Geodesic
Perturbation of a singular solution to the Liouville equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 118 (1999) no. 3, pp. 390-397

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We construct an asymptotic (with respect to a small parameter) solution of the Cauchy problem for the perturbed Liouville equation in the case where the unperturbed solution has singularities on timelike lines. We propose a modification of the Krylov–Bogoliubov method that, in particular, allows us to find the asymptotic corrections to the singularity lines. 
@article{TMF_1999_118_3_a8,
     author = {L. A. Kalyakin},
     title = {Perturbation of a singular solution to the {Liouville} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {390--397},
     publisher = {mathdoc},
     volume = {118},
     number = {3},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1999_118_3_a8/}
}
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L. A. Kalyakin. Perturbation of a singular solution to the Liouville equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 118 (1999) no. 3, pp. 390-397. http://geodesic.mathdoc.fr/item/TMF_1999_118_3_a8/