Search results for "Peregrine breather"
showing 7 items of 17 documents
Solutions to the NLS equation : differential relations and their different representations
2020
Solutions to the focusing nonlinear Schrödinger equation (NLS) of order N depending on 2N − 2 real parameters in terms of wronskians and Fredholm determinants are given. These solutions give families of quasirational solutions to the NLS equation denoted by vN and have been explicitly constructed until order N = 13. These solutions appear as deformations of the Peregrine breather PN as they can be obtained when all parameters are equal to 0. These quasi rational solutions can be expressed as a quotient of two polynomials of degree N (N + 1) in the variables x and t and the maximum of the modulus of the Peregrine breather of order N is equal to 2N + 1. Here we give some relations between sol…
Families of solutions of order nine to the NLS equation with sixteen parameters
2015
We construct new deformations of the Peregrine breather (P9) of order 9 with 16 real parameters. With this method, we obtain explicitly new families of quasi-rational solutions to the NLS equation in terms of a product of an exponential depending on t by a ratio of two polynomials of degree 90 in x and t; when all the parameters are equal to 0, we recover the classical P9 breather. We construct new patterns of different types of rogue waves as triangular configurations of 45 peaks as well as rings and concentric rings.
Patterns of deformations of P 3 and P 4 breathers solutions to the NLS equation
2016
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.
Determinant representation of NLS equation, Ninth Peregrine breather and multi-rogue waves
2012
This article 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 its deformations. Here we construct Peregrine breathers of order $N=9$ and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.
Tenth Peregrine breather solution of the NLS equation.
2012
We go on in this paper, in the study of the solutions of the focusing NLS equation. With a new representation given in a preceding paper, a very compact formulation without limit as a quotient of two determinants, we construct the Peregrine breather of order N=10. The explicit analytical expression of the Akhmediev's solution is completely given.
Tenth order solutions to the NLS equation with eighteen parameters
2015
We present here new solutions of the focusing one dimensional non linear Schrödinger equation which appear as deformations of the Peregrine breather of order 10 with 18 real parameters. With this method, we obtain new families of quasi-rational solutions of the NLS equation, and we obtain explicit quotients of polynomial of degree 110 in x and t by a product of an exponential depending on t. We construct new patterns of different types of rogue waves and recover the triangular configurations as well as rings and concentric as found for the lower orders.
The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation.
2013
We construct here explicitly new deformations of the Peregrine breather of order 5 with 8 real parameters. This gives new families of quasi-rational solutions of the NLS equation and thus one can describe in a more precise way the phenomena of appearance of multi rogue waves. With this method, we construct new patterns of different types of rogue waves. We get at the same time, the triangular configurations as well as rings isolated. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings.