Fission and fusion interaction phenomena of mixed lump kink solutions for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850161 ◽  
Author(s):  
Yaqing Liu ◽  
Xiaoyong Wen
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Water Wave ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Petviashvili Equation ◽  
Kink Soliton ◽  
Interaction Phenomena

In this paper, a generalized (3[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili (gBKP) equation is investigated by using the Hirota’s bilinear method. With the aid of symbolic computation, some new lump, mixed lump kink and periodic lump solutions are derived. Based on the derived solutions, some novel interaction phenomena like the fission and fusion interactions between one lump soliton and one kink soliton, the fission and fusion interactions between one lump soliton and a pair of kink solitons and the interactions between two periodic lump solitons are discussed graphically. Results might be helpful for understanding the propagation of the shallow water wave.

scholarly journals High-Order Breather Solutions, Lump Solutions, and Hybrid Solutions of a Reduced Generalized (3 + 1)-Dimensional Shallow Water Wave Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jing Wang ◽  
Biao Li
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
High Order ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Moving Path

We investigate a reduced generalized (3 + 1)-dimensional shallow water wave equation, which can be used to describe the nonlinear dynamic behavior in physics. By employing Bell’s polynomials, the bilinear form of the equation is derived in a very natural way. Based on Hirota’s bilinear method, the expression of N-soliton wave solutions is derived. By using the resulting N-soliton expression and reasonable constraining parameters, we concisely construct the high-order breather solutions, which have periodicity in x,y-plane. By taking a long-wave limit of the breather solutions, we have obtained the high-order lump solutions and derived the moving path of lumps. Moreover, we provide the hybrid solutions which mean different types of combinations in lump(s) and line wave. In order to better understand these solutions, the dynamic phenomena of the above breather solutions, lump solutions, and hybrid solutions are demonstrated by some figures.

scholarly journals Ablowitz-Kaup-Newel-Segur Formalism and N-Soliton Solutions of Generalized Shallow Water Wave Equation

2018 ◽  
Vol 20 ◽  
pp. 25-35
Author(s):  
Supratim Das
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Transform Method ◽  
Shallow Water Wave

We apply Ablowitz-Kaup-Newel-Segur hierarchy to derive the generalized shallow waterwave equation and we also investigate N-soliton solutions of the derived equation using InverseScattering Transform method and Hirota’s bilinear method.

scholarly journals Lump solutions to the (2+1)-dimensional shallow water wave equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1765-1769 ◽  
Author(s):  
Hong-Cai Ma ◽  
Ke Ni ◽  
Aiping Deng
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Symbolic Computation ◽  
Linear Form ◽  
Water Wave ◽  
The Other ◽  
Translation Invariance ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Free Parameters

Through symbolic computation with MAPLE, a class of lump solutions to the (2+1)-D shallow water wave equation is presented, making use of its Hirota bi-linear form. The resulting lump solutions contain six free parameters, two of which are due to the translation invariance of the (2+1)-D shallow water wave equation and the other four of which satisfy a non-zero determinant condition guaranteeing analyticity and rational localization of the solutions.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

scholarly journals New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 137-144 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Bo Xu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Shallow Water Waves ◽  
Bilinear Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

Lump Interaction Phenomena to a Nonlinear Mathematical Model Arising in Shallow Water Wave

2021 ◽  
Author(s):  
Xiao-Zhong Zhang ◽  
Abdullahi Yusuf ◽  
Tukur Abdulkadir Sulaiman ◽  
Mustafa Inc ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Shallow Water ◽  
Water Wave ◽  
Nonlinear Mathematical Model ◽  
Shallow Water Wave ◽  
Interaction Phenomena

Lump solutions and interaction behaviors to the (2+1)-dimensional extended shallow water wave equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850387 ◽  
Author(s):  
Wenguang Cheng ◽  
Tianzhou Xu
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Dynamic Properties ◽  
Quadratic Function ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Shallow Water Wave ◽  
Double Exponential ◽  
Combination Function ◽  
Bilinear Equation

In this paper, the exact solutions to the (2[Formula: see text]+[Formula: see text]1)-dimensional extended shallow water wave (SWW) equation are investigated by using its bilinear form and ansatz techniques. Following the method given by Ma [Phys. Lett. A 379 (2015) 1975–1978], two classes of lump solutions are constructed by searching for positive quadratic function solutions to the associated bilinear equation. Furthermore, two kinds of interaction solutions between a lump and solitary waves are presented by taking the solution of the associated bilinear equation as a linear combination function of a quadratic function and the double exponential function, one of which is the interaction solution between a lump and an exponentially decayed soliton, and the other one is the interaction solution between a lump and an exponentially decayed twin soliton. Finally, some figures are given to illustrate the dynamic properties of these obtained solutions.

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