Integrability of an extended (2+1)-dimensional shallow water wave equation with Bell polynomials

2013 ◽  
Vol 22 (5) ◽  
pp. 050509 ◽  
Author(s):  
Yun-Hu Wang ◽  
Yong Chen
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Bell Polynomials ◽  
Shallow Water Wave

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.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

scholarly journals On a shallow water wave equation

Nonlinearity ◽  
1994 ◽  
Vol 7 (3) ◽  
pp. 975-1000 ◽  
Author(s):  
P A Clarkson ◽  
E L Mansfield
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave

scholarly journals The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ying Wang ◽  
YunXi Guo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Sufficient Conditions ◽  
Global Weak Solution ◽  
Dissipative Term ◽  
Shallow Water Wave ◽  
Global Strong Solution ◽  
Special Cases ◽  
The Cauchy Problem

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that(1-∂x2)u0∈M+(R),u0∈H1(R),andu0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.

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