Exact solitary waves in a convecting fluid

The perturbed Korteweg-de Vries equation ut+ lambda 1uux+ lambda 2uxxx+ lambda 3uxxxx+ lambda 4 (uux)x+ lambda 5uxx=0, which describes the evolution of long shallow waves in a convecting fluid when the critical Rayleigh number slightly exceeds its critical value, admits two types of exact solitary wave solution.