Variational approximation and the use of collective coordinates

TL;DR

This paper introduces a collective coordinate approach for analyzing localized wave dynamics in PDEs, aligning with variational methods to accurately capture stationary states and dynamics, especially in non-variational equations.

Contribution

It proposes a natural projection method onto collective variables that yields ODEs matching the stationary states of the effective Lagrangian, improving upon previous projections.

Findings

The method accurately captures equilibria of the PDE.

It closely reproduces the wave dynamics.

Numerical results validate the approach's effectiveness.

Abstract

We consider propagating, spatially localised waves in a class of equations that contain variational and non-variational terms. The dynamics of the waves is analysed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered from applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a travelling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections.