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.

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.

scholarly journals New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 137-144 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Bo Xu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Shallow Water Waves ◽  
Bilinear Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

Sign in / Sign up

Export Citation Format

Share Document