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

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.''If this happens, a larger front speed is selected at a point at which the front profile is now marginally stable against this nonlinear invasion mode. We therefore refer to this as the nonlinear-marginal-stability mechanism. (Ben-Jacob et al. call it case-II marginal stability.) We present the results of detailed numerical studies that support our identification of the nonlinear-marginal-stability mechanism, and present the first examples of it for fronts in pattern-forming systems. In the neighborhood of a transition from linear to nonlinear marginal stability, the wavelength of the pattern generated by the front is only weakly dependent on the nonlinearities. We also analyze front propagation properties close to the threshold for instability at a pitchfork bifurcation. We conclude that linear marginal stability generally holds near a continuous transition (corresponding to a supercritical or forward bifurcation point), while front propagation close to a first-order transition (corresponding to a subcritical or inverted-bifurcation point) is generally governed by nonlinear marginal stability. These results are of importance for recent applications of the various approaches in fluid dynamics and other fields.Finally, we derive an expression for the rate of convergence of the front velocity to its asymptotic value. For the class of equations studied by Aronson and Weinberger, our expression reduces to a rigorous result by Bramson Mem. Am. Math. Soc. 285, 1 (1983), but it differs from the one often quoted in the pinch-point or saddle-point analysis. We argue that the latter one is only valid in a limited region of space, and show how to extend the usual analysis to arrive at our result. Several experimental systems to which our results are relevant are discussed.