scholarly journals New Lax Pairs and Darboux Transformation and Its Application to a Shallow Water Wave Model of Generalized KdV Type

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Huizhang Yang ◽  
Weiguo Rui
Keyword(s):  
Shallow Water ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Water Wave ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Wave Model ◽  
Lax Pairs ◽  
Shallow Water Wave ◽  
Generalized Kdv Equation

New Lax pairs of a shallow water wave model of generalized KdV equation type are presented. According to this Lax pair, we constructed a new spectral problem. By using this spectral problem, we constructed Darboux transformation with the help of a gauge transformation. Applying this Darboux transformation, some new exact solutions including double-soliton solution of the shallow water wave model of generalized KdV equation type are obtained. In order to visually show dynamical behaviors of these double soliton solutions, we plot graphs of profiles of them and discuss their dynamical properties.

Multiple soliton, fusion, breather, lump, mixed kink-lump and periodic solutions to the extended shallow water wave model in (2+1)-dimensions

2020 ◽  
pp. 2150138
Author(s):  
Hajar F. Ismael ◽  
Aly Seadawy ◽  
Hasan Bulut
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Soliton Solutions ◽  
Quadratic Function ◽  
Wave Model ◽  
Simple Method ◽  
Shallow Water Wave ◽  
Hirota Bilinear Form ◽  
The One ◽  
Physical Phenomena

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, [Formula: see text]-soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model.

Solitons and integrability for a (2+1)-dimensional generalized variable-coefficient shallow water wave equation

2017 ◽  
Vol 31 (03) ◽  
pp. 1750012 ◽  
Author(s):  
Ya-Le Wang ◽  
Yi-Tian Gao ◽  
Shu-Liang Jia ◽  
Zhong-Zhou Lan ◽  
Gao-Fu Deng ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Shallow Water Wave ◽  
Binary Bell Polynomials

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg–de Vries (KdV) equation and the Calogero–Bogoyavlenskii–Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

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 Diversity of Interaction Solutions of a Shallow Water Wave Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Jian-Ping Yu ◽  
Wen-Xiu Ma ◽  
Bo Ren ◽  
Yong-Li Sun ◽  
Chaudry Masood Khalique
Keyword(s):  
Fluid Dynamics ◽  
Wave Equation ◽  
General Theory ◽  
Shallow Water ◽  
Water Wave ◽  
Direct Method ◽  
Soliton Solutions ◽  
Interaction Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions

In this paper, we study the diversity of interaction solutions of a shallow water wave equation, the generalized Hirota–Satsuma–Ito (gHSI) equation. Using the Hirota direct method, we establish a general theory for the diversity of interaction solutions, which can be applied to generate many important solutions, such as lumps and lump-soliton solutions. This is an interesting feature of this research. In addition, we prove this new model is integrable in Painlevé sense. Finally, the diversity of interactive wave solutions of the gHSI is graphically displayed by selecting specific parameters. All the obtained results can be applied to the research of fluid dynamics.

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