Characteristics of lump-kink and their fission-fusion interactions, and rogue and breather wave solutions for a (3+1)-dimensional generalized shallow water equation

AbstractShallow water wave equation has increasing use in many applications for its success in eliminating spurious oscillation, and has been widely studied. In this paper, we investigate (3+1)-dimensional generalized shallow water equation system. Based on the $(G'/G)$-expansion method and the variable separation method, we choose $\xi (x,y,z,t) = f(y + cz) + ax + h(t)$ and suppose that ${a_i}(i = 1,2, \ldots,m)$ is an undetermined function about $x,y,z,t$ instead of a constant in eq. (3), which are different from those in previous literatures. With the aid of symbolic computation, we obtain a family of exact solutions of the (3+1)-dimensional generalized shallow water equation system in forms of the hyperbolic functions and the trigonometric functions. When the parameters take special values, in addition to traveling wave solutions, we also get the nontraveling wave solutions by using our method; these obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves. The $(G'/G)$-expansion method is a very general and powerful tool that will lead to further insights and improvements of the nonlinear models.

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Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation

Advances in Mathematical Physics ◽

10.1155/2015/291804 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Jingzhu Wu ◽

Xiuzhi Xing ◽

Xianguo Geng

Keyword(s):

Shallow Water ◽

Differential Operators ◽

Shallow Water Equation ◽

Asymptotic Properties ◽

Soliton Solutions ◽

Periodic Wave ◽

Periodic Wave Solution ◽

Bell Polynomials ◽

Periodic Wave Solutions ◽

Wave Solutions

The relations betweenDp-operators and multidimensional binary Bell polynomials are explored and applied to construct the bilinear forms withDp-operators of nonlinear equations directly and quickly. Exact periodic wave solution of a (3+1)-dimensional generalized shallow water equation is obtained with the help of theDp-operators and a general Riemann theta function in terms of the Hirota method, which illustrate that bilinearDp-operators can provide a method for seeking exact periodic solutions of nonlinear integrable equations. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

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Abundant exact analytical solutions and novel interaction phenomena of the generalized (3+1)-dimensional shallow water equation

Thermal Science ◽

10.2298/tsci191123103w ◽

2021 ◽

pp. 103-103

Author(s):

Xiaomin Wang ◽

Sudao Bilig ◽

Yueyang Feng

Keyword(s):

Shallow Water ◽

Analytical Solutions ◽

Shallow Water Equation ◽

Solution Process ◽

Soliton Solutions ◽

High Order ◽

Bilinear Method ◽

Exact Analytical Solutions ◽

Wave Solutions ◽

Interaction Phenomena

This paper reveals abundant exact analytical solutions to the generalized (3+1)-dimensional shallow water equation. The generalized bilinear method is used in the solution process and the obtained solutions include the high-order lump-type solutions, the three-wave solutions, the breather solutions. The interaction between the high-order lump-type solutions and the soliton solutions is also elucidated. These solutions have greatly enriched the generalized (3+1)-dimensional shallow water equation in open literature.

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Exact Traveling Wave Solutions and Bifurcations for a Shallow Water Equation Modeling Surface Waves of Moderate Amplitude

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501728 ◽

2016 ◽

Vol 26 (10) ◽

pp. 1650172

Author(s):

Wenjing Zhu ◽

Jibin Li

Keyword(s):

Shallow Water ◽

Traveling Wave ◽

Shallow Water Equation ◽

Traveling Wave Solutions ◽

Periodic Wave ◽

Phase Portraits ◽

Equation Modeling ◽

Wave System ◽

Wave Solutions ◽

Straight Line

In this paper, we consider the traveling wave solutions for a shallow water equation. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and explicit exact parametric representations for solitary wave solutions and smooth periodic wave solutions, as well as periodic peakon solutions. We show the existence of compacton solutions of the equation under different parameter conditions.

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Stability of solitary waves for a generalized higher-order Shallow water equation

Filomat ◽

10.2298/fil1405007d ◽

2014 ◽

Vol 28 (5) ◽

pp. 1007-1017 ◽

Cited By ~ 1

Author(s):

Nurhan Dündar ◽

Necat Polat

Keyword(s):

Solitary Wave ◽

Shallow Water ◽

Solitary Waves ◽

Shallow Water Equation ◽

Higher Order ◽

Solitary Wave Solutions ◽

Wave Solutions ◽

Existence And Stability ◽

Stability Of Solitary Waves

In this work, we consider solitary wave solutions of a generalized higher-order shallow water equation. We investigate the existence and stability of solitary waves of the equation.

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