Perturbation of a singular solution to the Liouville equation (Q1567790)

A formal asymptotic solution of the Cauchy problem \[ \partial^2_t u- \partial^2_xu+ 2e^u= \varepsilon F(u, \partial_x u,\partial_tu),\quad 0<\varepsilon\ll 1, \] \[ u|_{t= 0}= \psi_0(x; \varepsilon)\approx \sum^\infty_{n= 0} \varepsilon^n \psi_{0n}(x),\;\partial_t u|_{t=0}= \psi_1(x; \varepsilon)\approx \sum^\infty_{n= 0} \varepsilon^n \psi_{1n}(x) \] is constructed. It is assumed that the solution \[ \Phi(x, t)= \log 4{r_+'(x+ t)r_-'(x- t)\over [r+ (x+ t)+ r_-(x- t)]^2} \] of the unperturbed problem has singularities on the timelike lines (determined by the zeros of \(r_+(x+ t)+ r_-(x- t)\)).