91 20 11 v 1 1 6 D ec 1 99 9 Liouville equation under perturbation

Small perturbation of the Liouville equation under smooth initial data is considered. Asymptotic solution which is available for a long time interval is constructed by the two scale method. The Cauchy problem for the Liouville equation with a small perturbation ∂ 2 t u − ∂ 2 x u + 8 exp u = εF[u], 0 < ε ≪ 1, (0.1) u| t=0 = ψ 0 (x), ∂ t u| t=0 = ψ 1 (x), x ∈ R (0.2) is considered. The problem is not suit under soliton perturbation theory because the Liouville equation has no any soliton solution. Initial functions are here arbitrary , ones are smooth and decay rapidly at infinity ψ 0 , ψ 1 (x) = O(x −N), |x| → ∞, ∀ N. So we deal with a smooth solution; the case of singular solutions was considered in [1]. The perturbation operator is determined by two smooth functions F 1 , F 2 : F[u] = ∂ x F 1 (∂ x u, ∂ t u) + ∂ t F 2 (∂ x u, ∂ t u). (0.3) The purpose is to construct an asymptotic approach of the solution u(x, t; ε) as ε → 0 uniformly over long time interval {x ∈ R, 0 < t ≤ O(ε −1)}. Results. 1. The solution of the unperturbed problem (as ε = 0) decomposes asymptotically at infinity (as t → ∞) on two simple waves which travel on a decreasing background u(x, t; 0) = −4t + A 0 ± (s ±) + O((s ∓) −N), s ∓ → ∓∞. 2. The structure of the asymptotic solution as ε → 0 remains the same u(x, t; ε) ≈ −4t + A ± (s ± , τ) + O(ε) for long times t ≈ ε −1. The perturbation affects only a slow deformation of the waves A ± = A ± (s ± , τ) on the slow time scale τ = εt. 3. The deformation of the waves is described by the first order PDE's