Solitary and periodic wave solutions of nonlinear wave equations via the functional variable method

2018 ◽  
Vol 21 (1) ◽  
pp. 43-57 ◽  
Author(s):  
Kamruzzaman Khan ◽  
M. Ali Akbar
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Periodic Wave ◽  
Variable Method ◽  
Nonlinear Wave Equations ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Functional Variable ◽  
Functional Variable Method

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

Functional variable method to study nonlinear evolution equations

2013 ◽  
Vol 3 (3) ◽  
Author(s):  
Mostafa Eslami ◽  
Mohammad Mirzazadeh
Keyword(s):  
Exact Solutions ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Variable Method ◽  
Nonlinear Wave Equations ◽  
Wave Solutions ◽  
Functional Variable ◽  
Functional Variable Method

AbstractThe functional variable method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method presents a wider applicability for handling nonlinear wave equations. In this paper, the functional variable method is used to construct exact solutions of Davey-Stewartson equation, generalized Zakharov equation, K(m, n) equation with generalized evolution term, (2 + 1)-dimensional long-wave-short-wave resonance interaction equation and nonlinear Schrödinger equation with power law nonlinearity. The obtained solutions include solitary wave solutions, periodic wave solutions.

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.

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.

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.

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