The higher-order and multi-lump waves for a (3+1)-dimensional generalized variable-coefficient shallow water wave equation in a fluid

2021 ◽  
Author(s):  
Cong-Cong Hu ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Dan-Yu Yang
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Variable Coefficient ◽  
Higher Order ◽  
Shallow Water Wave

Bilinear form, bilinear Bäcklund transformation and dynamic features of the soliton solutions for a variable-coefficient (3+1)-dimensional generalized shallow water wave equation

2017 ◽  
Vol 31 (22) ◽  
pp. 1750126 ◽  
Author(s):  
Qian-Min Huang ◽  
Yi-Tian Gao
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Bilinear Form ◽  
Water Wave ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Backlund Transformation ◽  
Spectral Parameters ◽  
Shallow Water Wave

Under investigation in this letter is a variable-coefficient (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave equation. Bilinear form and Bäcklund transformation are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Stability of the solitons is studied numerically. Soliton amplitude is determined by the spectral parameters. Soliton velocity is not only related to the spectral parameters, but also to the variable coefficients. Phase shifts are the only difference between the two-soliton solutions and the superposition of the two relevant one-soliton solutions. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

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.

Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation

2016 ◽  
Vol 71 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Zhong-Zhou Lan ◽  
Yi-Tian Gao ◽  
Jin-Wei Yang ◽  
Chuan-Qi Su ◽  
Da-Wei Zuo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Conservation Law ◽  
Water Wave ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Shallow Water Wave

AbstractUnder investigation in this article is a (2+1)-dimensional generalised variable-coefficient shallow water wave equation, which describes the interaction of the Riemann wave propagating along the y axis with a long-wave propagating along the x axis in a fluid, where x and y are the scaled space coordinates. Bilinear forms, Bäcklund transformation, Lax pair, and infinitely many conservation law are derived based on the binary Bell polynomials. Multi-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 multi-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.

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