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 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 
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.
with dj = j, 1 ≤ j ≤ N, the solution of NLS Equation (15) takes the form
and 
can be easily verified from the recursive formulae given in [
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 [
, to shorten the formulation one prefers to give that inspired by Akhmediev et al. in [
