# Two Parameters Deformations of Ninth Peregrine Breather Solution of the NLS Equation and Multi-Rogue Waves

Authors: Pierre Gaillard
Journal: Journal of Mathematics
 (2013)

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## Abstract

This paper is a continuation of a recent paper on the solutions of the focusing NLS equation. The representation in terms of a quotient of two determinants gives a very efficient method of determination of famous Peregrine breathers and their deformations. Here we construct Peregrine breathers of orderN=9 and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.


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## Body

## 1. Introduction

From the fundamental work of Zakharov and Shabat in 1972 who solved the nonlinear Schrödinger equation (NLS) using the inverse scattering method, a lot of studies have been carried out on this equation. Its and Kotlyarov studied the case of periodic and almost periodic algebrogeometric solutions to the focusing NLS equation and constructed these solutions in 1976 [1]. Peregrine constructed the first quasi-rational solutions of NLS equation in 1983, nowadays called worldwide Peregrine breathers. In 1985, Akhmediev et al.  obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [2]. Other families of higher order were constructed in a series of articles by Akhmediev et al. [3, 4] using Darboux transformations.In 2010, it has been shown in [5] that rational solutions of the NLS equation can be written as a quotient of two Wronskians. Recently, in [6] a new representation of the solutions of the NLS equation has been constructed in terms of a ratio of two Wronskian determinants of even order 2N composed of elementary functions; the related solutions of NLS are 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 [7] and also higher order Peregrine’s breathers constructed by Akhmediev et al. [3, 8].Recently, another representation of the solutions of the focusing NLS equation, as a ratio of two determinants, has been given in [9] using generalized Darboux transform.A new approach has been done in [10], 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.Here, we construct the breather of orderN=9, 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 the NLS Equation in terms ofθ Functions

The solution of the NLS equation(1)ivt+vxx+2|v|2v=0
is given in terms of truncated theta function by (see [11])
(2)v(x,t)=θ3(x,t)θ1(x,t)exp(2it-iφ),
where
(3)θr(x,t)=∑k∈{0;1}2n‍exp{∑μ>ν,μ,ν=12n‍ln(γν-γμγν+γμ)2kμkν+(∑ν=12n‍iκνx-2δνt+xr,ν+∑μ=1,μ≠ν2n‍ln|γν+γμγν-γμ|+πiϵν+eν∑ν=12n‍)kν},r=1,3.
In this formula, κν, δν, γν, and x(r,ν) are functions of the parameters λj, 1≤j≤2n; they are defined by the formulas
(4)κν=21-λν2,δν=κνλν,γν=1-λν1+λν,xr,ν=(r-1)lnγν-iγν+i,r=1,3.
The parameters -1<λν<1, ν=1,…,2n, are real numbers such that
(5)-1<λn+1<λn+2<⋯<λ2n<0<λn<λn-1<⋯<λ1<1,λn+j=-λj,j=1,…,n.
Condition (5) implies that
(6)κj+n=κj,δj+n=-δj,γj+n=γj-1,xr,j+n=xr,j,j=1,…,n.
Complex numbers eν are defined in the following way:
(7)e1=ia-b,en+1=ia+b,eν=0∀ν≠1,n+1,a,b∈R.ϵν∈{0;1}, φ, ν=1⋯2n are arbitrary real numbers.


### 2.2. Relation betweenθ and Fredholm Determinant

The functionθr defined in (3) can be rewritten with a summation in terms of subsets of [1,…,2n],
(8)θr(x,t)=∑J⊂{1,…,2n}‍∏ν∈J‍(-1)ϵν∏ν∈J,μ∉J‍|γν+γμγν-γμ|×exp{∑ν∈J‍iκνx-2δνt+xr,ν+eν}.
We choose ϵj in formula (3) as ϵj=0 for 1≤j≤n, and ϵj=1 for n+1≤j≤2n.LetI be the unit matrix and A=(ajk)1≤j,k≤2n the matrix defined by
(9)aνμ=(-1)ϵν∏η≠ν‍|γν+γηγν-γη|exp(iκνx-2δνt+xr,ν+eν).
Then det(I+Ar) has the following form:
(10)det(I+Ar)=∑J⊂{1,…,2n}‍∏ν∈J‍(-1)ϵν∏ν∈J,μ∉J‍|γν+γμγν-γμ|×exp(iκνx-2δνt+xr,ν+eν).
From the beginning of this section, θ has the same expression as in (10), so we have clearly the equality
(11)θr=det(I+Ar).
Then the solution of NLS equation takes the form
(12)v(x,t)=det(I+A3(x,t))det(I+A1(x,t))exp(2it-iφ).


### 2.3. Link between Fredholm Determinants and Wronskians

We consider the following functions:(13)ϕν∶=sin(κνx2+iδνt-ixr,ν2+γνy-ieν2),1≤ν≤n,ϕνr=cos(κνx2+iδνt-ixr,ν2+γνy-ieν2),n+1≤ν≤2n.
We use the following notations:
(14)Θν=κνx2+iδνt-ixr,ν2+γνy-ieν2,1≤j≤2n.Wr(y)=W(ϕ1,…,ϕ2n) is the wronskian Wr(y)=det[(∂yμ-1ϕν)ν,μ∈[1,…,2n]].We consider matrixDr=(dνμ)ν,μ∈[1,…,2n] defined by
(15)dνμ=(-1)ϵν∏η≠ν‍|γν+γμγν-γμ|exp(iκνx-2δνt+xr,ν-ieν),1≤ν≤2n,1≤μ≤2n.
Then we have the following statement.Theorem 1.
Consider(16)det(I+Dr)=kr(0)×Wr(ϕ1,…,ϕ2n)(0),
where
(17)kr(y)=22nexp(i∑ν=12nΘν)∏ν=22n∏μ=1ν-1(γν-γμ).Proof.
We start to remove factor(2i)-1eiΘν in each row ν in the wronskian Wr(y) for 1≤ν≤2n.
Then(18)Wr=∏ν=12n‍eiΘν(2i)-n(2)-n×Wr1,
with(19)Wr1=|(1-e-2iΘ1)iγ1(1+e-2iΘ1)…(iγ1)2n-1(1+(-1)2ne-2iΘ1)(1-e-2iΘ2)iγ2(1+e-2iΘ2)…(iγ2)2n-1(1+(-1)2ne-2iΘ2)⋮⋮⋮⋮(1-e-2iθ2n)iγ2n(1+e-2iΘ2n)…(iγ2n)2n-1(1+(-1)2ne-2iΘ2n)|.The determinant Wr1 can be written as
(20)Wr1=det(αjkej+βjk),
where αjk=(-1)k(iγj)k-1, ej=e-2iθj, and βjk=(iγj)k-1, 1≤j≤n, 1≤k≤2n, αjk=(-1)k-1(iγj)k-1,  ej=e-2iθj, and βjk=(iγj)k-1, n+1≤j≤2n, 1≤k≤2n.
DenotingU=(αij)i,j∈[1,…,2n], V=(βij)i,j∈[1,…,2n], the determinant of U is clearly equal to
(21)det(U)=(i)2n(2n-1)/2∏2n≥l>m≥1‍(γl-γm).Then we use the following lemma.Lemma 2.
LetA=(aij)i,j∈[1,…,n], B=(bij)i,j∈[1,…,n], and (Hij)i,j∈[1,…,n], the matrix formed by replacing the jth row of A by the ith row of B. Then
(22)det(aijxi+bij)=det(A)×det(δijxi+det(Hij)det(A)).Proof.
ForA~=(a~ji)i,j∈[1,…,n], the transposed matrix in the cofactors of A, we have the well-known formula A×A~t=detA×I.
So it is clear thatdet(A~)=(det(A))n-1.
The general term of the product(cij)i,j∈[1,…,n]=(aijxi+bij)i,j∈[1,…,n]×(a~ji)i,j∈[1,…,n] can be written as
(23)cij=∑s=1n‍(aisxi+bis)×a~js=xi∑s=1n‍aisa~js+∑s=1n‍bisa~js=δijdet(A)xi+det(Hij).
We get
(24)det(cij)=det(aijxi+bij)×(det(A))n-1=(det(A))n×det(δijxi+det(Hij)det(A)).
Thus, det(aijxi+bij)=det(A)×det(δijxi+det(Hij)/det(A)).According to the relation (22) of the previous lemma, we get
(25)det(αijei+βij)=det(αij)×det(δijei+det(Hij)det(αij)),
where (Hij)i,j∈[1,…,n] is the matrix formed by replacing the jth row of U by the ith row of V defined previously.We computedet(Hij) and we get
(26)det(Hij)=(-1)(2n(2n+1)/2)+1(i)2n(2n-1)/2×∏2n≥l>m≥1,l≠j,m≠j‍(γl-γm)×∏l<j‍(γk-γl)×∏l>j‍(γk-γl).
We can simplify the quotient q=det(Hij)/det(αij):(27)q=(-1)ϵ(j)∏l≠j(γl+γk)(-1)k-1∏l≠k(γl-γk)=(-1)ϵ(j)-k+12γk∏l≠k(γl+γk)(γj+γk)∏l≠k(γl-γk)=(-1)ϵ(j)2γk(γj+γk)∏l≠k‍|γl+γkγl-γk|.
So detδjkej+(det(Hjk)/det(αjk)) can be expressed as
(28)det(δjkej+det(Hjk)det(αjk))=∏j=12n‍e-2iΘjdet(δjk+(-1)ϵ(j)2γk(γj+γk)∏l≠k‍|γl+γkγl-γk|e2iΘj).
Then dividing each column k by γk, 1≤k≤n, and multiplying each row j by γj, 1≤j≤n, we get
(29)det(δjkej+det(Hjk)det(αjk))=∏j=12n‍e-2iΘjdet(δjk+(-1)ϵ(j)2γj(γj+γk)∏l≠k‍|γl+γkγl-γk|e2iΘj)
and therefore the wronskian
(30)Wr(ϕ1,…,ϕn)(0)
can be written as
(31)∏j=12n‍eiΘj|y=0(2)-2n(i)2n(2n-2)/2∏j=22n‍∏i=1j-1‍(γj-γi)×∏j=12n‍e-2iΘj|y=0det(I+Dr).
It follows that
(32)det(I+Dr)=kr(0)Wr(ϕ1,…,ϕ2n)(0).
So, the solution of NLS equation takes the form
(33)v(x,t)=W3(0)W1(0)exp(2it-iφ).


### 2.4. Wronskian Representation of Solutions of NLS Equation

From the previous section, we get the following result.Theorem 3.
Functionv defined by
(34)v(x,t)=W3(ϕ1,…,ϕ2n)(0)W1(ϕ1,…,ϕ2n)(0)exp(2it-iφ)
is a smooth solution of the focusing NLS equation depending on two real parameters, a and b.


### 2.5. 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 the termcj/2  by  dj.We consider the parameterλj written in the form
(35)λj=1-2ϵ2dj2,1≤j≤N.
When ϵ goes to 0, we realize limited expansions at order p, for 1≤j≤N, of the terms
(36)κj=4djϵ(1-ϵ2dj2)1/2,δj=4djϵ(1-2ϵ2dj2)(1-ϵ2dj2)1/2,γj=djϵ(1-ϵ2dj2)-1/2,xr,j=(r-1)ln1+iϵdj(1-ϵ2dj2)-1/21-iϵdj(1-ϵ2dj2)-1/2,κN+j=4djϵ(1-ε2dj2)1/2,δN+j=-4djϵ(1-2ϵ2dj2)(1-ϵ2dj2)1/2,γN+j=1(djϵ)(1-ϵ2dj2)1/2,xr,N+j=(r-1)ln1-iϵdj(1-ϵ2dj2)-1/21+iϵdj(1-ϵ2dj2)-1/2.
The parameters a and a, for 1≤N, are chosen in the form(37)a=a~ϵM-1,b=b~ϵM-1,1≤j≤N,M=2N.
Then we have the following result.Theorem 4.
With the parametersλj defined by (35), a and b chosen as in (37), for 1≤j≤N, the function v defined by
(38)v(x,t)=exp(2it-iφ)limϵ→0W3(0)W1(0)
is a quasi-rational solution of the NLS equation (1)
(39)ivt+vxx+2|v|2v=0,
depending on two parameters.


## 2.1. Solutions of the NLS Equation in terms ofθ Functions

The solution of the NLS equation(1)ivt+vxx+2|v|2v=0
is given in terms of truncated theta function by (see [11])
(2)v(x,t)=θ3(x,t)θ1(x,t)exp(2it-iφ),
where
(3)θr(x,t)=∑k∈{0;1}2n‍exp{∑μ>ν,μ,ν=12n‍ln(γν-γμγν+γμ)2kμkν+(∑ν=12n‍iκνx-2δνt+xr,ν+∑μ=1,μ≠ν2n‍ln|γν+γμγν-γμ|+πiϵν+eν∑ν=12n‍)kν},r=1,3.
In this formula, κν, δν, γν, and x(r,ν) are functions of the parameters λj, 1≤j≤2n; they are defined by the formulas
(4)κν=21-λν2,δν=κνλν,γν=1-λν1+λν,xr,ν=(r-1)lnγν-iγν+i,r=1,3.
The parameters -1<λν<1, ν=1,…,2n, are real numbers such that
(5)-1<λn+1<λn+2<⋯<λ2n<0<λn<λn-1<⋯<λ1<1,λn+j=-λj,j=1,…,n.
Condition (5) implies that
(6)κj+n=κj,δj+n=-δj,γj+n=γj-1,xr,j+n=xr,j,j=1,…,n.
Complex numbers eν are defined in the following way:
(7)e1=ia-b,en+1=ia+b,eν=0∀ν≠1,n+1,a,b∈R.ϵν∈{0;1}, φ, ν=1⋯2n are arbitrary real numbers.


## 2.2. Relation betweenθ and Fredholm Determinant

The functionθr defined in (3) can be rewritten with a summation in terms of subsets of [1,…,2n],
(8)θr(x,t)=∑J⊂{1,…,2n}‍∏ν∈J‍(-1)ϵν∏ν∈J,μ∉J‍|γν+γμγν-γμ|×exp{∑ν∈J‍iκνx-2δνt+xr,ν+eν}.
We choose ϵj in formula (3) as ϵj=0 for 1≤j≤n, and ϵj=1 for n+1≤j≤2n.LetI be the unit matrix and A=(ajk)1≤j,k≤2n the matrix defined by
(9)aνμ=(-1)ϵν∏η≠ν‍|γν+γηγν-γη|exp(iκνx-2δνt+xr,ν+eν).
Then det(I+Ar) has the following form:
(10)det(I+Ar)=∑J⊂{1,…,2n}‍∏ν∈J‍(-1)ϵν∏ν∈J,μ∉J‍|γν+γμγν-γμ|×exp(iκνx-2δνt+xr,ν+eν).
From the beginning of this section, θ has the same expression as in (10), so we have clearly the equality
(11)θr=det(I+Ar).
Then the solution of NLS equation takes the form
(12)v(x,t)=det(I+A3(x,t))det(I+A1(x,t))exp(2it-iφ).


## 2.3. Link between Fredholm Determinants and Wronskians

We consider the following functions:(13)ϕν∶=sin(κνx2+iδνt-ixr,ν2+γνy-ieν2),1≤ν≤n,ϕνr=cos(κνx2+iδνt-ixr,ν2+γνy-ieν2),n+1≤ν≤2n.
We use the following notations:
(14)Θν=κνx2+iδνt-ixr,ν2+γνy-ieν2,1≤j≤2n.Wr(y)=W(ϕ1,…,ϕ2n) is the wronskian Wr(y)=det[(∂yμ-1ϕν)ν,μ∈[1,…,2n]].We consider matrixDr=(dνμ)ν,μ∈[1,…,2n] defined by
(15)dνμ=(-1)ϵν∏η≠ν‍|γν+γμγν-γμ|exp(iκνx-2δνt+xr,ν-ieν),1≤ν≤2n,1≤μ≤2n.
Then we have the following statement.Theorem 1.
Consider(16)det(I+Dr)=kr(0)×Wr(ϕ1,…,ϕ2n)(0),
where
(17)kr(y)=22nexp(i∑ν=12nΘν)∏ν=22n∏μ=1ν-1(γν-γμ).Proof.
We start to remove factor(2i)-1eiΘν in each row ν in the wronskian Wr(y) for 1≤ν≤2n.
Then(18)Wr=∏ν=12n‍eiΘν(2i)-n(2)-n×Wr1,
with(19)Wr1=|(1-e-2iΘ1)iγ1(1+e-2iΘ1)…(iγ1)2n-1(1+(-1)2ne-2iΘ1)(1-e-2iΘ2)iγ2(1+e-2iΘ2)…(iγ2)2n-1(1+(-1)2ne-2iΘ2)⋮⋮⋮⋮(1-e-2iθ2n)iγ2n(1+e-2iΘ2n)…(iγ2n)2n-1(1+(-1)2ne-2iΘ2n)|.The determinant Wr1 can be written as
(20)Wr1=det(αjkej+βjk),
where αjk=(-1)k(iγj)k-1, ej=e-2iθj, and βjk=(iγj)k-1, 1≤j≤n, 1≤k≤2n, αjk=(-1)k-1(iγj)k-1,  ej=e-2iθj, and βjk=(iγj)k-1, n+1≤j≤2n, 1≤k≤2n.
DenotingU=(αij)i,j∈[1,…,2n], V=(βij)i,j∈[1,…,2n], the determinant of U is clearly equal to
(21)det(U)=(i)2n(2n-1)/2∏2n≥l>m≥1‍(γl-γm).Then we use the following lemma.Lemma 2.
LetA=(aij)i,j∈[1,…,n], B=(bij)i,j∈[1,…,n], and (Hij)i,j∈[1,…,n], the matrix formed by replacing the jth row of A by the ith row of B. Then
(22)det(aijxi+bij)=det(A)×det(δijxi+det(Hij)det(A)).Proof.
ForA~=(a~ji)i,j∈[1,…,n], the transposed matrix in the cofactors of A, we have the well-known formula A×A~t=detA×I.
So it is clear thatdet(A~)=(det(A))n-1.
The general term of the product(cij)i,j∈[1,…,n]=(aijxi+bij)i,j∈[1,…,n]×(a~ji)i,j∈[1,…,n] can be written as
(23)cij=∑s=1n‍(aisxi+bis)×a~js=xi∑s=1n‍aisa~js+∑s=1n‍bisa~js=δijdet(A)xi+det(Hij).
We get
(24)det(cij)=det(aijxi+bij)×(det(A))n-1=(det(A))n×det(δijxi+det(Hij)det(A)).
Thus, det(aijxi+bij)=det(A)×det(δijxi+det(Hij)/det(A)).According to the relation (22) of the previous lemma, we get
(25)det(αijei+βij)=det(αij)×det(δijei+det(Hij)det(αij)),
where (Hij)i,j∈[1,…,n] is the matrix formed by replacing the jth row of U by the ith row of V defined previously.We computedet(Hij) and we get
(26)det(Hij)=(-1)(2n(2n+1)/2)+1(i)2n(2n-1)/2×∏2n≥l>m≥1,l≠j,m≠j‍(γl-γm)×∏l<j‍(γk-γl)×∏l>j‍(γk-γl).
We can simplify the quotient q=det(Hij)/det(αij):(27)q=(-1)ϵ(j)∏l≠j(γl+γk)(-1)k-1∏l≠k(γl-γk)=(-1)ϵ(j)-k+12γk∏l≠k(γl+γk)(γj+γk)∏l≠k(γl-γk)=(-1)ϵ(j)2γk(γj+γk)∏l≠k‍|γl+γkγl-γk|.
So detδjkej+(det(Hjk)/det(αjk)) can be expressed as
(28)det(δjkej+det(Hjk)det(αjk))=∏j=12n‍e-2iΘjdet(δjk+(-1)ϵ(j)2γk(γj+γk)∏l≠k‍|γl+γkγl-γk|e2iΘj).
Then dividing each column k by γk, 1≤k≤n, and multiplying each row j by γj, 1≤j≤n, we get
(29)det(δjkej+det(Hjk)det(αjk))=∏j=12n‍e-2iΘjdet(δjk+(-1)ϵ(j)2γj(γj+γk)∏l≠k‍|γl+γkγl-γk|e2iΘj)
and therefore the wronskian
(30)Wr(ϕ1,…,ϕn)(0)
can be written as
(31)∏j=12n‍eiΘj|y=0(2)-2n(i)2n(2n-2)/2∏j=22n‍∏i=1j-1‍(γj-γi)×∏j=12n‍e-2iΘj|y=0det(I+Dr).
It follows that
(32)det(I+Dr)=kr(0)Wr(ϕ1,…,ϕ2n)(0).
So, the solution of NLS equation takes the form
(33)v(x,t)=W3(0)W1(0)exp(2it-iφ).


## 2.4. Wronskian Representation of Solutions of NLS Equation

From the previous section, we get the following result.Theorem 3.
Functionv defined by
(34)v(x,t)=W3(ϕ1,…,ϕ2n)(0)W1(ϕ1,…,ϕ2n)(0)exp(2it-iφ)
is a smooth solution of the focusing NLS equation depending on two real parameters, a and b.


## 2.5. 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 the termcj/2  by  dj.We consider the parameterλj written in the form
(35)λj=1-2ϵ2dj2,1≤j≤N.
When ϵ goes to 0, we realize limited expansions at order p, for 1≤j≤N, of the terms
(36)κj=4djϵ(1-ϵ2dj2)1/2,δj=4djϵ(1-2ϵ2dj2)(1-ϵ2dj2)1/2,γj=djϵ(1-ϵ2dj2)-1/2,xr,j=(r-1)ln1+iϵdj(1-ϵ2dj2)-1/21-iϵdj(1-ϵ2dj2)-1/2,κN+j=4djϵ(1-ε2dj2)1/2,δN+j=-4djϵ(1-2ϵ2dj2)(1-ϵ2dj2)1/2,γN+j=1(djϵ)(1-ϵ2dj2)1/2,xr,N+j=(r-1)ln1-iϵdj(1-ϵ2dj2)-1/21+iϵdj(1-ϵ2dj2)-1/2.
The parameters a and a, for 1≤N, are chosen in the form(37)a=a~ϵM-1,b=b~ϵM-1,1≤j≤N,M=2N.
Then we have the following result.Theorem 4.
With the parametersλj defined by (35), a and b chosen as in (37), for 1≤j≤N, the function v defined by
(38)v(x,t)=exp(2it-iφ)limϵ→0W3(0)W1(0)
is a quasi-rational solution of the NLS equation (1)
(39)ivt+vxx+2|v|2v=0,
depending on two parameters.


## 3. Quasi-Rational Solutions of Order 9

Wa have already constructed in [6] solutions for the cases N=1 until N=8, and this method gives the same results. We do not reproduce it here. We only give solutions of NLS equation in the case N=9.Because of the length of the expressions of polynomialsN and D in the solutions v of the NLS equation defined by
(40)v(x,t)=N(x,t)D(x,t)exp(2it-iφ),
we only give them in the appendix. In the following cases, we only give the plots for the modulus of v in the (x,t) coordinates.Fora1=0, b1=0, we obtain Akhmediev’s breather; we get the expected amplitude of 2N+1=19 for the spike (Figure 1).Figure 1
Solution of NLS,N=9, a1=0, b1=0.If we choosea1=0, b1=1012, we obtain Figure 2.Figure 2
Solution of NLS,N=9, a1=0, b1=1012.If we choosea1=1016, b1=-1012, we have Figure 3.Figure 3
Solution of NLS,N=9, a1=1016, b1=-1012.It can be noted that Figures2 and 3 are closely analogous with Figure 2(b) in paper [12] of Kedziora et al. In that work (N=8), it was pointed out that the shift (here corresponding to a and b nonzero) pulls out a ring of 2N-1 fundamental rogue elements, corresponding to  15 of them there and to 17 here. It leaves behind a rogue wave of order N-2, that is, 6 there (amplitude = 13) and 7  here (amplitude = 15). Of course, Figure 1 here is analogous with Figure 2(a) there (amplitudes 19 and 17, resp.).


## 4. Conclusion

The method used in the present paper provides a powerful tool as the explicit analytical formulation of the ninth order shows it. To my knowledge, it is the first time that the analytical expression of the Peregrine breather of order nine is presented.It confirms the conjecture about the shape of the breather in the(x,t) coordinates, the maximum of amplitude equal to 2N+1, and the degree of polynomials in x and t here equal to N(N+1). For a and b nonzero, the maximum is less than that, as discussed above and seen in Figures 2 and 3.



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*Source: 520214-2013-10-24.xml*