Lax pair, Bäcklund transformation and conservation laws for the (2+1)-dimensional extended shallow water wave equation

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

Modern Physics Letters B ◽

10.1142/s0217984917501263 ◽

2017 ◽

Vol 31 (22) ◽

pp. 1750126 ◽

Cited By ~ 3

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.

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Group properties and conservation laws for nonlocal shallow water wave equation

Applied Mathematics and Computation ◽

10.1016/j.amc.2011.03.020 ◽

2011 ◽

Vol 218 (3) ◽

pp. 974-979 ◽

Cited By ~ 5

Author(s):

Farshad Rezvan ◽

Emrullah Yaşar ◽

Teoman Özer

Keyword(s):

Wave Equation ◽

Conservation Laws ◽

Shallow Water ◽

Water Wave ◽

Shallow Water Wave

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Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation

Open Physics ◽

10.1515/phys-2018-0049 ◽

2018 ◽

Vol 16 (1) ◽

pp. 364-370 ◽

Cited By ~ 13

Author(s):

Dumitru Baleanu ◽

Mustafa Inc ◽

Abdullahi Yusuf ◽

Aliyu Isa Aliyu

Keyword(s):

Exact Solution ◽

Wave Equation ◽

Conservation Laws ◽

Shallow Water ◽

Water Wave ◽

Airy Function ◽

One Dimensional ◽

Shallow Water Wave ◽

Differential Substitution ◽

Conservation Theorem

AbstractIn this article, the generalized shallow water wave (GSWW) equation is studied from the perspective of one dimensional optimal systems and their conservation laws (Cls). Some reduction and a new exact solution are obtained from known solutions to one dimensional optimal systems. Some of the solutions obtained involve expressions with Bessel function and Airy function [1,2,3]. The GSWW is a nonlinear self-adjoint (NSA) with the suitable differential substitution, Cls are constructed using the new conservation theorem.

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Dynamical solitary interactions between lump waves and different forms of n-solitons (n→∞) for the (2+1)-dimensional shallow water wave equation

Partial Differential Equations in Applied Mathematics ◽

10.1016/j.padiff.2021.100026 ◽

2021 ◽

Vol 3 ◽

pp. 100026

Author(s):

Fahad Sameer Alshammari ◽

Md Fazlul Hoque ◽

Harun-Or-Roshid

Keyword(s):

Wave Equation ◽

Shallow Water ◽

Water Wave ◽

Shallow Water Wave

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

Mathematics ◽

10.3390/math9121439 ◽

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.

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