title: Coupling fields and underlying space curvature: an augmented Lagrangian approach.
abstract: We demonstrate a systematic implementation of coupling between a scalar field and the geometry of the space which carries the field. This naturally gives rise to a feedback mechanism between the field and the geometry. We develop a systematic model for the feedback in a general form, inspired by a specific implementation in the context of molecular dynamics (the so-called Rahman-Parrinello molecular dynamics, or RP-MD). We use a generalized Lagrangian that allows for the coupling of the space's metric tensor to the scalar field, and add terms motivated by RP-MD. We present two implementations of the scheme: one in which the metric is only time-dependent (which gives rise to an ordinary differential equation for its temporal evolution), and the other with spatiotemporal dependence (wherein the metric's evolution is governed by a partial differential equation). Numerical results are reported for the (1+1)-dimensional model with a nonlinearity of the sine-Gordon type.
authors.affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.
authors.forename: P G
authors.initials: PG
authors.lastname: Kevrekidis
authors.forename: F L
authors.initials: FL
authors.lastname: Williams
authors.forename: A R
authors.initials: AR
authors.lastname: Bishop
authors.forename: I G
authors.initials: IG
authors.lastname: Kevrekidis
authors.forename: B A
authors.initials: BA
authors.lastname: Malomed