scholarly journals A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Jie Li ◽  
Chaudry Masood Khalique
Keyword(s):  
Three Dimensional ◽  
Lump Solution ◽  
Lump Solutions ◽  
Solution Structures ◽  
New Class ◽  
Contour Plots ◽  
Bilinear Formulation ◽  
Hirota Bilinear ◽  
Interesting Solution ◽  
Ito Equation

The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a special presented lump solution are made to shed light on the characteristic of the resulting lump solutions.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

2021 ◽  
pp. 2150437
Author(s):  
Liyuan Ding ◽  
Wen-Xiu Ma ◽  
Yehui Huang
Keyword(s):  
Symbolic Computation ◽  
Three Dimensional ◽  
Order Derivative ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Dynamical Behaviors ◽  
Contour Plots ◽  
Hirota Bilinear ◽  
Ito Equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

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.

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.

Lump solutions to a (2+1)-dimensional fourth-order nonlinear PDE possessing a Hirota bilinear form

2021 ◽  
pp. 2150160
Author(s):  
Wen-Xiu Ma ◽  
Solomon Manukure ◽  
Hui Wang ◽  
Sumayah Batwa
Keyword(s):  
Nonlinear Equation ◽  
Fourth Order ◽  
Bilinear Form ◽  
Three Dimensional ◽  
Nonlinear Pde ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
Bilinear Formulation ◽  
Order Nonlinear Equation ◽  
Hirota Bilinear

Through the Hirota bilinear formulation, a (2+1)-dimensional combined fourth-order nonlinear equation is proposed, which possesses lump solutions. Two classes of lump solutions are presented explicitly in terms of the coefficients in the combined nonlinear equation. A set of examples of equations is provided to show the diversity of the considered combined nonlinear equation, together with three-dimensional plots, [Formula: see text]-curves and [Formula: see text]-curves of two specific lump solutions in two cases of the combined equation.

Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

2019 ◽  
Vol 20 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jianqing Lü ◽  
Sudao Bilige ◽  
Xiaoqing Gao
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Lump Solution ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Contour Plots ◽  
Special Case ◽  
Interaction Phenomenon

AbstractIn this paper, with the help of symbolic computation system Mathematica, six kinds of lump solutions and two classes of interaction solutions are discussed to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation via using generalized bilinear form with a dependent variable transformation. Particularly, one special case are plotted as illustrative examples, and some contour plots with different determinant values are presented. Simultaneously, we studied the trajectory of the interaction solution.

scholarly journals Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

2021 ◽  
Author(s):  
Hongcai Ma ◽  
Shupan Yue ◽  
Yidan Gao ◽  
Aiping Deng
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bilinear Method ◽  
Test Function ◽  
Lump Solutions ◽  
Hirota Bilinear ◽  
Test Function Method

Abstract Exact solutions of a new (2+1)-dimensional nonlinear evolution equation are studied. Through the Hirota bilinear method, the test function method and the improved tanh-coth and tah-cot method, with the assisstance of symbolic operations, one can obtain the lump solutions, multi lump solutions and more soliton solutions. Finally, by determining different parameters, we draw the three-dimensional plots and density plots at different times.

scholarly journals A Study on Lump and Interaction Solutions to a (3 + 1)-Dimensional Soliton Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Xi-zhong Liu ◽  
Zhi-Mei Lou ◽  
Xian-Min Qian ◽  
Lamine Thiam
Keyword(s):  
Interaction Process ◽  
Lump Solution ◽  
Soliton Equation ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Finite Period ◽  
Bilinear Formulation ◽  
Dimensional Soliton

Based on bilinear formulation of a (3 + 1)-dimensional soliton equation, lump solution and related interaction solutions are investigated. The lump solutions of the soliton equation are classified into three cases with nonsingularity conditions being given. The interaction solutions between lump and a stripe soliton are obtained in eight cases, which have interesting fusing and fission behaviors with changing time. The interaction solutions of the soliton equation between a lump and a resonant pair of stripe solitons are also given, and we find that the lump just exist for a finite period during the interaction process.

Lump-type, breather and interaction solutions to the (3+1)-dimensional generalized KdV-type equation

2020 ◽  
Vol 34 (29) ◽  
pp. 2050329
Author(s):  
Pengfei Han ◽  
Taogetusang
Keyword(s):  
Free Choice ◽  
Three Dimensional ◽  
Type Equation ◽  
Type Model ◽  
Physical Explanation ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Contour Plots ◽  
Dimensional Soliton ◽  
Hirota Bilinear

The [Formula: see text]-dimensional generalized Korteweg-de Vries (KdV)-type model equation is investigated based on the Hirota bilinear method. Diversity of exact solutions for this equation are obtained with the help of symbolic computation. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting three-dimensional plots and contour plots. The obtained results are useful in gaining the understanding of high dimensional soliton-like structures equation related to mathematical physics branches, natural sciences and engineering areas.

scholarly journals Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-5 ◽  
Author(s):  
Yanni Zhang ◽  
Jing Pang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Three Dimensional ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Contour Plots ◽  
Dimensional Variable ◽  
Hirota Bilinear

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.

Lump solutions to nonlinear PDEs involving Hirota derivative Dt2DxDy

2020 ◽  
Vol 34 (18) ◽  
pp. 2050197
Author(s):  
Fudong Wang ◽  
Wen Xiu Ma
Keyword(s):  
Nonlinear Pde ◽  
Nonlinear Pdes ◽  
Bilinear Method ◽  
Lump Solutions ◽  
New Class ◽  
Derivative Term ◽  
Pde Systems ◽  
Derivatives Of ◽  
Hirota Bilinear ◽  
Temporal Variable

This paper aims to study lump solutions to a class of (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear PDE systems, which involve the fourth-order Hirota derivative term: [Formula: see text]. This Hirota derivative term generates higher-order derivatives of the temporal variable. Lump solutions to the resulting new class of nonlinear PDE systems are studied in detail via the Hirota bilinear method.

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