Multiple-soliton solutions, soliton-type solutions and rational solutions for the $$\varvec{(3+1)}$$ ( 3 + 1 ) -dimensional generalized shallow water equation in oceans, estuaries and impoundments

2016 ◽  
Vol 86 (1) ◽  
pp. 667-675 ◽  
Author(s):  
Zhi-Fang Zeng ◽  
Jian-Guo Liu ◽  
Bin Nie
Keyword(s):  
Shallow Water ◽  
Shallow Water Equation ◽  
Soliton Solutions ◽  
Rational Solutions ◽  
Multiple Soliton Solutions

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 Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Yaning Tang ◽  
Pengpeng Su
Keyword(s):  
Shallow Water ◽  
Shallow Water Equation ◽  
Sufficient Conditions ◽  
Wronskian Technique ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Wronskian Determinant ◽  
Linear Partial Differential Equations ◽  
Hirota Bilinear Equation ◽  
Hirota Bilinear

Based on the Hirota bilinear method and Wronskian technique, two different classes of sufficient conditions consisting of linear partial differential equations system are presented, which guarantee that the Wronskian determinant is a solution to the corresponding Hirota bilinear equation of a (3+1)-dimensional generalized shallow water equation. Our results show that the nonlinear equation possesses rich and diverse exact solutions such as rational solutions, solitons, negatons, and positons.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
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.

scholarly journals On solitons: Propagation of shallow water waves for the fifth-order KdV hierarchy integrable equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 828-842
Author(s):  
Aly R. Seadawy ◽  
Shafiq U. Rehman ◽  
Muhammad Younis ◽  
Syed T. R. Rizvi ◽  
Ali Althobaiti
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Sound Propagation ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Capillary Waves ◽  
Nonlinear Phenomena ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
Fifth Order

Abstract This article studies the fifth-order KdV (5KdV) hierarchy integrable equation, which arises naturally in the modeling of numerous wave phenomena such as the propagation of shallow water waves over a flat surface, gravity–capillary waves, and magneto-sound propagation in plasma. Two innovative integration norms, namely, the G ′ G 2 \left(\frac{{G}^{^{\prime} }}{{G}^{2}}\right) -expansion and ansatz approaches, are used to secure the exact soliton solutions of the 5KdV type equations in the shapes of hyperbolic, singular, singular periodic, shock, shock-singular, solitary wave, and rational solutions. The constraint conditions of the achieved solutions are also presented. Besides, by selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in three-dimensional, two-dimensional, and contour graphs. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex nonlinear phenomena.

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

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.

Sign in / Sign up

Export Citation Format

Share Document