ON TRAVELING-WAVE SOLUTIONS OF THE KURAMOTO-SIVASHINSKY EQUATION

Previous numerical results indicate that the Kuramoto-Sivashinsky equa tion admits three classes of non-periodic traveling-wave solutions, na mely regular shocks, oscillatory shocks, and solitary waves. However, it has been shown that regular (monotonic) shocks cease to exist in th e weak-shock limit, due to the radiation of oscillatory waves of expon entially small (with respect to the shock strength) but growing (in sp ace) amplitude. Here, oscillatory shocks and solitary waves are constr ucted by asymptotic analysis. It thus transpires that, in the weak-sho ck limit, oscillatory shocks can only be antisymmetric, otherwise osci llatory and monotonic waves of exponentially small but growing (in spa ce) amplitude would inevitably be excited. Under certain conditions, h owever, the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave. The pred ictions of the asymptotic theory are supported by numerical results.