Competition between generic and nongeneric fronts inenvelope equations

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Physical Review A



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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)].