Breather molecules and localized interaction solutions in the (2+1)-dimensional BLMP equation

2021 ◽  
Author(s):  
Jiaxin Qi ◽  
Hongli An ◽  
Peng Jin
Keyword(s):  
Interaction Solutions

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

N-lump and interaction solutions of localized waves to the (2+1)-dimensional generalized KDKK equation

2021 ◽  
pp. 104312
Author(s):  
Xuejun Zhou ◽  
Onur Alp Ilhan ◽  
Jalil Manafian ◽  
Gurpreet Singh ◽  
Nalbiy Salikhovich Tuguz
Keyword(s):  
Interaction Solutions ◽  
Localized Waves

scholarly journals Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1268
Author(s):  
Yarong Xia ◽  
Ruoxia Yao ◽  
Xiangpeng Xin ◽  
Yan Li
Keyword(s):  
Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Three Dimensional ◽  
Dynamical Evolution ◽  
Symmetry Transformation ◽  
Nonlocal Symmetry ◽  
Similarity Reduction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Dependent Variables

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

2021 ◽  
pp. 2150277
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Dynamic Behavior ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Bilinear Transformation ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Ytsf Equation ◽  
Line Soliton

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

Localized interaction solution and its dynamics of the extended Hirota–Satsuma–Ito equation

2021 ◽  
pp. 2150313
Author(s):  
Jian-Ping Yu ◽  
Wen-Xiu Ma ◽  
Chaudry Masood Khalique ◽  
Yong-Li Sun
Keyword(s):  
Numerical Simulations ◽  
Rogue Wave ◽  
The Other ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Physical Phenomena ◽  
Wave Phenomena ◽  
Hirota Bilinear ◽  
Ito Equation

In this research, we will introduce and study the localized interaction solutions and th eir dynamics of the extended Hirota–Satsuma–Ito equation (HSIe), which plays a key role in studying certain complex physical phenomena. By using the Hirota bilinear method, the lump-type solutions will be firstly constructed, which are almost rationally localized in all spatial directions. Then, three kinds of localized interaction solutions will be obtained, respectively. In order to study the dynamic behaviors, numerical simulations are performed. Two interesting physical phenomena are found: one is the fission and fusion phenomena happening during the procedure of their collisions; the other is the rogue wave phenomena triggered by the interaction between a lump-type wave and a soliton wave.

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