Exact solitary waves in a convecting fluid

The perturbed Korteweg-de Vries equation u t + lambda 1 uu x + lambda 2 u xxx + lambda 3 u xxxx + lambda 4 (uu x ) x + lambda 5 u xx =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.