Search results for "numbers : 33Q55"
showing 10 items of 19 documents
Multi-parameters rational solutions to the mKdV equation
2021
N-order solutions to the modified Korteweg-de Vries (mKdV) equation are given in terms of a quotient of two wronskians of order N depending on 2N real parameters. When one of these parameters goes to 0, we succeed to get for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N real parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 6.
Solutions of the LPD equation and multi-parametric rogue waves
2022
Quasi-rational solutions to the Lakshmanan Porsezian Daniel equation are presented. We construct explicit expressions of these solutions for the first orders depending on real parameters. We study the patterns of these configurations in the (x, t) plane in function of the different parameters. We observe in the case of order 2, three rogue waves which move according to the two parameters. In the case of order 3, six rogue waves are observed with specific configurations moving according to the four parameters.
Other patterns for the first and second order rational solutions to the KPI equation
2022
We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree 2N (N + 1) − 2 in x, y and t by a polynomial of degree 2N (N + 1) in x, y and t, depending on 2N − 2 real parameters for each positive integer N. We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x, y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of…
Multi-parametric families solutions to the Burgers equation
2021
We construct 2N real parameter solutions to the Burgers' equation in terms of determinant of order N and we call these solutions, N order solutions. We deduce general expressions of these solutions in terms of exponentials and study the patterns of these solutions in functions of the parameters for N = 1 until N = 4.
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 …
N order solutions with multi-parameters to the Boussinesq and KP equations and the degenerate rational case
2021
From elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these parameters goes to 0, we get rational solutions as a quotient of a polynomial of degree N (N + 1) − 2 in x and t, by a polynomial of degree N (N + 1) in x and t for each positive integer N depending on 3N parameters. We restrict ourself to give the explicit expressions of these rational solutions for N = 1 until N = 3 to shortened the paper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from 1 to 3.
N-order rational solutions to the Johnson equation depending on 2N - 2 parameter
2017
International audience; We construct rational solutions of order N depending on 2N-2 parameters. They can be written as a quotient of 2 polynomials of degree 2N(N+1) in x, t and 2N(N+1) in y 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.
Solutions to the Gardner equation with multiparameters and the rational case
2022
We construct solutions to the Gardner equation in terms of trigonometric and hyperbolic functions, depending on several real parameters. Using a passage to the limit when one of these parameters goes to 0, we get, for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 3. We easily deduce solutions to the mKdV equation in terms of wronskians as well as rational solutions depending on 2N real parameters.
From particular polynomials to rational solutions to the KPI equation
2022
We construct solutions to the Kadomtsev-Petviashvili equation (KPI) from particular polynomials. We obtain rational solutions written as a second derivative with respect to the variable x of a logarithm of a determinant of order n. So we get with this method an infinite hierarchy of rational solutions to the KPI equation. We give explicitly the expressions of these solutions for the first five orders.
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…