THE NONEXISTENCE OF A CERTAIN CLASS OF TRAVELING-WAVE SOLUTIONS OF THE KURAMOTO-SIVASHINSKY EQUATION

In a previous paper we considered numerically travelling wave solutions of the Kuramoto-Sivashinsky equation, and found solutions in each of three categories, these being regular shocks, oscillatory shocks and solitary waves. Here we show that there are no regular shocks when a certain parameter, representing the effect of short-wave stability, is asymptotically small, even though an asymptotic expansion in this parameter appears to establish at least asymptotic existence. The explanation is because of the presence of a term exponentially small in this parameter. The calculation of this term requires "asymptotics beyond all orders" and our calculation is based on a method used by Kruskal and Segur for a related problem.