# Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters

## Abstract

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the (*x*, *t*) plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending on *t*. This formulation gives in the case of the order 3 and 4, solutions with, respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others.

## Keywords

NLS equation Peregrine breathers Rogue waves## Introduction

The rogue waves phenomenon currently exceed the strict framework of the study of ocean’s waves [1, 2, 3, 4] and play a significant role in other fields; in nonlinear optics [5, 6], Bose–Einstein condensate [7], superfluid helium [8], atmosphere [9], plasmas [10], capillary phenomena [11] and even finance [12]. In the following, we consider the one-dimensional focusing nonlinear equation of Schrödinger (NLS) to describe the phenomena of rogue waves. The first results concerning the NLS equation date from the Seventies. Precisely, in 1972 Zakharov and Shabat solved it using the inverse scattering method [13, 14]. The case of periodic and almost periodic algebro-geometric solutions to the focusing NLS equation was first constructed in 1976 by Its and Kotlyarov [15, 16]. The first quasi rational solutions of NLS equation were constructed in 1983 by Peregrine [17]. In 1986 Akhmediev, Eleonskii and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [18, 19, 20]. Other analogues of the Peregrine breathers of order 3 and 4 were constructed in a series of articles by Akhmediev et al. [21, 22, 23] using Darboux transformations. The present paper presents multi-parametric families of quasi rational solutions of NLS of order *N* in terms of determinants of order 2*N* dependent on \(2N-2\) real parameters. The aim of this paper was to try to distinguish among all the possible configurations obtained by different choices of parameters, one those which have a characteristic in order to try to give a classification of these solutions.

## Expression of solu-tions of NLS equation in terms of a ratio of two determinants

*N*which we call solution of order

*N*depending on \(2N-2\) real parameters. It is given in the following result [24, 25, 26, 27]:

### **Theorem 1**

*The functions v defined by*

*are quasi-rational solution of the NLS Eq.*(1)

*depending on*\(2N-2\)

*parameters*\(\tilde{a}_{j}\), \(\tilde{b}_{j}\), \(1\le j \le N-1,\)

*where*

*The functions f and g are defined in*(9), (10), (11), (12).

## Patterns of quasi rational solutions to the NLS equation

*N*depending on \(2N-2\) parameters \(\tilde{a}_{j}\), \(\tilde{b}_{j}\) (for \(1\le j \le N-1\)) have been already explicitly constructed and can be written as

*x*;

*t*).

### Patterns of quasi-rational solutions of order 3 with 4 parameters

#### \(P_{3}\) breather

#### Triangles

#### Rings

#### Arcs

### Patterns of quasi rational solutions of order 4 with 6 parameters

#### \(P_{4}\) breather

#### Triangles

#### Rings

#### Arcs

*b*), we obtain arc configuration with 10 peaks.

^{1}

#### Triangles inside rings

*b*), we obtain ring with inside triangle.

## Conclusion

We have presented here patterns of modulus of solutions to the NLS focusing equation in the (*x*, *t*) plane. This study can be useful at the same time for hydrodynamics as well for nonlinear optics; many applications in these fields have been realized, as it can be seen in recent works of Chabchoub et al. [30] or Kibler et al. [31]. This study tries to bring all possible types of patterns of quasi-rational solutions to the NLS equation. We see that we can obtain \(2^{N-1}\) different structures at the order *N*. Parameters *a* or *b* give the same type of structure. For \(a_{1}\ne 0\) (and other parameters equal to 0), we obtain triangular rogue wave; for \(a_{j}\ne 0\) (\(j\ne 1\) and other parameters equal to 0) we get ring rogue wave; in the other choices of parameters, we get in particular arc structures (or claw structure). This type of study has been realized in preceding works. Akhmediev et al, study the order \(N=2\) in [32], \(N=3\) in [33]; the case \(N=4\) was studied in particular (\(N=5,\, 6\) were also studied) in [34, 35] showing triangle and arc patterns; only one type of ring was presented. The extrapolation was done until the order \(N=9\) in [36]. Ohta and Yang [37] presented the study of the case cas \(N=3\) with rings and triangles. Recently, Ling and Zhao [38] presented the cases \(N=2,\, 3, \,4\) with rings, triangle and also claw structures.

In the present study, one sees appearing richer structures, in particular the appearance of a triangle of 3 peaks inside a ring of 7 peaks in the case of order \(N=4\); to the best of my knowledge, it is the first time that this configuration for order 4 is presented. In this way, we try to bring a better understanding to the hierarchy of NLS rogue wave solutions. It will be relevant to go on this study with higher orders.

## Footnotes

- 1.
In the following notations 2

*A*4 / 3*I*, I meaning Reversed.

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