Analytical soliton solutions to the generalized (3+1)-dimensional shallow water wave equation

2021 ◽  
Author(s):  
Sachin Kumar ◽  
Dharmendra Kumar
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Evolutionary Dynamics ◽  
Evolution Equations ◽  
Soliton Solutions ◽  
Physical Property ◽  
Nonlinear Evolution Equations ◽  
Ocean Engineering ◽  
Shallow Water Wave ◽  
Wave Solutions

In this paper, the soliton solutions and dynamical wave structures for the generalized (3+1)-dimensional shallow water wave (SWW) equation, which is an important physical property in ocean engineering and hydrodynamics, are presented. The generalized exponential rational function (GERF) method is used to investigate the closed-form wave solutions of the generalized SWW equation, which is used to describe the evolutionary dynamics of SWW. We successfully archive a variety of soliton solutions such as exponential solutions, kink wave solutions, non-topological solutions, periodic singular solutions, and topological solutions. These newly established results are also important for understanding the wave-propagation and dynamics of exact solutions of the equation, which is of great significance in physical oceanography and chemical oceanography. Eventually, it is shown that the proposed GERF technique is effective, robust, and straightforward and is also used to solve other types of higher-dimensional nonlinear evolution equations. In our work, we have used Mathematica extensively for such complicated algebraic calculations.

Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics

2016 ◽  
Vol 131 (5) ◽  
Author(s):  
Mohammad Mirzazadeh ◽  
Mehmet Ekici ◽  
Abdullah Sonmezoglu ◽  
Sami Ortakaya ◽  
Mostafa Eslami ◽  
...  
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Wave Dynamics ◽  
Shallow Water Wave

scholarly journals Diversity of Interaction Solutions of a Shallow Water Wave Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Jian-Ping Yu ◽  
Wen-Xiu Ma ◽  
Bo Ren ◽  
Yong-Li Sun ◽  
Chaudry Masood Khalique
Keyword(s):  
Fluid Dynamics ◽  
Wave Equation ◽  
General Theory ◽  
Shallow Water ◽  
Water Wave ◽  
Direct Method ◽  
Soliton Solutions ◽  
Interaction Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions

In this paper, we study the diversity of interaction solutions of a shallow water wave equation, the generalized Hirota–Satsuma–Ito (gHSI) equation. Using the Hirota direct method, we establish a general theory for the diversity of interaction solutions, which can be applied to generate many important solutions, such as lumps and lump-soliton solutions. This is an interesting feature of this research. In addition, we prove this new model is integrable in Painlevé sense. Finally, the diversity of interactive wave solutions of the gHSI is graphically displayed by selecting specific parameters. All the obtained results can be applied to the research of fluid dynamics.

The lump, lumpoff and rouge wave solutions of a (3+1)-dimensional generalized shallow water wave equation

2019 ◽  
Vol 33 (17) ◽  
pp. 1950190
Author(s):  
Jin-Jie Yang ◽  
Shou-Fu Tian ◽  
Wei-Qi Peng ◽  
Zhi-Qiang Li ◽  
Tian-Tian Zhang
Keyword(s):  
Shallow Water ◽  
Bilinear Form ◽  
Water Wave ◽  
Rational Solution ◽  
Bell Polynomials ◽  
Ocean Engineering ◽  
Shallow Water Wave ◽  
Propagation Behavior ◽  
Wave Solutions ◽  
Lump Wave

We consider the (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave (GSWW) equation. By virtue of the binary Bell polynomials theory, we obtain the bilinear form of the equation. Then its lump wave solutions, a kind of rational solution localized in all directions of the space, are derived by employing its bilinear form at the special situation for [Formula: see text]. Furthermore, it is worth noting that the lump wave solutions can interact with single-stripe soliton waves and double-stripe solution waves to generate lumpoff waves and a kind of predictable rouge waves, respectively. Especially, it is interesting that we can predict when and where the peculiar rouge waves will occur. Moreover, in order to understand the dynamics and propagation of the lump waves and the interaction solution, some graphic analyses are exhibited by selecting special parameters. The results of this work can be used to understand the propagation behavior of these solutions of the GSWW equation, which is of great significance for ocean engineering.

New Exact Solutions of (2+1)-Dimensional Generalization of Shallow Water Wave Equation by (G′/G)-Expansion Method

2010 ◽  
Vol 20-23 ◽  
pp. 1516-1521 ◽  
Author(s):  
Bang Qing Li ◽  
Mei Ping Xu ◽  
Yu Lan Ma
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Function Solution ◽  
Shallow Water Wave

Extending a symbolic computation algorithm, namely, (G′/G)-expansion method, a new series of exact solutions are constructed for (2+1)-dimensional generalization of shallow water wave equation. These solutions included hyperbolic function solution, trigonometric function solution and rational function solution. The procedure can illustrate that the new algorithm is concise, powerful and straightforward, and it can also be applied to find exact solutions for other high dimensional nonlinear evolution equations.

scholarly journals Some exact solutions to the generalized Korteweg–de Vries equation and the system of shallow water wave equations

2013 ◽  
Vol 18 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Ihsan Timuçin Dolapci ◽  
Ahmet Yıldırım
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Evolution Equations ◽  
Transform Method ◽  
Shallow Water Wave ◽  
Solitary Solutions

In this paper, we establish exact solutions for nonlinear evolution equations in mathematical physics. The exp-transform method is proposed to seek solitary solutions, periodic solutions and compaction-like solutions of nonlinear differential equations. The generalized KdV equation and the system of the shallow water wave equation are chosen to illustrate the effectiveness and convenience of the method.

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.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

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.

Completing the Study of Traveling Wave Solutions for Three Two-Component Shallow Water Wave Models

2020 ◽  
Vol 30 (03) ◽  
pp. 2050036 ◽  
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Two Component ◽  
Wave Models

For three two-component shallow water wave models, from the approach of dynamical systems and the singular traveling wave theory developed in [Li & Chen, 2007], under different parameter conditions, all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are derived. More than 19 explicit exact parametric representations are obtained. Of more interest is that, for the integrable two-component generalization of the Camassa–Holm equation, it is found that its [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions. In addition, its [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions. The new results complete a recent study by Dutykh and Ionescu-Kruse [2016].

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