scholarly journals Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

2021 ◽  
Author(s):  
Gao-Fu Deng ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Ting-Ting Jia ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Hybrid Solutions ◽  
The One ◽  
Hybrid Waves ◽  
Positive Integers

Abstract Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2+1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H -soliton, M -breather and J -lump solutions have been constructed via the modified Pfaffian technique, where H , M and J are the positive integers. One-breather solutions with a real function ϕ ( y ) are derived, where y is the scaled space variable, we notice that ϕ ( y ) influences the shapes of the background planes. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not affected by one soliton after interaction. One-lump solutions with ϕ ( y ) are obtained with the condition, where k 1 R and k 1 I are the real constants, we notice that the one lump consists of two low valleys and one high peak, as well as the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ ( y ) is changed from a linear function to a periodic function.

scholarly journals High-Order Breather Solutions, Lump Solutions, and Hybrid Solutions of a Reduced Generalized (3 + 1)-Dimensional Shallow Water Wave Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jing Wang ◽  
Biao Li
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
High Order ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Moving Path

We investigate a reduced generalized (3 + 1)-dimensional shallow water wave equation, which can be used to describe the nonlinear dynamic behavior in physics. By employing Bell’s polynomials, the bilinear form of the equation is derived in a very natural way. Based on Hirota’s bilinear method, the expression of N-soliton wave solutions is derived. By using the resulting N-soliton expression and reasonable constraining parameters, we concisely construct the high-order breather solutions, which have periodicity in x,y-plane. By taking a long-wave limit of the breather solutions, we have obtained the high-order lump solutions and derived the moving path of lumps. Moreover, we provide the hybrid solutions which mean different types of combinations in lump(s) and line wave. In order to better understand these solutions, the dynamic phenomena of the above breather solutions, lump solutions, and hybrid solutions are demonstrated by some figures.

Hybrid waves for a (2 + 1)-dimensional extended shallow water wave equation

2021 ◽  
Vol 33 (11) ◽  
pp. 117120
Author(s):  
Gao-Fu Deng ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Ting-Ting Jia ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave ◽  
Hybrid Waves

scholarly journals Lump solutions to the (2+1)-dimensional shallow water wave equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1765-1769 ◽  
Author(s):  
Hong-Cai Ma ◽  
Ke Ni ◽  
Aiping Deng
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Symbolic Computation ◽  
Linear Form ◽  
Water Wave ◽  
The Other ◽  
Translation Invariance ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Free Parameters

Through symbolic computation with MAPLE, a class of lump solutions to the (2+1)-D shallow water wave equation is presented, making use of its Hirota bi-linear form. The resulting lump solutions contain six free parameters, two of which are due to the translation invariance of the (2+1)-D shallow water wave equation and the other four of which satisfy a non-zero determinant condition guaranteeing analyticity and rational localization of the solutions.

scholarly journals Camassa-Holm equation on shallow water wave equation

2017 ◽  
Vol 5 (12) ◽  
pp. 7758-7764
Author(s):  
Sh. Hajrulla, L. Bezati, F. Hoxha
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Weak Solution ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Kdv Equation ◽  
Shallow Water Wave ◽  
Holm Equation ◽  
The Cauchy Problem

In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption  Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution  to the Cauchy problem for  Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

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 On a shallow water wave equation

Nonlinearity ◽  
1994 ◽  
Vol 7 (3) ◽  
pp. 975-1000 ◽  
Author(s):  
P A Clarkson ◽  
E L Mansfield
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave
Sign in / Sign up

Export Citation Format

Share Document