scholarly journals PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION

2017 ◽  
Vol 7 (1) ◽  
pp. 278-290
Author(s):  
Shaolong Xie ◽  
◽  
Xiaochun Hong ◽  
Tao Jiang ◽  
◽  
...  
Keyword(s):  
Dynamical System ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Wave Solutions ◽  
Holm Equation

scholarly journals Propagation of stochastic travelling waves of cooperative systems with noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hao Wen ◽  
Jianhua Huang ◽  
Yuhong Li
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Wave Speed ◽  
Travelling Waves ◽  
Equilibrium Points ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Cooperative System ◽  
Wave Solutions ◽  
Suitable Marker

<p style='text-indent:20px;'>We consider the cooperative system driven by a multiplicative It\^o type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

scholarly journals Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Bin He ◽  
Qing Meng ◽  
Jinhua Zhang
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component

By using the bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation:mt=-3mux-mxu+kρρx,  ρt=-ρxu+2ρux,the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.

scholarly journals Periodic Loop Solutions and Their Limit Forms for the Kudryashov-Sinelshchikov Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Bin He ◽  
Qing Meng ◽  
Jinhua Zhang ◽  
Yao Long
Keyword(s):  
Dynamical Systems ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

The Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. We show that the limit forms of periodic loop solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions. Also, some new exact travelling wave solutions are presented through some special phase orbits.

scholarly journals Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hang Zheng ◽  
Yonghui Xia ◽  
Yuzhen Bai ◽  
Guo Lei
Keyword(s):  
Dynamical System ◽  
Equilibrium Points ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
System Method ◽  
Dynamical System Method ◽  
Nonzero Equilibrium ◽  
Derivative Order

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

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