Singular solution of the Liouville equation under perturbation

is considered. We are interesting for asymptotics of the perturbed solution u(x, t; ε) as ε→ 0. Perturbation theory for integrable equations remains a very attractive task. As a rule a perturbation of smooth solutions such as a single soliton were usually considered. We intend here to discuss a perturbation of a singular solution under assumption that the perturbed solution has singularities as well. A simple well known instance of this kind is a chock wave under weak perturbation as given by the Hopf equation with a small perturbation term ut+uux = εf(u), 0 < ε≪ 1. In this paper we consider a more complicated problem namely a perturbation of the singular solution of Liouville equation. We deal with the singularities studied by Pogrebkov and Polivanov, [1].