Two-parameter determinant representation of seventh order rogue wave solutions of the NLS equation

Abstract

We present a new representation of solutions of focusing nonlinear Schrödinger equation (NLS) equation as a quotient of two determinants. We construct families of quasi-rational solutions of the NLS equation depending on two parameters, a and b. We construct, for the first time, analytical expressions of Peregrine breather of order 7 and multi-rogue waves by deformation of parameters. These expressions make possible to understand the behavior of the solutions. In the case of the Peregrine breather of order 7, it is shown for great values of parameters a or b the appearance of the Peregrine breather of order 5.

PACS

35Q55; 37K10

Introduction

One of the most direct approaches to model the evolution of deep water waves is the use of nonlinear Schrödinger equation (NLS)[1, 2]. This equation was first solved in 1972 by Zakharov and Shabat[3] using inverse scattering method. Its and Kotlyarov constructed in 1976 periodic and almost periodic solutions of the focusing NLS equation[4]. It is in 1979 that Ma found the first breather-type solution of the NLS equation[5]; this solution breathes temporally but is spatially localized. In 1983, Peregrine[6] gave the first example of quasi-rational solutions of the NLS equation which was localized both in space and time. Akhmediev, Eleonski, and Kulagin obtained the first higher order analog of the Peregrine breather[7] in 1986. Other families of higher order were constructed by Akhmediev et al.[8, 9] using Darboux transformations; these solutions breathe spatially and are localized in time. The Peregrine solution appears to be a limiting case of a breather when the spatial period is taken to be infinite.

The notion of rogue waves first appeared in studies of ocean waves; then it moved to other domains of physics like optics[10], Bose-Einstein condensates[11], etc. A lot of experiments about solutions of NLS equation have been realized; in particular, the basic Peregrine soliton has been studied very recently in[12, 13]; furthermore, the NLS equation accurately describes physical rogue waves of relatively high order according to the work[14].

Actually, there is growing interest in studying higher order rational solutions. In 2010, rational solutions of the NLS equation have been written as a quotient of two Wronskians[15]. In 2011, it has been constructed in[16] a new representation of the solutions of the NLS equation in terms of a ratio of two Wronskian determinants of even order 2N composed of elementary functions.

Recently, Guo, Ling, and Liu gave another representation of the solutions of the focusing NLS equation as a ratio of two determinants[17] using generalized Darboux transform. Ohta and Yang have realized a new approach in[18] which gives a determinant representation of solutions of the focusing NLS equation, obtained from Hirota bilinear method.

Here we present a new representation of quasi-rational solutions of NLS expressed as a quotient of two determinants which do not involve limits introduced in[19]. It is important to note that the formulations given in[16, 19] depend only on two parameters as mentioned in the title; this remark was first made by VB Matveev in March 2012. With this method, we obtain very easily explicit analytical expressions of the solutions of the NLS equation as deformations of the Peregrine breather of order 7 depending on two parameters, a and b. The following orders will be the object of other publications.

These explicit solutions are important since it makes it possible to understand the behavior of rogue waves. In particular, it shows the appearance for great values of parameters a or b of the Peregrine breather of order 5. Precisely, according to the explicit formulation of the solution of order 7, the factor of a² or b² is exactly the analytical expression of the Peregrine breather of order 5. This fact was first pointed out for the orders 3 and 4 by VB Matveev in March 2012. It was also noticed and highlighted by Akhmediev et al. in the article[20].

This result, in addition to being the first time that explicit expressions of deformations of order 7 of the solutions of the equation of NLS are given, makes it possible to understand the phenomenon of appearance of rogue waves and their asymptotic behavior.

The method given in this paper is a very powerful tool, better than that presented in[16, 21], and leads to new results about NLS equation.

Expression of solutions of NLS equation in terms of a ratio of two determinants

We recall in this section the result exposed in[19]. We consider the following focusing NLS equation:

$$\begin{array}{l}i{v}_t+v_{\mathit{\text{xx}}}+2|v{|}^{2}v=0.\end{array}$$

(1)

We use the following notations:

κ _ν ,δ _ν , and γ _ν are functions of the parameters λ _ν and ν=1,…,2N, satisfying the relations

$$\begin{array}{l}0<{\lambda }_j<1,{\lambda }_{N+j}=-{\lambda }_j,\\ 1\le j\le \mathrm{N.}\end{array}$$

(2)

They are given by the following equations:

$$\begin{array}{l}\begin{array}{c}{\kappa }_j=2\sqrt{1-{\lambda }_j^2},{\delta }_j={\kappa }_j{\lambda }_j,\\ {\gamma }_j=\sqrt{\frac{1-{\lambda }_j}{1+{\lambda }_j}},1\le j\le N,\end{array}\end{array}$$

(3)

and

$$\begin{array}{l}\begin{array}{c}{\kappa }_{N+j}={\kappa }_j,{\delta }_{N+j}=-{\delta }_j,\\ {\gamma }_{N+j}=1/{\gamma }_j,1\le j\le N,\end{array}\end{array}$$

(4)

The terms x_r,ν(r=1, 3) are defined by

$$\begin{array}{l}{x}_{r,\nu }=(r-1)ln\frac{{\gamma }_{\nu }-i}{{\gamma }_{\nu }+i},\\ 1\le \nu \le 2\mathrm{N.}\end{array}$$

(5)

The parameters e _ν are defined by

$$\begin{array}{l}{e}_j=i{a}_j-b_j,e_{N+j}=i{a}_j+b_j,\\ 1\le j\le N,\end{array}$$

(6)

where a _j and b _j , for 1≤j≤N, are arbitrary real numbers.

We then define the following expressions:

$$\begin{array}{cc}{A}_{\nu }& ={\kappa }_{\nu }x/2+i{\delta }_{\nu }t-i{x}_{3,\nu }/2-i{e}_{\nu }/2,\\ B_{\nu }& ={\kappa }_{\nu }x/2+i{\delta }_{\nu }t-i{x}_{1,\nu }/2-i{e}_{\nu }/2,\end{array}$$

for 1≤ν≤2N, with κ _ν , δ _ν , and x_r,ν defined previously in Equations 3, 4, and 5.

The parameters e _ν are defined in Equation 6. For the following reduction, we choose a _j and b _j in the form

$$\begin{array}{l}{a}_j=\widetilde{a_1}{j}^{2N-1}{\epsilon }^{2N-1},\\ b_j=\widetilde{b_1}{j}^{2N-1}{\epsilon }^{2N-1},\\ 1\le j\le \mathrm{N.}\end{array}$$

(7)

We consider the parameter λ _j written in the form

$$\begin{array}{l}{\lambda }_j=1-2j^2{\epsilon }^2,1\le j\le \mathrm{N.}\end{array}$$

(8)

Below we use the following functions for 1≤j≤2N:

$$\begin{array}{l}{f}_{4j+1,k}={\gamma }_k^{4j-1}sin{A}_k,\\ f_{4j+2,k}={\gamma }_k^{4j}cos{A}_k,\\ f_{4j+3,k}=-{\gamma }_k^{4j+1}sin{A}_k,\\ f_{4j+4,k}=-{\gamma }_k^{4j+2}cos{A}_k,\end{array}$$

(9)

1≤k≤N, and

$$\begin{array}{l}{f}_{4j+1,k}={\gamma }_k^{2N-4j-2}cos{A}_k,\\ f_{4j+2,k}=-{\gamma }_k^{2N-4j-3}sin{A}_k,\\ f_{4j+3,k}=-{\gamma }_k^{2N-4j-4}cos{A}_k,\\ f_{4j+4,k}={\gamma }_k^{2N-4j-5}sin{A}_k,\end{array}$$

(10)

for N+1≤k≤2N.

We define the functions g_j,k for 1≤j≤2N, 1≤k≤2N in the same way; we replace only the term A _k by B _k :

$$\begin{array}{l}{g}_{4j+1,k}={\gamma }_k^{4j-1}sin{B}_k,\\ g_{4j+2,k}={\gamma }_k^{4j}cos{B}_k,\\ g_{4j+3,k}=-{\gamma }_k^{4j+1}sin{B}_k,\\ g_{4j+4,k}=-{\gamma }_k^{4j+2}cos{B}_k,\end{array}$$

(11)

for 1≤k≤N, and

$$\begin{array}{l}{g}_{4j+1,k}={\gamma }_k^{2N-4j-2}cos{B}_k,\\ g_{4j+2,k}=-{\gamma }_k^{2N-4j-3}sin{B}_k,\\ g_{4j+3,k}=-{\gamma }_k^{2N-4j-4}cos{B}_k,\\ g_{4j+4,k}={\gamma }_k^{2N-4j-5}sin{B}_k,\end{array}$$

(12)

for N+1≤k≤2N.

All functions f_j,k and g_j,k and their derivatives depend on ε and can all be prolonged by continuity when ε=0. Then they have the following result:

Theorem 1

The function v defined by

$$\begin{array}{l}v(x,t)=exp(2\mathit{\text{it}}-\mathrm{i\phi })\\ \times \frac{det\left(\right(n_{\mathit{\text{jk}}{)}_{j,k\in [1,2N]}})}{det\left(\right(d_{\mathit{\text{jk}}{)}_{j,k\in [1,2N]}})}\end{array}$$

(13)

is a quasi-rational solution of the NLS Equation 1:

$$i{v}_t+v_{\mathit{\text{xx}}}+2|v{|}^{2}v=0,$$

where

$$\begin{array}{l}{n}_{j1}=f_{j,1}(x,t,0),\\ n_{\mathit{\text{jk}}}=\frac{{\partial }^{2k-2}{f}_{j,1}}{\partial {\epsilon }^{2k-2}}(x,t,0),\\ n_{\mathit{\text{jN}}+1}=f_{j,N+1}(x,t,0),\\ n_{\mathit{\text{jN}}+k}=\frac{{\partial }^{2k-2}{f}_{j,N+1}}{\partial {\epsilon }^{2k-2}}(x,t,0),\\ d_{j1}=g_{j,1}(x,t,0),\\ d_{\mathit{\text{jk}}}=\frac{{\partial }^{2k-2}{g}_{j,1}}{\partial {\epsilon }^{2k-2}}(x,t,0),\\ d_{\mathit{\text{jN}}+1}=g_{j,N+1}(x,t,0),\\ d_{\mathit{\text{jN}}+k}=\frac{{\partial }^{2k-2}{g}_{j,N+1}}{\partial {\epsilon }^{2k-2}}(x,t,0),\\ 2\le k\le N,1\le j\le 2\mathrm{N.}\end{array}$$

The functions f and g are defined in Equations 9, 10, 11, and 12.

Proof

We do not have the space to reproduce the proof here. The reader can find it in[19].

In other words, the solutions of the NLS equation can be written as

$$v(x,t)=exp(2\mathit{\text{it}}-\mathrm{i\phi })\times Q(x,t)$$

with Q(x,t) defined by

□

Quasi-rational solutions of order 7

We have already constructed in[16, 19, 21] solutions of the NLS equation for the cases N=1 until N=6. Here, we give new solutions of the NLS equation in the case N=7. We start to give some plots of these solutions in the (x,t) plan for various values of the parameters.

In the following, for more simplicity, we note ã₁=a and${\tilde{b}}_1=b$.

Figures

If we choose a=0 and b=0, we recognize the famous Peregrine breather of order 7 in Figure1. If we choose a=0 and b=10⁸, we obtain the following plot in Figure2. If we choose a=10¹² and b=−10⁸, we obtain the following regular plot in Figure3.

Figure 1

Solution of NLS, N = 7, a = 0, b = 0.

Figure 2

Solution of NLS, N = 7, a = 0, b = 10 ⁸ .

Figure 3

Solution of NLS, N =7, a = 10 ¹² , b = −10 ⁸ .

We saw in these plots that when parameters a and b increase, there is the appearance of circular configurations or more exactly annular configurations. This phenomenon was also highlighted in the article[22]. Indeed, the previous images given by Figures2 and3 are closely analogous to Figure2 parameter b in that paper which gives the case of order 8 as an example. This fact is also highlighted in the works of the author[23]. In the work[22], it was pointed out that the shift (here corresponding to a and b nonzero) pulls out or ‘emits’ a ring of 2N−1 fundamental (N=1) rogue elements, corresponding to 15 of them there and to 13 here. It leaves behind a rogue wave of order N−2, i.e., 6 there (amplitude=13) and 5 (amplitude=11) in this paper. Of course, Figure1 here is analogous to Figure2 parameter a there (amplitudes 15 and 17, respectively).

Asymptotic behavior

In all these figures, for N=7, we see that when the parameters a and b tend towards the infinite, there is the appearance of the Peregrine breather of order 5. This remark was made the first time by VB Matveev in March 2012 for orders 3 and 4[24]. It was also reported in[20].

In fact, if one identifies the factors of a² or b² (even of a²+b²) in the analytical expression of Peregrine of order 7 given in Additional file1, one finds exactly that of the Peregrine breather of order 5. We do not have enough space in this text to detail these calculations. The reader who would like to find the expression of the Peregrine breather of order 5 will be able to refer to the article[21].

This explains why when one or the other of parameters a or b (even both) tend towards infinite in the case of the Peregrine breather of order N=7, it appears in the center of the figures’ Peregrine breather of order N−2=5.

Analytic expressions

N takes the value 7; we make the following change of variables X=2x and T=4t. The solution of the NLS equation can be written in the form

$$\begin{array}{l}{v}_N(x,t)=\frac{n(x,t)}{d(x,t)}exp\left(2\mathit{\text{it}}\right)\\ =(1-{\alpha }_N\frac{G_N(2x,4t)+i{H}_N(2x,4t)}{Q_N(2x,4t)})e^{2\mathit{\text{it}}}\end{array}$$

with

$$\begin{array}{c}{G}_N(X,T)=\sum _{k=0}^{N(N+1)}{\mathbf{\text{g}}}_k\left(T\right)X^k\\ H_N(X,T)=\sum _{k=0}^{N(N+1)}{\mathbf{\text{h}}}_k\left(T\right)X^k\\ Q_N(X,T)=\sum _{k=0}^{N(N+1)}{\mathbf{\text{q}}}_k\left(T\right)X^k\end{array}$$

The analytic expression of the Peregrine breather of order 7 is obtained as a particular case when a=0 and b=0. Because of the length of the coefficients of the polynomials G _N , H _N , and Q _N , we present them in Additional file1.

Conclusions

We presented here for the first time to my knowledge the analytical expressions of the Peregrine breather of order 7 and its associated deformations. The conjecture about the form of the breather of order N in coordinates (x,t) is checked, also that the maximum of amplitude equal to 2N+1; the degree of the polynomials in x and t is quite equal to N(N+1) as that presented in[8]. For a and b nonzero, the maximum is less than that, as discussed above and seen in Figures2 and3. This new formulation gives a powerful novel method to obtain the nonsingular solutions of the NLS equation. We will present Peregrine breathers of a higher order in other publications. We can also imagine to obtain deformations with 2N−2 parameters for Peregrine breathers of order N. Some advances were made in this direction recently[24–26].

Also, we explained why, in the case of order N=7, when one or the other of the parameters a or b (even both) tend towards infinite in the case of Peregrine breather of order N, it appears in the center of the figures’ Peregrine breather of order N−2. This phenomenon was checked for the orders of N=3 until N=7. It would be important to prove this conjecture in the general case.