Front Propagation into Unstable States II: Linear Versus Nonlinear Marginal Stability and Rate of Convergence.
01 January 1989
In an earlier paper, we have 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., it is known from the work of Aronson and Weinberger 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 wave length of the pattern generated by the front is only weakly dependent on the nonlinearities. We also analyze front propagation properties close to thresholds for instability of a pitch fork 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, 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.