Search results for "4735Fg"
showing 8 items of 8 documents
2N+1 highest amplitude of the modulus of the N-th order AP breather and other 2N-2 parameters solutions to the NLS equation
2015
We construct here new deformations of the AP breather (Akhmediev-Peregrine breather) of order N (or AP N breather) with 2N −2 real parameters. Other families of quasi-rational solutions of the NLS equation are obtained. We evaluate the highest amplitude of the modulus of AP breather of order N ; we give the proof that the highest amplitude of the AP N breather is equal to 2N + 1. We get new formulas for the solutions of the NLS equation, different from these already given in previous works. New solutions for the order 8 and their deformations according to the parameters are explicitly given. We get the triangular configurations as well as isolated rings at the same time. Moreover, the appea…
Families of solutions to the CKP equation with multi-parameters
2020
We construct solutions to the CKP (cylindrical Kadomtsev-Petviashvili)) equation in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions are called solutions of order N ; they depend on 2N − 1 parameters. They can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters. We explicitly construct the expressions up to order 5 and we study the patterns of their modulus in plane (x, y) and their evolution according to time and parameters.
From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N − 2 parameters
2017
International audience; We have already constructed solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants and wronskians of order 2N. These solutions have been called solutions of order N and they depend on 2N −1 parameters. We construct here N-order rational solutions. We prove that they can be written as a quotient of 2 polynomials of degree 2N(N +1) in x, y and t depending on 2N−2 parameters. We explicitly construct the expressions of the rational solutions of order 4 depending on 6 real parameters and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, b1, b2, b3.
An attempt to classification of the quasi rational solutions to the NLS equation
2015
Based on a representation in terms of determinants of order 2N , an attempt to classification of quasi rational solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) is given and several conjectures about the structure of the solutions are also formulated. These solutions can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N (N + 1) in x and t depending on 2N −2 parameters. It is remarkable to mention that in this representation, when all parameters are equal to 0, we recover the PN breathers.
Families of solutions to the KPI equation and the structure of their rational representations of order N
2018
We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions called solutions of order N depend on 2N − 1 parameters. They can also be written as a quotient of two polynomials of degree 2N (N + 1) in x, y and t depending on 2N − 2 parameters. The maximum of the modulus of these solutions at order N is equal to 2(2N + 1) 2. We explicitly construct the expressions until the order 6 and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters.
8-parameter solutions of fifth order to the Johnson equation
2019
We give different representations of the solutions of the Johnson equation with parameters. First, an expression in terms of Fredholm determinants is given; we give also a representation of the solutions written as a quotient of wronskians of order 2N. These solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polyno-mials of degree 2N (N +1) in x, t and 4N (N +1) in y depending on 2N −2 parameters. Here, we explicitly construct the expressions of the rational solutions of order 5 depending on 8 real parameters and we study the patterns of their modulus in the plane (x, y) and their …
From first to fourth order rational solutions to the Boussinesq equation
2020
Rational solutions to the Boussinesq equation are constructed as a quotient of two polynomials in x and t. For each positive integer N , the numerator is a polynomial of degree N (N + 1) − 2 in x and t, while the denominator is a polynomial of degree N (N + 1) in x and t. So we obtain a hierarchy of rational solutions depending on an integer N called the order of the solution. We construct explicit expressions of these rational solutions for N = 1 to 4.
First and second order rational solutions to the Johnson equation and rogue waves
2018
Rational solutions to the Johnson equation are constructed as a quotient of two polynomials in x, y and t depending on several real parameters. We obtain an infinite hierarchy of rational solutions written in terms of polynomials of degrees 2N (N + 1) in x, and t, 4N (N + 1) in y, depending on 2N − 2 real parameters for each positive integer N. We construct explicit expressions of the solutions in the cases N = 1 and N = 2 which are given in the following. We study the evolution of the solutions by constructing the patterns of their modulus in the (x, y) plane, and this for different values of parameters.