of D_{3}, and for the component j of, for the component j of of D_{1}. This algorithm can be continued until the columns C_{N}, C_{2N} of D_{3} and, of D_{1}.

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

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.

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

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

Equation (15) plays a fundamental role in optics and is the object of active research as recent work [

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 [

Please see

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 [

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 [

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

with