Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence

Abstract

In an earlier paper, we developed a general physical picture for the linear-marginal-stability mechanism governing the dynamics of front propagation into linearly unstable states. The main conclusion from this approach and the expressions for the resulting front velocity are similar to those obtained along different lines for the space-time evolution of instabilities in plasma physics and fluid dynamics with the so-called pinch-point analysis (a special type of saddle-point analysis). However, as stressed by Ben-Jacob et al. [Physica 14D, 348 (1985)], it is known from the work of Aronson and Weinberger [in Partial Differential Equations and Related Topics, edited by J. A. Goldstein (Springer, Heidelberg, 1975); Adv. Math. 30, 33 (1978)] on a class of simple model equations that exceptions can occur to the linear-marginal-stability velocity selection. In this paper, we generalize these observations and incorporate such exceptions into our general picture of front propagation into unstable states. We show that a breakdown of linear marginal stability occurs if the linear-marginal-stability front profile becomes unstable against a particular nonlinear ‘‘invasion mode.’’