Geodesic
Symmetries of Kadomtsev--Petviashvili equation, isomonodromic deformations, and nonlinear generalizations of the special functions of wave catastrophes
Teoretičeskaâ i matematičeskaâ fizika, Tome 97 (1993) no. 2, pp. 213-226

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A special solution of the Kadomtsev–Petviashvili equation $$u_{tx} + u_{xxxx} + 3u_{yy} + 3(u^2)_{xx} = 0,$$ that is a “nonlinear” analog of the special function of wave catastrophe corresponding to a singularity of swallowtail type is considered. On the basis of a symmetry analysis it is shown that the solution must simultaneously satisfy nonlinear ordinary differential equations with respect to all three independent variables. After “dressing” of the corresponding $\Psi$ function, equations with respect to a spectral parameter arise in a regular manner, and this indicates the possibility of applying the method of isomonodromic deformation. 
@article{TMF_1993_97_2_a3,
     author = {B. I. Suleimanov and I. T. Habibullin},
     title = {Symmetries of {Kadomtsev--Petviashvili} equation, isomonodromic deformations, and nonlinear generalizations of the special functions of wave catastrophes},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {213--226},
     publisher = {mathdoc},
     volume = {97},
     number = {2},
     year = {1993},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1993_97_2_a3/}
}
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B. I. Suleimanov; I. T. Habibullin. Symmetries of Kadomtsev--Petviashvili equation, isomonodromic deformations, and nonlinear generalizations of the special functions of wave catastrophes. Teoretičeskaâ i matematičeskaâ fizika, Tome 97 (1993) no. 2, pp. 213-226. http://geodesic.mathdoc.fr/item/TMF_1993_97_2_a3/