Stationary travelling-wave solutions of an unstable KdV–Burgers equation

Both “solitary” and “periodic” stationary travelling wave solutions are investigated numerically for an unstable Korteweg–de Vries–Burgers equation ut+uux+uxxx−η(u+uxx)=0 (η>0). A family of stationary solitary wave solutions whose members are distinguished by the number of “humps” is found for a given η. Corresponding to each solitary wave thus found, a family of stationary periodic waves with the same number of “humps” exists under periodic condition and ends up in the infinite periodicity to the corresponding solitary wave. The numerical results are consistent with the theoretical estimates based on the conservation properties.