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.

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.

Dynamics of localized waves in a (3+1)-dimensional nonlinear evolution equation

2019 ◽  
Vol 33 (09) ◽  
pp. 1950101 ◽  
Author(s):  
Yunfei Yue ◽  
Yong Chen
Keyword(s):  
Evolution Equation ◽  
Soliton Solution ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Dark Soliton ◽  
Interaction Solutions ◽  
Localized Waves

In this paper, a (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equation is studied via the Hirota method. Soliton, lump, breather and rogue wave, as four types of localized waves, are derived. The obtained N-soliton solutions are dark solitons with some constrained parameters. General breathers, line breathers, two-order breathers, interaction solutions between the dark soliton and general breather or line breather are constructed by choosing suitable parameters on the soliton solution. By the long wave limit method on the soliton solution, some new lump and rogue wave solutions are obtained. In particular, dark lumps, interaction solutions between dark soliton and dark lump, two dark lumps are exhibited. In addition, three types of solutions related with rogue waves are also exhibited including line rogue wave, two-order line rogue waves, interaction solutions between dark soliton and dark lump or line rogue wave.

scholarly journals Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

2021 ◽  
Author(s):  
Hongcai Ma ◽  
Shupan Yue ◽  
Yidan Gao ◽  
Aiping Deng
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bilinear Method ◽  
Test Function ◽  
Lump Solutions ◽  
Hirota Bilinear ◽  
Test Function Method

Abstract Exact solutions of a new (2+1)-dimensional nonlinear evolution equation are studied. Through the Hirota bilinear method, the test function method and the improved tanh-coth and tah-cot method, with the assisstance of symbolic operations, one can obtain the lump solutions, multi lump solutions and more soliton solutions. Finally, by determining different parameters, we draw the three-dimensional plots and density plots at different times.

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