scholarly journals A variety of exact analytical solutions of extended shallow water wave equations via improved (G’/G) -expansion method

2017 ◽  
Vol 5 (1) ◽  
pp. 21 ◽  
Author(s):  
Faisal Hawlader ◽  
Dipankar Kumar
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Wave Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Arbitrary Parameter ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Shallow Water Wave Equations

In this present work, we have established exact solutions for (2+1) and (3+1) dimensional extended shallow-water wave equations in-volving parameters by applying the improved (G’/G) -expansion method. Abundant traveling wave solutions with arbitrary parameter are successfully obtained by this method, and these wave solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The improved (G’/G) -expansion method is simple and powerful mathematical technique for constructing traveling wave, solitary wave, and periodic wave solutions of the nonlinear evaluation equations which arise from application in engineering and any other applied sciences. We also present the 3D graphical description of the obtained solutions for different cases with the aid of MAPLE 17.

Exact Traveling Wave Solutions and Bifurcations of Classical and Modified Serre Shallow Water Wave Equations

2019 ◽  
Vol 29 (12) ◽  
pp. 1950153
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Wave Equations ◽  
Wave Theory ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Shallow Water Wave Equations

Using the dynamical systems analysis and singular traveling wave theory developed by Li and Chen [2007] to the classical and modified Serre shallow water wave equations, it is shown that, in different regions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) can be obtained. More than 28 explicit and exact parametric representations are precisely derived. It is demonstrated that, more interestingly, the modified Serre equation has uncountably infinitely many smooth solitary wave solutions and uncountably infinitely many pseudo-peakon solutions. Moreover, it is found that, differing from the well-known peakon solution of the Camassa–Holm equation, the modified Serre equation has four new forms of peakon solutions.

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

2020 ◽  
pp. 72-86
Author(s):  
Md. Mamunur Rashid ◽  
Whida Khatun
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Gordon Equation ◽  
Expansion Method ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Shallow Water Wave Equations ◽  
Klein Gordon

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.

scholarly journals SOLITON SOLUTIONS OF SHALLOW WATER WAVE EQUATIONS BY MEANS OF G'/G EXPANSION METHOD

2014 ◽  
Vol 4 (3) ◽  
pp. 221-229 ◽  
Author(s):  
Marwan Alquran ◽  
◽  
Aminah Qawasmeh
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

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

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

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.

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

2004 ◽  
Vol 16 (1) ◽  
pp. 167-178 ◽  
Author(s):  
Chongsheng Cao ◽  
Darryl D. Holm ◽  
Edriss S. Titi
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
One Dimensional ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Wave Models
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