[Paper Review] Existence and Stability of Traveling Waves for a Class of Nonlocal Nonlinear Equations

H.A. Erbay, S. Erbay+1 more|arXiv (Cornell University)|Jul 1, 2014

Advanced Mathematical Physics Problems28 references4 citations

TL;DR

This paper establishes the existence and orbital stability of traveling wave solutions for a general class of nonlocal nonlinear wave equations with two dispersive operators, L and B. It proves that for Klein-Gordon-type equations (L = I), traveling waves are orbitally stable or unstable via blow-up depending on wave velocity, with a complete characterization for p > 1 and c² < (p−1)/(p+3).

ABSTRACT

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: $ u_{tt}-Lu_{xx}=B(\pm |u|^{p-1}u)_{xx}$, $ p&gt;1$. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators $L$ and $B$. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case $L=I$, corresponding to a class of Klein-Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up.

Motivation & Objective

• To establish the existence of traveling wave solutions for a general class of nonlocal nonlinear wave equations with two coercive pseudo-differential operators L and B.
• To investigate the orbital stability and instability of these traveling waves using energy methods and concentration-compactness principles.
• To provide a complete characterization of wave velocity regimes for orbital stability or instability by blow-up in the case L = I (Klein-Gordon-type equations).
• To extend results to a broad class of equations, including Boussinesq-type and double-dispersion equations, via a unified nonlocal framework.
• To analyze blow-up behavior using Levine’s Lemma by constructing a functional H(t) whose finite-time blow-up implies instability.

Proposed method

• Formulate the nonlocal wave equation as $ u_{tt} - L u_{xx} = B( ext{sign}(u)|u|^{p-1}u)_{xx} $, with L and B as coercive pseudo-differential operators with smooth, decaying symbols.
• Apply the traveling wave ansatz $ u(x,t) = ilde{ heta}_c(x - ct) $, reducing the PDE to an ODE in the moving frame.
• Use variational methods and the concentration-compactness principle to prove existence of nontrivial solitary wave solutions.
• Define a functional $ H(t) = rac{1}{2} ig floor B^{-1/2} v(t) ig floor_{L^2}^2 $ and apply Levine’s Lemma to show finite-time blow-up under specific energy and momentum conditions.
• Employ conservation laws for energy $ ilde{ ext{E}}(u,w) $ and momentum $ ilde{ ext{M}}(u,w) $ to bound initial data and derive inequalities for instability.
• Derive a differential inequality $ H(t)H''(t) - (1+ u)(H'(t))^2 o 0 $ with $ u = (p+3)/4 - 1 $, confirming blow-up when $ H''(t) > (p+1) u $.

Experimental results

Research questions

• RQ1 For what values of wave velocity c does the traveling wave solution of the nonlocal wave equation exist and remain orbitally stable?
• RQ2 Under what conditions does the solution exhibit instability via finite-time blow-up in the L² norm of $ B^{-1/2}v $?
• RQ3 How does the interplay between two dispersive operators L and B affect the existence and stability of solitary waves?
• RQ4 Can the stability results be extended to a general class of nonlocal equations beyond specific models like Boussinesq or double-dispersion equations?
• RQ5 What role do the conservation laws and the functional $ ilde{I}_c(u) $ play in determining the threshold for instability?

Key findings

• Traveling wave solutions exist for the general nonlocal wave equation $ u_{tt} - L u_{xx} = B( ext{sign}(u)|u|^{p-1}u)_{xx} $ under coercivity and decay conditions on L and B.
• For the case $ L = I $, the paper provides an almost complete characterization: orbital stability holds when $ c^2 < rac{p-1}{p+3} $, and instability via blow-up occurs when $ c^2 > rac{p-1}{p+3} $.
• Blow-up of the solution is proven using Levine’s Lemma by showing $ H''(t) > (p+1) u $ and $ H(t)H''(t) - (1+ u)(H'(t))^2 o 0 $, implying finite-time blow-up of $ H(t) $.
• The instability threshold is sharp: for initial data with $ ilde{ ext{E}}(U_0) + c ilde{ ext{M}}(U_0) < d(c) - u $, blow-up occurs in finite time.
• The functional $ ilde{I}_c(u) $ satisfies $ rac{p+1}{p-1}d(c) < ilde{I}_c(u) $, which is crucial in estimating the second derivative of H(t).
• The method applies uniformly to all equations in the class, including Boussinesq-type, double-dispersion, and nonlocal convolution models, unifying stability analysis across diverse dispersive wave systems.