Approximmated Solution for a higher-order PDE

In Soliton perturbation theory is used to derive the details of an evolving solitary wave, governed by (1). An analytical expression is found for the solitary wave tail. An excellent comparison is found between the theoretical and numerical solutions, for the solitary wave tail. Moreover, the perturbation theory shows that the tail vanishes when is satisfied and confirms the existence of the two-parameter family of higher-order asymptotic embedded solitons