Previous numerical results indicate that the Kuramoto-Sivashinsky equation admits three classes of non-periodic traveling-wave solutions, namely regular shocks, oscillatory shocks, and solitary waves. However, it has been shown that regular (monotonic) shocks cease to exist in the weak-shock limit, due to the radiation of oscillatory waves of exponentially small (with respect to the shock strength) but growing (in space) amplitude. Here, oscillatory shocks and solitary waves are constructed by asymptotic analysis. It thus transpires that, in the weak-shock limit, oscillatory shocks can only be antisymmetric, otherwise oscillatory and monotonic waves of exponentially small but growing (in space) amplitude would inevitably be excited. Under certain conditions, however, the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave. The predictions of the asymptotic theory are supported by numerical results.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics