scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

scholarly journals Some Smooth and Nonsmooth Traveling Wave Solutions for KP-MEW(2, 2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Liping He ◽  
Yuanhua Lin ◽  
Hongying Zhu
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Integral Constant ◽  
Wave Solutions ◽  
Planar Dynamical Systems

In this paper, we consider the KP-MEW(2, 2) equation by the theory of bifurcations of planar dynamical systems when integral constant is considered. The periodic peakon solution and peakon and smooth periodic solutions are given.

scholarly journals Traveling Wave Solutions of a Generalized Zakharov-Kuznetsov Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Wenbin Zhang ◽  
Jiangbo Zhou
Keyword(s):  
Dynamical System ◽  
Solitary Wave ◽  
Periodic Solutions ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Planar Dynamical System ◽  
Wave Solutions

We employ the bifurcation theory of planar dynamical system to investigate the traveling-wave solutions of the generalized Zakharov-Kuznetsov equation. Four important types of traveling wave solutions are obtained, which include the solitary wave solutions, periodic solutions, kink solutions, and antikink solutions.

ON NONLINEAR WAVE EQUATIONS WITH BREAKING LOOP-SOLUTIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 519-537 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Level Curves ◽  
Straight Line ◽  
Fixed Parameter ◽  
Unstable Manifolds

Using analytic methods from the dynamical systems theory, some new nonlinear wave equations are investigated, which have exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions. It is shown that these parametric representations are associated with some families of open level-curves of traveling wave systems corresponding to such nonlinear wave equations, each of which lies in an area bounded by a singular straight line and the stable and the unstable manifolds of a saddle point of such a system.

Analysis of Straight-Line Motion of Towed Ships

1991 ◽  
Vol 35 (04) ◽  
pp. 304-313
Author(s):  
Fotis Andrea Papoulias
Keyword(s):  
Dynamic Response ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Loss Of Stability ◽  
Nonlinear Analyses ◽  
Surface Ship ◽  
Numerical Integrations ◽  
Straight Line ◽  
Linear And Nonlinear ◽  
Dynamic Loss

The problem of dynamic loss of stability in steady towing of a surface ship is considered. The two coordinates of the towing point and the towline length are the main bifurcation parameters. Bifurcation theory techniques are used in order to compute equilibrium and periodic solutions. The results are confirmed by numerical integrations. It is shown that both linear and nonlinear analyses are required to thoroughly understand, predict, and evaluate the system dynamic response.

Smooth Exact Traveling Wave Solutions Determined by Singular Nonlinear Traveling Wave Systems: Two Models

2019 ◽  
Vol 29 (04) ◽  
pp. 1950047
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Shengfu Deng
Keyword(s):  
Traveling Wave ◽  
Singular System ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Regular System ◽  
Wave System ◽  
Optical Metamaterials ◽  
Wave Solutions ◽  
Straight Line ◽  
Kink Wave

For a singular nonlinear traveling wave system of the first class, if there exist two node points of the associated regular system in the singular straight line, then the dynamics of the solutions of the singular system will be very complex. In this paper, two representative nonlinear traveling wave system models (namely, the traveling wave system of Green–Naghdi equations and the traveling wave system of the Raman soliton model for optical metamaterials) are investigated. It is shown that, if there exist two node points of the associated regular system in the singular straight line, then the singular system has no peakon, periodic peakon and compacton solutions, but rather, it has smooth periodic wave, solitary wave and kink wave solutions.

Index Evaluation for Dynamical Systems and Its Application to Locating All the Zeros of a Vector Function

1983 ◽  
Vol 50 (4a) ◽  
pp. 858-862 ◽  
Author(s):  
C. S. Hsu ◽  
R. S. Guttalu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Vector Function ◽  
Evaluation Method ◽  
Point Mapping ◽  
Strongly Nonlinear ◽  
Index Evaluation

An index evaluation method is discussed in this paper. It can also serve as the basis of a procedure to locate all the zeros of a vector function. An application of the procedure is made to a strongly nonlinear point-mapping dynamical system in order to locate all the periodic solutions of period one and period two, 41 in total number.

Phase Space Reconstruction

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Statistical Mechanics ◽  
State Space ◽  
Phase Plane ◽  
Material Point ◽  
The Other ◽  
Straight Line ◽  
Mathematical Construction

In this chapter we introduce an important concept concerning the study of both discrete and continuous dynamical systems, the concept of phase space or “state space”. It is an abstract mathematical construction with important applications in statistical mechanics, to represent the time evolution of a dynamical system in geometric shape. This space has as many dimensions as the number of variables needed to define the instantaneous state of the system. For instance, the state of a material point moving on a straight line is defined by its position and velocity at each instant, so that the phase space for this system is a plane in which one axis is the position and the other one the velocity. In this case, the phase space is also called “phase plane”. It is later applied in many chapters of the book.

Peakons of a Generalized Camassa-Holm Equation with Bifurcation Theory of Dynamical System

2012 ◽  
Vol 252 ◽  
pp. 36-39 ◽  
Author(s):  
Min Sun ◽  
Jing Li ◽  
Ting Ting Quan
Keyword(s):  
Dynamical System ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Solitary Wave Solution ◽  
Qualitative Theory ◽  
Bifurcation Method ◽  
Holm Equation ◽  
Linear Transform ◽  
Parameter Condition

In this paper, the peakons and bifurcations in a generalized Camassa-Holm equation are studied by using the bifurcation method and qualitative theory of dynamical systems. First, the averaged equation is obtained by introducing linear transform and traveling wave transform to the generalized Camassa-Holm equation. Then, we applied the bifurcation theory of planar dynamical system and maple software to investigate the averaged equation. The phase portrait of the system under a parameter condition is obtained. Finally, we get the peakons from the limit of general single solitary wave solution.

scholarly journals Existence of random invariant periodic curves via random semiuniform ergodic theorem

2016 ◽  
Vol 17 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Kenneth Uda
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Dissipative Structure ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Compact Set

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system on a cylinder [Formula: see text] has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

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