Low-dimensional modelling of a generalized Burgers equation

TL;DR

This paper develops low-dimensional models for a generalized Burgers equation with time-dependent diffusion, using centre manifold theory, extending classical similarity solutions and providing simplified representations of complex nonlinear PDEs.

Contribution

It introduces l-mode and 2-mode centre manifold models for the generalized Burgers equation with time-dependent coefficients, extending existing similarity solution methods.

Findings

Derived low-dimensional models using centre manifold theory

Extended similarity solutions to time-dependent coefficients

Provided simplified models capturing key dynamics of the generalized Burgers equation

Abstract

Burgers equation is one of the simplest nonlinear partial differential equations-it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a time-dependent function. Using a Wayne's transformation and centre manifold theory, we derive l-mode and 2-mode centre manifold models of the generalised Burgers equations for bounded smooth time dependent coefficients. These modellings give some interesting extensions to existing results such as the similarity solutions using the similarity method.