Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion

TL;DR

This paper investigates the long-term behavior of a discretized KdV-Burgers type equation with fast dispersion and slow diffusion, using averaging methods based on invariants of the fast system to analyze its dynamics.

Contribution

It introduces a novel approach to analyze the limit behavior of a fast-slow discretized PDE system via invariants and averaging, addressing the challenge of non-trivial fast-slow variable separation.

Findings

Computed characteristic features of the fast motion.

Demonstrated the effectiveness of averaging based on invariants.

Provided insights into the long-term dynamics of the discretized system.

Abstract

The long time behavior of the dynamics of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. The discretization is based on a model developed by Goodman and Lax, that is composed of a fast system drifted by a slow forcing term. A natural split to fast and slow state variables is, however, not available. Our approach views the limit behavior as an invariant measure of the fast motion drifted by the slow component, where the known constants of motion of the fast system are employed as slowly evolving observables; averaging equations for the latter lead to computation of characteristic features of the motion. Such computations are presented in the paper.