HOMOCLINIC MANIFOLDS, CENTER MANIFOLDS AND EXACT SOLUTIONS OF FOUR-DIMENSIONAL TRAVELING WAVE SYSTEMS FOR TWO CLASSES OF NONLINEAR WAVE EQUATIONS

2011 ◽  
Vol 21 (02) ◽  
pp. 527-543 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Wave Equations ◽  
Periodic Wave ◽  
Nonlinear Wave Equations ◽  
Center Manifolds ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Gap Soliton ◽  
Hyperbolic Equilibrium

For the Lax KdV5 equation and the KdV–Sawada–Kotera–Ramani equation, their corresponding four-dimensional traveling wave systems are studied by using Congrove's method. Exact explicit gap soliton, embedded soliton, periodic and quasi-periodic wave solutions are obtained. The existence of homoclinic manifolds to three kinds of equilibria including a hyperbolic equilibrium, a center-saddle and an equilibrium with zero pair of eigenvalues is revealed. The bifurcation conditions of equilibria are given.

Travelling And Periodic Wave Solutions Of Some Nonlinear Wave Equations

2004 ◽  
Vol 59 (7-8) ◽  
pp. 389-396 ◽  
Author(s):  
A. H. Khater ◽  
M. M. Hassan
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Pacs 02.30

We present the mixed dn-sn method for finding periodic wave solutions of some nonlinear wave equations. Introducing an appropriate transformation, we extend this method to a special type of nonlinear equations and construct their solutions, which are not expressible as polynomials in the Jacobi elliptic functions. The obtained solutions include the well known kink-type and bell-type solutions as a limiting cases. Also, some new travelling wave solutions are found. - PACS: 02.30.Jr; 03.40.Kf

The periodic wave solutions for two systems of nonlinear wave equations

2003 ◽  
Vol 12 (12) ◽  
pp. 1341-1348 ◽  
Author(s):  
Wang Ming-Liang ◽  
Wang Yue-Ming ◽  
Zhang Jin-Liang
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Periodic Wave ◽  
Nonlinear Wave Equations ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Systems

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS (II): AN EXAMPLE OF GCKdV EQUATIONS

2009 ◽  
Vol 19 (06) ◽  
pp. 1995-2007 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
XIAOHUA ZHAO
Keyword(s):  
Traveling Wave ◽  
Wave Equations ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Kink Wave ◽  
Coupled Kdv Equations

By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.

BIFURCATIONS OF TRAVELING WAVE AND BREATHER SOLUTIONS OF A GENERAL CLASS OF NONLINEAR WAVE EQUATIONS

2005 ◽  
Vol 15 (09) ◽  
pp. 2913-2926 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Traveling Wave ◽  
General Class ◽  
Wave Equations ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Wave Equations ◽  
Breaking Wave ◽  
Wave Solutions ◽  
Kink Wave

Bifurcations of a general class of traveling wave solutions are analyzed. In particular, the existence of solitary wave, kink and anti-kink wave solutions, and uncountably infinite periodic wave solutions and breather solutions of a general class of traveling wave equations is proved. Also, the existence of breaking wave solution is discussed in detail. Under different parametric conditions, several sufficient conditions for the existence of these solutions are derived. Sufficient simulation results are provided to visualize the theoretical results.

scholarly journals Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type forK(m,n)Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

By using the integral bifurcation method, we study the nonlinearK(m,n)equation for all possible values ofmandn. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions ofK(n,n),K(2n−1,n),K(3n−2,n),K(4n−3,n), andK(m,1)equations are chosen to illustrate with the concrete features.

scholarly journals Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher-Order Nonlinear Wave Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Sixth Order ◽  
Nonlinear Wave Equations ◽  
Lax Equation ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a general sixth-order KdV equation.

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS

2007 ◽  
Vol 17 (11) ◽  
pp. 4049-4065 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Periodic Wave ◽  
Reaction Diffusion Equations ◽  
Nonlinear Wave Equations ◽  
Fundamental Properties ◽  
Wave Solutions ◽  
Kink Wave

The existence of solitary wave, kink wave and periodic wave solutions of a class of singular reaction–diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of traveling wave solutions determined by some bounded orbits of the traveling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

BIFURCATIONS OF TRAVELING WAVE SOLUTIONS FOR FOUR CLASSES OF NONLINEAR WAVE EQUATIONS

2005 ◽  
Vol 15 (12) ◽  
pp. 3973-3998 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Wave Equations ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave

Four large classes of nonlinear wave equations are studied, and the existence of solitary wave, kink and anti-kink wave, and uncountably many periodic wave solutions is proved. The analysis is based on the bifurcation theory of dynamical systems. Under some parametric conditions, various sufficient conditions for the existence of the aforementioned wave solutions are derived. Moreover, all possible exact parametric representations of solitary wave, kink and anti-kink wave, and periodic wave solutions are obtained and classified.

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