scholarly journals Multiparametric Rational Solutions of Order N to the KPI Equation and the Explicit Case of Order 3

2021 ◽  
pp. 58-71
Author(s):  
P. Gaillard
Keyword(s):  
Rational Solutions ◽  
Petviashvili Equation ◽  
Explicit Expressions

We present multiparametric rational solutions to the Kadomtsev-Petviashvili equation (KPI). These solutions of order N depend on 2N − 2 real parameters. Explicit expressions of the solutions at order 3 are given. They can be expressed 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. We study the patterns of their modulus in the (x,y) plane for different values of time t and parameters.

scholarly journals Other Families of Rational Solutions to the KPI Equation

2021 ◽  
pp. 27-34
Author(s):  
Pierre Gaillard
Keyword(s):  
Positive Integer ◽  
Rational Solutions ◽  
Infinite Hierarchy ◽  
Petviashvili Equation ◽  
Two Peaks ◽  
Explicit Expressions

Aims / Objectives: 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. Place and Duration of Study: Institut de math´ematiques de Bourgogne, Universit´e de Bourgogne Franche-Cont´e between January 2020 and January 2021. Conclusion: 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 two peaks in the case of order 2.

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

2018 ◽  
Vol 95 (2) ◽  
pp. 1027-1033 ◽  
Author(s):  
Jian-Guo Liu ◽  
Mostafa Eslami ◽  
Hadi Rezazadeh ◽  
Mohammad Mirzazadeh
Keyword(s):  
Variable Coefficient ◽  
Rational Solutions ◽  
Lump Solutions ◽  
Petviashvili Equation

scholarly journals Weakly Coupled B-Type Kadomtsev-Petviashvili Equation: Lump and Rational Solutions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Na Xiong ◽  
Wen-Tao Li ◽  
Biao Li ◽  
Zine El Abiddine Fellah
Keyword(s):  
Numerical Analysis ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Positive Semidefinite ◽  
Nonlinear Evolution Equations ◽  
Rational Solutions ◽  
Weakly Coupled ◽  
Petviashvili Equation ◽  
Trajectory Equation ◽  
Semidefinite Matrix

Through the method of Z N -KP hierarchy, we propose a new ( 3 + 1 )-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.

Multiple breathers and high-order rational solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2021 ◽  
Author(s):  
Na Liu ◽  
Xinhua Tang ◽  
Weiwei Zhang
Keyword(s):  
Soliton Solutions ◽  
High Order ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Dynamical Characteristics ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Wave Limit ◽  
Hirota Bilinear ◽  
Bkp Equation

This paper is devoted to obtaining the multi-soliton solutions, high-order breather solutions and high-order rational solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation by applying the Hirota bilinear method and the long-wave limit approach. Moreover, the interaction solutions are constructed by choosing appropriate value of parameters, which consist of four waves for lumps, breathers, rouges and solitons. Some dynamical characteristics for the obtained exact solutions are illustrated using figures.

Sign in / Sign up

Export Citation Format

Share Document