PERIODIC WAVES AND THEIR LIMITS FOR THE CAMASSA–HOLM EQUATION

2006 ◽  
Vol 16 (08) ◽  
pp. 2261-2274 ◽  
Author(s):  
ZHENGRONG LIU ◽  
ALI MOHAMMED KAYED ◽  
CAN CHEN
Keyword(s):  
Periodic Wave ◽  
Periodic Waves ◽  
Implicit Functions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Holm Equation ◽  
Implicit Expressions ◽  
Periodic Cusp Waves ◽  
Theoretical Results

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation [Formula: see text] We investigate the periodic wave solutions of form u = φ(ξ) which satisfy φ(ξ + T) = φ(ξ), here ξ = x - ct and c, T are constants. Their six implicit expressions and two explicit expressions are obtained. We point out that when the initial values are changed, the periodic waves may become three waves, periodic cusp waves, smooth solitary waves and peakons. Our results give an explanation to the appearance of periodic cusp waves and peakons. Moreover, three sets of graphs of the implicit functions are drawn, and three sets of numerical simulations are displayed. The identity of these graphs and simulations imply the correctness of our theoretical results.

scholarly journals Peakon, Solitary and Smooth Periodic Wave Solutions for a Modification of theK(2,2)Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Periodic Wave ◽  
Point Of View ◽  
Phase Portraits ◽  
Periodic Waves ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We consider a modification of theK(2,2)equationut=2uuxxx+2kuxuxx+2uuxusing the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, some peakons, solitary, and smooth periodic waves are found and their exact parametric representations are presented. Also, the coexistence of peakon and solitary wave solutions is investigated.

scholarly journals EXACT PERIODIC WAVE SOLUTIONS OF A SINGULAR INTEGRABLE EQUATION

2011 ◽  
Vol 16 (1) ◽  
pp. 315-325
Author(s):  
Shaolong Xie ◽  
Bin Gao
Keyword(s):  
Dynamical Systems ◽  
Periodic Wave ◽  
Integrable Equation ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Fixed Parameter ◽  
Periodic Cusp Waves

In this paper, theory of dynamical systems is employed to investigate periodic waves of a singular integrable equation. These periodic waves contain smooth periodic waves, periodic cusp waves and periodic cusp loop waves. Under fixed parameter conditions, their exact parametric expressions are given.

scholarly journals Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhengyong Ouyang
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Periodic Waves ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.

scholarly journals Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Blow Up ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Bbm Equation

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary wave solutions.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

scholarly journals Numerical calculation of N -periodic wave solutions to coupled KdV–Toda-type equations

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200752
Author(s):  
Yingnan Zhang ◽  
Xingbiao Hu ◽  
Jianqing Sun
Keyword(s):  
Toda Lattice ◽  
Kdv Equation ◽  
Direct Method ◽  
Periodic Wave ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Numerical Process ◽  
Relativistic Toda Lattice ◽  
Ito Equation

In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N  = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

scholarly journals Traveling Wave Solutions for the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

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 Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical System ◽  
Numerical Simulation ◽  
Solitary Wave ◽  
Planar Graphs ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Periodic Cusp Waves

We employ the approaches of both dynamical system and numerical simulation to investigate a generalized KdV equation, which is presented by Yin (2012). Some peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions are obtained, and the planar graphs of the compactons and the periodic cusp waves are simulated.

scholarly journals On periodic waves of the nonlinear systems

1998 ◽  
Vol 20 (4) ◽  
pp. 11-19
Author(s):  
Le Xuan Can
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Electric Cables ◽  
Necessary And Sufficient

The paper is concerned with the solvability and approximate solution of the nonlinear partial differential equation describing the periodic wave propagation. Necessary and sufficient conditions for the existence of the periodic wave solutions are obtained. An approximate method for solving the equation is presented. As an illustrative example, the equation of periodic waves of the electric cables is considered.

Sign in / Sign up

Export Citation Format

Share Document