Singular Solution of the Liouville Equation under Perturbation

Abstract

The Cauchy problem for the Liouville equation with a small perturbation is considered. We are interested in the asymptotics of the perturbed solution under the assumption that one has singularity. The main goal is to study both the asymptotic approximation of the singular lines and the asymptotic approximation of the solution everywhere outside the narrow neighbourhood of the singular lines. The Cauchy problem for the Liouville equation with a small perturbation ∂ 2 t u−∂ 2 u+8 exp u = eF[u], 0 <e � 1, (0.1) u|t=0 = ψ0(x; e) ,∂ tu|t=0 = ψ1(x; e) ,x ∈ R (0.2) is considered. We are interesting for asymptotics of the perturbed solution u(x, t; e )a s e → 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 = ef (u), 0 <e � 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].