scholarly journals Stability and Instability of Traveling Wave Solutions to Nonlinear Wave Equations

2021 ◽  
Author(s):  
John Anderson ◽  
Samuel Zbarsky
Keyword(s):  
Plane Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Linear Instability ◽  
Nonlinear Wave Equations ◽  
Null Condition ◽  
Semilinear Systems ◽  
Wave Solutions ◽  
Stability And Instability ◽  
The Stability

Abstract In this paper, we study the stability and instability of plane wave solutions to semilinear systems of wave equations satisfying the null condition. We identify a condition that allows us to prove the global nonlinear asymptotic stability of the plane wave. The proof of global stability requires us to analyze the geometry of the interaction between the background plane wave and the perturbation. When this condition is not met, we are able to prove linear instability assuming an additional genericity condition. The linear instability is shown using a geometric optics ansatz.

EXACT SOLUTIONS AND THEIR DYNAMICS OF TRAVELING WAVES IN THREE TYPICAL NONLINEAR WAVE EQUATIONS

2009 ◽  
Vol 19 (07) ◽  
pp. 2249-2266 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Nonlinear Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Wave Equations ◽  
Visual Illusion ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Dynamical Analysis ◽  
Nonlinear Wave Equations ◽  
Wave Solutions

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

scholarly journals Instability of periodic traveling waves for the symmetric regularized long wave equation

2015 ◽  
Vol 219 ◽  
pp. 235-268
Author(s):  
Jaime Angulo Pava ◽  
Carlos Alberto Banquet Brango
Keyword(s):  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Linear Instability ◽  
Nonlinear Instability ◽  
Periodic Traveling Waves ◽  
Wave Solutions ◽  
Long Wave ◽  
Linearized Problem ◽  
Periodic Traveling Wave ◽  
The Stability

AbstractWe prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

scholarly journals Traveling Wave Solutions of the Benjamin-Bona-Mahony Water Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. R. Seadawy ◽  
A. Sayed
Keyword(s):  
Traveling Wave ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Dispersive Media ◽  
Wave Solutions ◽  
De Vries ◽  
The Stability

The modeling of unidirectional propagation of long water waves in dispersive media is presented. The Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations are derived from water waves models. New traveling solutions of the KdV and BBM equations are obtained by implementing the extended direct algebraic and extended sech-tanh methods. The stability of the obtained traveling solutions is analyzed and discussed.

BREAKING WAVE SOLUTIONS TO THE SECOND CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS

2009 ◽  
Vol 19 (04) ◽  
pp. 1289-1306 ◽  
Author(s):  
JIBIN LI ◽  
XIAOHUA ZHAO ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Nonlinear Wave Equations ◽  
Breaking Wave ◽  
Dynamical Behaviors ◽  
Wave Solutions

The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.

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 Instability of periodic traveling waves for the symmetric regularized long wave equation

2015 ◽  
Vol 219 ◽  
pp. 235-268
Author(s):  
Jaime Angulo Pava ◽  
Carlos Alberto Banquet Brango
Keyword(s):  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Linear Instability ◽  
Nonlinear Instability ◽  
Periodic Traveling Waves ◽  
Wave Solutions ◽  
Long Wave ◽  
Linearized Problem ◽  
Periodic Traveling Wave ◽  
The Stability

AbstractWe prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

scholarly journals Traveling Wave Solutions of Two Nonlinear Wave Equations by (G′/G)-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yazhou Shi ◽  
Xiangpeng Li ◽  
Ben-gong Zhang
Keyword(s):  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Wave Equations ◽  
Hyperbolic Function ◽  
Exact Traveling Wave Solutions ◽  
Function Solution ◽  
Wave Solutions

We employ the (G′/G)-expansion method to seek exact traveling wave solutions of two nonlinear wave equations—Padé-II equation and Drinfel’d-Sokolov-Wilson (DSW) equation. As a result, hyperbolic function solution, trigonometric function solution, and rational solution with general parameters are obtained. The interesting thing is that the exact solitary wave solutions and new exact traveling wave solutions can be obtained when the special values of the parameters are taken. Comparing with other methods, the method used in this paper is very direct. The (G′/G)-expansion method presents wide applicability for handling nonlinear wave equations.

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.

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