A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

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A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

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New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

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.

Abundant solitary wave solutions for the fractional coupled Jaulent–Miodek equations arising in applied physics

This article explores the abundant solitary wave solutions of the conformable coupled Jaulent–Miodek (JM) equations appearing in applied physics. The aforesaid coupled equations belong to the family of shallow-water wave equations. Two recent modified integration schemes are used for the first time to produce a novel solitary wave, trigonometric and other solutions with some free parameters in the conformable derivative sense. In particular, the modified Kudryashov and [Formula: see text]-expansion schemes are used to illustrate the wave propagations through aforesaid solutions of the JM equations. Furthermore, a comparison is made with some recent results and the dynamics of the obtained solutions are displayed for the reader via soft computation. The outcomes reveal that the methods are effective and provide a direct way of finding novel solutions.

Solitary Wave Solutions for the Shallow Water Wave Equations and the Generalized Klein-Gordon Equation Using Exp(-ϕ(η) )-Expansion Method

In this article, we investigate the exact and solitary wave solutions for the shallow water wave equations and the generalized Klein-Gordon equation using the exp -expansion method. A wave transformation is applied to convert the problem into the form of an ordinary differential equation. By using this method, we found the explicit solitary wave solutions in terms of the hyperbolic functions, trigonometric functions, exponential functions and rational functions. The extracted solution plays a significant role in many physical phenomena such as electromagnetic waves, nonlinear lattice waves, ion sound waves in plasma, nuclear physics, shallow water waves and so on. It is noted that the method is reliable, straightforward and an effective mathematical tool for analytic treatment of nonlinear systems of partial differential equation in mathematical physics and engineering.

On New Solutions of System of Shallow Water Wave Equations

Obtaining analytical solutions of nonlinear differential equations and nonlinear systems of partial and ordinary differential equations is an important topic in various fields of Mathematics. Many techniques are available in the literature. In this note, the enhanced modified simple equation method (EMSEM) is applied to system of shallow water wave equations.