Variational approximation and the use of collective coordinates

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.

To appear in PRE (2013)

References in corpus (4)

• The emergence of a coherent structure for coherent structures: localized states in nonlinear systems
• Variational approximations to homoclinic snaking in continuous and discrete systems
• Variational approximations to homoclinic snaking
• Reaction-diffusion front crossing a local defect

Cited by in corpus (11)

• Embedded solitons in second-harmonic-generating lattices
• The Swift-Hohenberg equation with a nonlocal nonlinearity
• Snakes and ghosts in a parity-time-symmetric chain of dimers
• Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model
• Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation
• On multichannel solutions of nonlinear Schrödinger equations: algorithm, analysis and numerical explorations
• Analysis of multistability in discrete quantum droplets and bubbles
• Variational approximations using Gaussian ansatz, false instability, and its remedy in nonlinear Schrödinger lattices
• Unidirectional segregation of Bright-Bright soliton through a $\mathcal{P}\mathcal{T}$-symmetric potential
• Reservoir Engineering for Classical Nonlinear Fields
• Stopping a reaction-diffusion front