Traveling Wave Solutions to Riesz Time-Fractional Camassa–Holm Equation in Modeling for Shallow-Water Waves

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
S. Saha Ray ◽  
S. Sahoo
Keyword(s):  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Present Method ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Analysis Method ◽  
Shallow Water Waves ◽  
New Approach ◽  
Homotopy Analysis

In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics, viz., Riesz time-fractional Camassa–Holm (CH) equation by modified homotopy analysis method (MHAM). As a result, new types of solutions are obtained. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. The main aim of this paper is to employ a new approach, which enables us successful and efficient derivation of the analytical solutions for the Riesz time-fractional CH equation.

Breather degeneration and lump superposition for the (3 + 1)-dimensional nonlinear evolution equation

2021 ◽  
pp. 2150250X
Author(s):  
Wei Tan ◽  
Miao Li
Keyword(s):  
Evolution Equation ◽  
Exponential Type ◽  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Nonlinear Phenomena ◽  
Degradation Behavior ◽  
Shallow Water Waves ◽  
Perturbation Parameters ◽  
Logarithmic Type

This paper is devoted to the study of a (3 + 1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. The breather solutions with different structures are obtained based on the bilinear form with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions, and we also study general lump soliton, lumpoff solution and superposition phenomenon between lump soliton and breather solution. Besides, some theorems about the superposition between lump soliton and [Formula: see text]-soliton ([Formula: see text] is a nonnegative integer) are given. Some examples, including lump-[Formula: see text]-exponential type, lump-[Formula: see text]-logarithmic type, higher-order lump-type [Formula: see text]-soliton, are given to illustrate the correctness of the theorems and corollaries described. Finally, some novel nonlinear phenomena, such as emergence of lump soliton, degeneration of breathers, fission and fusion of lumpoff, superposition of lump-[Formula: see text]-solitons, etc., are analyzed and simulated.

scholarly journals On the Bivariate Spectral Homotopy Analysis Method Approach for Solving Nonlinear Evolution Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. S. Motsa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Analysis Method ◽  
Partial Differential ◽  
New Approach ◽  
Homotopy Analysis ◽  
Spectral Homotopy Analysis Method

This paper presents a new application of the homotopy analysis method (HAM) for solving evolution equations described in terms of nonlinear partial differential equations (PDEs). The new approach, termed bivariate spectral homotopy analysis method (BISHAM), is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.

scholarly journals Degeneration of Solitons for a the (3+1)-dimensional Generalized Nonlinear Evolution Equation for the Shallow Water Waves

2021 ◽  
Author(s):  
longxing li ◽  
Long-Xing Li
Keyword(s):  
Evolution Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Shallow Water Waves ◽  
Hirota Bilinear Form ◽  
Wave Limit

Abstract A the (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves is investigated with different methods. Based on symbolic computation and Hirota bilinear form, Nsoliton solutions are constructed. In the process of degeneration of N-soliton solutions, T-breathers are derived by taking complexication method. Then rogue waves will emerge during the degeneration of breathers by taking the parameter limit method. Through full degeneration of N-soliton, M-lump solutions are derived based on long wave limit approach. In addition, we also find out that the partial degeneration of N-soliton process can generate the hybrid solutions composed of soliton, breather and lump.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

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