THE GAP LEMMA AND GEOMETRIC CRITERIA FOR INSTABILITY OF VISCOUS SHOCKPROFILES

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operato r about the wave accumulates at the imaginary axis, since the Evans fu nction has in general been constructed only away from the essential sp ectrum. A notable case in which this difficulty occurs is in the stabi lity analysis of viscous shock profiles. Here we prove a general theor em, the ''gap lemma,'' concerning the analytic continuation of the Eva ns function associated with the point spectrum of a traveling wave int o the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previ ously. We demonstrate the power of this method by analyzing the stabil ity of certain undercompressive Viscous sock waves. A necessary geomet ric condition for stability is determined in terms of the sign of a ce rtain Melnikov integral of the associated viscous profile. This sign c an easily be evaluated numerically. We also compute it analytically fo r solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of un stable viscous shock waves. We also show that (strong) heteroclinic un dercompressive waves are sometimes unstable. Similar stability conditi ons are also derived for Lax and overcompressive shocks and for n x n conservation laws, n greater than or equal to 2. (C) 1998 John Wiley & Sons, Inc.