Competition between generic and nongeneric fronts inenvelope equations
Document Type
Article
Journal/Book Title/Conference
Physical Review A
Volume
44
Publication Date
1991
First Page
3636
Last Page
3652
Abstract
Arguments are presented for understanding the selection of the speed and the nature of the fronts that join stable and unstable states on the supercritical side of first-order phase transitions. It is suggested that from compact support, nonpositive-definite initial conditions, observable front behavior occurs only when the asymptotic spatial structure of a trajectory in the Galilean ordinary differential equation (ODE) corresponds to the most unstable temporal mode in the governing partial differential equation (PDE). This selection criterion distinguishes between a ‘‘nonlinear’’ front, which has its origin in the first-order nature of the bifurcation, and a ‘‘linear’’ front. The nonlinear front has special properties as a strongly heteroclinic trajectory in the ODE and as an integrable trajectory in the PDE. Many of the characteristics of the linear front are obtained from a steepest-descent linear analysis originally due to Kolmogorov, Petrovsky, and Piscounov [Bull. Univ. Moscow, Ser. Int., Sec. A 1, 1 (1937)]. Its connection with global stability arguments, and in particular with arguments based on a Lyapunov functional where it exists, is pursued. Finally, the point of view and results are compared and contrasted with those of van Saarloos [Phys. Rev. A 37, 211 (1988); 39, 6367 (1989)].
Recommended Citation
J. Powell, A.C. Newell and C.K.R.T. Jones. “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636–3652, 1991.
