Lump and rational solutions for weakly coupled generalized Kadomtsev–Petviashvili equation

2021 ◽  
pp. 2150449
Author(s):  
Hongyu Wu ◽  
Jinxi Fei ◽  
Wenxiu Ma
Keyword(s):  
Wave Crest ◽  
Rational Solutions ◽  
Kp Hierarchy ◽  
Lump Solutions ◽  
Soliton Theory ◽  
Weakly Coupled ◽  
Petviashvili Equation ◽  
Dimensional Equation ◽  
Trajectory Equation ◽  
Hirota Bilinear

Through the [Formula: see text]-KP hierarchy, we present a new (3+1)-dimensional equation called weakly coupled generalized Kadomtsev–Petviashvili (wc-gKP) equation. Based on Hirota bilinear differential equations, we get rational solutions to wc-gKP equation, and further we obtain lump solutions by searching for a symmetric positive semi-definite matrix. We do some numerical analysis on the trajectory of rational solutions and fit the trajectory equation of wave crest. Some graphics are illustrated to describe the properties of rational solutions and lump solutions. The method used in this paper to get lump solutions by constructing a symmetric positive semi-definite matrix can be applied to other integrable equations as well. The results expand the understanding of lump and rational solutions in soliton theory.

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.

Solitary wave solutions, lump solutions and interactional solutions to the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation

2020 ◽  
Vol 34 (07) ◽  
pp. 2050053
Author(s):  
Min Gao ◽  
Hai-Qiang Zhang
Keyword(s):  
Solitary Wave ◽  
Solitary Wave Solution ◽  
Lump Solution ◽  
Solitary Wave Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Dimensional Equation ◽  
Hirota Bilinear ◽  
Bkp Equation

In this paper, we investigate a [Formula: see text]-dimensional B-type Kadomtsev-Petviashvili (BKP) equation, which is a generalization of the [Formula: see text]-dimensional equation. Based on the Hirota bilinear method and the limit technique of long wave, we systematically construct a family of exact solutions of BKP equation including the [Formula: see text]-solitary wave solution, lump solution as well as interaction solution between lump waves and solitary waves.

A (2+1)-dimensional shallow water equation and its explicit lump solutions

2019 ◽  
Vol 33 (07) ◽  
pp. 1950038 ◽  
Author(s):  
Solomon Manukure ◽  
Yuan Zhou
Keyword(s):  
Shallow Water Equation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Contour Plots ◽  
Dimensional Equation ◽  
New Equation ◽  
Necessary And Sufficient ◽  
Hirota Bilinear

We introduce a new (2+1)-dimensional equation by modifying the potential form of the Calogero–Bogoyavlenskii–Schiff (CBS) equation. By applying the Hirota bilinear method, we construct explicit lump solutions to this new equation and establish necessary and sufficient conditions that guarantee that the solutions are analytic and rationally localized in all directions in space. We also depict the evolution of the profiles of some selected lump solutions with three-dimensional and contour plots. It is immediately observed that the lump solutions generated are solitary wave type solutions as is the case with the KP equation.

Rational solutions and lump solutions to the (3+1)-dimensional Mel’nikov equation

2019 ◽  
Vol 34 (03) ◽  
pp. 2050033 ◽  
Author(s):  
Xuelin Yong ◽  
Xiaoyu Li ◽  
Yehui Huang ◽  
Wen-Xiu Ma ◽  
Yong Liu
Keyword(s):  
Three Dimensional ◽  
Explicit Representation ◽  
Hirota Bilinear Method ◽  
Matrix Elements ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
First Order ◽  
Special Value ◽  
Hirota Bilinear

In this paper, explicit representation of general rational solutions for the (3[Formula: see text]+[Formula: see text]1)-dimensional Mel’nikov equation is derived by employing the Hirota bilinear method. It is obtained in terms of determinants whose matrix elements satisfy some differential and difference relations. By selecting special value of the parameters involved, the first-order and second-order lump solutions are given and their dynamic characteristics are illustrated by two- and three-dimensional figures.

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 Abundant lump solutions and interaction solutions of a (3+1)-D Kadomtsev-Petviashvili equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2437-2445 ◽  
Author(s):  
Xiaoqing Gao ◽  
Sudao Bilige ◽  
Jianqing Lü ◽  
Yuexing Bai ◽  
Runfa Zhang ◽  
...  
Keyword(s):  
Lump Solution ◽  
Solution Properties ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Contour Plots ◽  
Hirota Bilinear

In this paper, abundant lump solutions and two types of interaction solutions of the (3+1)-D Kadomtsev-Petviashvili equation are obtained by the Hirota bilinear method. Some contour plots with different determinant values are sequentially given to show that the corresponding lump solution tends to zero when the deter-minant approaches to zero. The interaction solutions with special parameters are plotted to elucidate the solution properties.

Study of lump and lump-kink solitons of a coupled reduced Hirota bilinear equation

2020 ◽  
Vol 34 (22) ◽  
pp. 2050224
Author(s):  
Shun Wang ◽  
Chuanzhong Li ◽  
Zhenli Wang
Keyword(s):  
Sufficient Conditions ◽  
Quadratic Function ◽  
Quadratic Functions ◽  
Bulk Solution ◽  
Lump Solutions ◽  
Weakly Coupled ◽  
Hirota Bilinear Equation ◽  
Necessary And Sufficient ◽  
Hirota Bilinear ◽  
Bilinear Equation

By symbolic computation and searching for the solutions of the positive quadratic functions of the related bilinear equations, two kinds of lump solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional weakly coupled Hirota bilinear equation are derived, and the practicability of this method is verified. Then we add an exponential function to the original positive quadratic function, and obtain a new solution of the Hirota bilinear equation. The interaction between the lump solutions and lump-kink solutions is included in the new solution. On this basis, we give the possibility of adding multiple exponential functions. Finally, we give the coupled reduced Hirota bilinear equation lump-kink solitons by combining the above two methods. In order to ensure the analyticity and reasonable localization of the block, two sets of necessary and sufficient conditions are given for the parameters involved in the solution. The local characteristics and energy distribution of bulk solution are analyzed and explained.

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.

scholarly journals Lump solutions for the dimensionally reduced variable coefficient B-type Kadomtsev-Petviashvili equation

2021 ◽  
pp. 39-39
Author(s):  
Yanni Zhang ◽  
Xin Zhao ◽  
Jing Pang
Keyword(s):  
Nonlinear Equations ◽  
Present Method ◽  
Variable Coefficient ◽  
Solution Process ◽  
Solution Properties ◽  
Lump Solutions ◽  
Petviashvili Equation ◽  
Bilinear Formulation ◽  
Hirota Bilinear

Based on Hirota bilinear formulation, the lump solutions to dimensionally reduced generalized variable coefficient B-type Kadomtsev-Petviashvili equation are obtained. The solution process is figured out and the solution properties are illustrated graphically. The present method can be extended to other nonlinear equations.

scholarly journals Symmetry Breaking Soliton, Breather, and Lump Solutions of a Nonlocal Kadomtsev–Petviashvili System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Hong-Yu Wu ◽  
Jin-Xi Fei ◽  
Zheng-Yi Ma ◽  
Jun-Chao Chen ◽  
Wen-Xiu Ma
Keyword(s):  
Symmetry Breaking ◽  
Nonlinear Waves ◽  
Time Reversal ◽  
Space Variable ◽  
Dispersive Media ◽  
Explicit Solutions ◽  
Lump Solutions ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

The Kadomtsev–Petviashvili equation is one of the well-studied models of nonlinear waves in dispersive media and in multicomponent plasmas. In this paper, the coupled Alice-Bob system of the Kadomtsev–Petviashvili equation is first constructed via the parity with a shift of the space variable x and time reversal with a delay. By introducing an extended Bäcklund transformation, symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions for this system are presented through the established Hirota bilinear form. According to the corresponding constants in the involved ansatz function, a few fascinating symmetry breaking structures of the presented explicit solutions are shown.

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