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Weakly Nonlinear Dynamics of Interface Propagation
Author(s) -
Gross L. K.
Publication year - 2002
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.01415
Subject(s) - nonlinear system , bifurcation , boundary value problem , amplitude , instability , ordinary differential equation , parametric statistics , mathematical analysis , asymptotic analysis , physics , mathematics , classical mechanics , mechanics , differential equation , statistics , quantum mechanics
A simple conceptual description of condensed‐phase combustion, explosive solidification, and certain other exothermic phenomena can be given via a free‐boundary model with a nonlinear kinetic condition at the free boundary. For a wide range of parametric regimes, the reaction front exhibits a great variety of spatial patterns and instabilities. In [1], we did a linear stability analysis of interfaces that move along a two‐dimensional semi‐infinite strip with thermally insulated edges. Here we use the normal‐mode method to perform a weakly nonlinear analysis for the development of transverse instabilities in the strip. The asymptotic analysis leads to the derivation of ordinary differential equations of Landau‐Stuart type for the slowly varying amplitudes of linearly unstable modes. We focus on a strip in which two eigenmodes lose stability at the same value of a parameter related to the activation energy. Such a case gives rise to nontrivial couplings between the amplitude equations, and the two unstable modes compete for dominance. Based on the bifurcation analysis of the amplitude equations, we classify the front configurations that will emerge for any given choice of the kinetics parameter.