[Paper Review] Non linear travelling waves with diffusion

Monica De Angelis, Enrico Mazziotti|arXiv (Cornell University)|Mar 4, 2012

Differential Equations and Numerical Methods2 references3 citations

TL;DR

This paper analyzes a third-order parabolic operator modeling nonlinear wave equations with small diffusion, deriving explicit solutions for the undisturbed hyperbolic case and rigorously estimating the error between the full solution and the wave approximation. It proves that for any $ k < 1 $, diffusion effects remain of order $ \varepsilon^k $ over time intervals $ [0, T_\varepsilon] $, with $ T_\varepsilon \sim \frac{a}{2} \ln(\varepsilon^{k-1}) $, establishing long-time validity of wave-like behavior in models like superconductivity and viscoelasticity.

ABSTRACT

A third order parabolic operator L_εtypical of a non linear wave operator cal L_0 perturbed by viscous terms, is analyzed. Some particular solutions related to L_0 are explicitly determined and the initial value problem for L_εis considered. The parabolic-hyperbolic behaviour is analyzed and rigorous approximations of the solution are achieved.

Motivation & Objective

• To analyze the behavior of nonlinear traveling waves perturbed by small diffusion terms in partial differential equations.
• To rigorously estimate the solution of the initial value problem for the perturbed operator $ \mathcal{L}_\varepsilon $, especially as $ \varepsilon \to 0 $.
• To quantify the time interval over which wave-like solutions dominate despite viscous effects, using the perturbed sine-Gordon equation as a key model.
• To establish existence, uniqueness, and convergence of solutions via integral equation formulations and fixed-point techniques.

Proposed method

• Derives the fundamental solution $ K_\varepsilon(x,t) $ of the linearized operator $ \mathcal{L}_\varepsilon $ using Laplace transforms and inverse transforms.
• Reduces the nonlinear problem to an integral equation with kernel $ K_\varepsilon $, enabling analysis of the remainder term $ v = u - w $.
• Applies fixed-point theory in a Banach space $ \mathcal{B}_\eta $ to prove existence and uniqueness of the solution $ v $ to the remainder problem.
• Uses Gronwall’s inequality on the $ L^\infty $-norm of $ v $ to derive bounds on the error in terms of $ \varepsilon $.
• Employs explicit traveling wave solutions of the sine-Gordon equation as reference states $ w $, particularly $ w(x,t) = 2\arctan(e^{(x-t)/a}) $.
• Estimates the third-order derivative $ w_{xxt} $ to bound the source term $ F_w $, enabling control of the perturbation.

Experimental results

Research questions

• RQ1 How does the addition of a small diffusion term $ \varepsilon \partial_{xxt} $ affect the long-term behavior of nonlinear wave solutions?
• RQ2 Over what time interval $ [0, T_\varepsilon] $ can the solution of the perturbed equation be approximated by the undisturbed hyperbolic wave solution with controlled error?
• RQ3 What is the precise asymptotic order of the diffusion error as $ \varepsilon \to 0 $, and how does it depend on the initial data and model parameters?
• RQ4 Can the remainder term $ v = u - w $ be rigorously bounded using integral equation and fixed-point methods?

Key findings

• The fundamental solution $ K_\varepsilon(x,t) $ is smooth and shares key properties with the heat kernel, enabling rigorous analysis of the perturbed problem.
• For any $ k < 1 $, the solution $ u $ remains within $ \varepsilon^k $-neighborhood of the wave solution $ w $ over the time interval $ [0, T_\varepsilon] $, where $ T_\varepsilon = \frac{a}{2} \ln\left( \frac{1}{\beta \varepsilon^{1-k}} \right) $.
• The error $ r_\varepsilon(t) = \sup_x |v(x,t,\varepsilon)| $ satisfies $ r_\varepsilon(t) \leq \beta e^{2T/a} \varepsilon $, proving $ \mathcal{O}(\varepsilon) $-error for finite $ T $.
• The bound $ |w_{xxt}| \leq \beta $, with $ \beta $ depending only on $ a $, ensures the source term $ F_w $ satisfies the required Lipschitz and boundedness conditions.
• The analysis confirms that wave behavior dominates for $ \varepsilon \to 0 $, with diffusion effects remaining small and controllable over exponentially growing time intervals.
• The method applies to models such as the perturbed sine-Gordon equation in superconductivity and viscoelasticity, where wave propagation and diffusion coexist.