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.

scholarly journals Abundant Symmetry-Breaking Solutions of the Nonlocal Alice–Bob Benjamin–Ono System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wang Shen ◽  
Zheng-Yi Ma ◽  
Jin-Xi Fei ◽  
Quan-Yong Zhu ◽  
Jun-Chao Chen
Keyword(s):  
Symmetry Breaking ◽  
Gravity Waves ◽  
Bilinear Form ◽  
Time Reversal ◽  
Internal Gravity Waves ◽  
Explicit Solutions ◽  
Time Reversal Symmetry ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

The Benjamin–Ono equation is a useful model to describe the long internal gravity waves in deep stratified fluids. In this paper, the nonlocal Alice–Bob Benjamin–Ono system is induced via the parity and time-reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry-breaking soliton, breather, and lump solutions for this system are obtained through the derived Hirota bilinear form. By taking suitable constants in the involved ansatz functions, abundant fascinating symmetry-breaking structures of the related explicit solutions are shown.

Lump solutions to the BKP equation by symbolic computation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640028 ◽  
Author(s):  
Jing-Yun Yang ◽  
Wen-Xiu Ma
Keyword(s):  
Symbolic Computation ◽  
Kp Equation ◽  
Lump Solutions ◽  
Positivity Condition ◽  
Independent Variables ◽  
Computation Process ◽  
Starting Point ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Bkp Equation

Lump solutions are rationally localized in all directions in the space. A general class of lump solutions to the (2+1)-dimensional B-Kadomtsev–Petviashvili (BKP) equation is presented through symbolic computation with Maple. The Hirota bilinear form of the equation is the starting point in the computation process. Like the KP equation, the resulting lump solutions contain six arbitrary parameters. Two of the parameters are due to the translation invariances of the BKP equation with the independent variables, and the other four need to satisfy a nonzero determinant condition and the positivity condition, which guarantee analyticity and rational localization of the solutions.

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.

Non-traveling lump solutions and mixed lump–kink solutions to (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation

2019 ◽  
Vol 33 (22) ◽  
pp. 1950262 ◽  
Author(s):  
Jing Wang ◽  
Hong-Li An ◽  
Biao Li
Keyword(s):  
Variable Coefficient ◽  
Quadratic Function ◽  
Lump Solution ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Hyperbolic Cosine ◽  
Hirota Bilinear Form ◽  
Dimensional Variable ◽  
Hyperbolic Cosine Function ◽  
Hirota Bilinear

Through Hirota bilinear form and symbolic computation with Maple, we investigate some non-traveling lump and mixed lump–kink solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Caudrey–Doddy–Gibbon–Kotera–Sawada equation by an extended method. Firstly, the non-traveling lump solutions are directly obtained by taking the function [Formula: see text] as a quadratic function. Secondly, we can get the interaction solutions for a lump solution and one kink solution by taking [Formula: see text] as a combination of quadratic function and exponential function. Finally, the interaction solutions between a lump solution and a pair of kinks solution can be derived by taking [Formula: see text] as a combination of quadratic function and hyperbolic cosine function. The dynamic phenomena of the above three types of exact solutions are demonstrated by some figures.

Lump solution and lump-type solution to a class of mathematical physics equation

2020 ◽  
Vol 34 (10) ◽  
pp. 2050096
Author(s):  
Yanfang Sun ◽  
Jinting Ha ◽  
Huiqun Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Three Dimensional ◽  
Mathematical Physics ◽  
Lump Solution ◽  
Explicit Solutions ◽  
Type Solution ◽  
Hirota Bilinear Form ◽  
Dynamical Features ◽  
Hirota Bilinear

Based on the Hirota bilinear form, lump-type and lump solutions to a class of mathematical physics equation are explored. Specific examples are discussed to show the richness of the considered partial differential equation. In addition, a few of the analyses and three-dimensional plots of some explicit solutions are made to show the dynamical features in detail.

scholarly journals Exact Solutions to a Generalized Bogoyavlensky-Konopelchenko Equation via Maple Symbolic Computations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Shou-Ting Chen ◽  
Wen-Xiu Ma
Keyword(s):  
Nonlinear Equation ◽  
Traveling Wave ◽  
Computer Algebra System ◽  
Traveling Wave Solutions ◽  
Explicit Solutions ◽  
Symbolic Computations ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Bilinear Equation

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky-Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two-wave solutions, and polynomial solutions.

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.

Dynamics of abundant solutions to the generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2020 ◽  
pp. 2150110
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Soliton Solutions ◽  
Point Of View ◽  
Complex Conjugate ◽  
Bell Polynomial ◽  
Localized Solutions ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

The present paper is devoted to discussion of (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation from point of view of their multi-soliton solutions and localized solutions associated with multi-soliton solutions. Firstly, taking advantage of the Bell-polynomial approach, we construct the Hirota bilinear form of (1.1). Based on that, the multi-soliton solutions are also singled out. Subsequently, the (3+1)-dimensional gBKP equation is also found to allow fruitful localized solutions, including breather, lump, rogue wave, and hybrid solutions. These results obtained in this work adequately illustrate the effectiveness of the long wave limit method and complex conjugate technique, which are expected to be employed to obtain more abundant exact solutions.

scholarly journals Controllable symmetry breaking solutions for a nonlocal Boussinesq system

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Jinxi Fei ◽  
Zhengyi Ma ◽  
Weiping Cao
Keyword(s):  
Symmetry Breaking ◽  
Boussinesq Equation ◽  
Rogue Wave ◽  
Symmetry Transformation ◽  
Boussinesq System ◽  
Residual Symmetry ◽  
Hirota Bilinear Form ◽  
Wave Symmetry ◽  
Generalized Boussinesq Equation ◽  
Hirota Bilinear

AbstractThe generalized Boussinesq equation is a useful model to describe the water wave. In this paper, with the coupled Alice-Bob (AB) systems, the nonlocal Boussinesq system can be obtained via the parity and time reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry breaking rogue wave, symmetry breaking soliton and symmetry breaking breather solutions for a nonlocal Boussinesq system are obtained through the derived Hirota bilinear form. The residual symmetry and finite symmetry transformation of the nonlocal AB-Boussinesq system are also studied.

scholarly journals Evolution Behaviour of Kink Breathers and Lump-M-Solitons (M → ∞) for the (3+1)-Dimensional Hirota-Satsuma-Ito-Like Equation

2021 ◽  
Author(s):  
Long-Xing Li
Keyword(s):  
Exponential Type ◽  
Bilinear Form ◽  
Nonlinear Dynamic ◽  
Wave Solution ◽  
Three Dimensional ◽  
Lump Solutions ◽  
Hirota Bilinear Form ◽  
The Given ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

Abstract In this paper, some novel lump solutions and interaction phenomenon between lump and kink M-soliton are investigated. Firstly, we study the evolution and degeneration behaviour of kink breather wave solution with difffferent forms for the (3+1)-dimensional Hirota-Satsuma-Ito-like equation by symbolic computation and Hirota bilinear form. In the process of degeneration of breather waves, some novel lump solutions are derived by the limit method. In addition, M-fifissionable soliton and the interaction phenomenon between lump solutions and kink M-solitons (lump-M-solitons) are investigated, the theorem and corollary about the conditions for the existence of the interaction phenomenon are given and proved further. The lump-M-solitons with difffferent types is studied to illustrate the correctness and availability of the given theorem and corollary, such as lump-cos type, lump-cosh-exponential type, lump cosh-cos-cosh type. Several three-dimensional fifigures are drawn to better depict the nonlinear dynamic behaviours including the oscillation of breather wave, the emergence of lump, the evolution behaviour of fission and fusion of lump-M-solitons and so on.

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