Wronskian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave ()
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 CN, C2N 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 dj = 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.
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
