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Symmetries of Kadomtsev-Petviashvili equation, isomonodromic deformations, and “nonlinear” generalizations of the special functions of wave catastrophes. (English. Russian original) Zbl 0801.35127
Theor. Math. Phys. 97, No. 2, 1250-1258 (1993); translation from Teor. Mat. Fiz. 97, No. 2, 213-226 (1993).
Summary: 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.

MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 58J70 Invariance and symmetry properties for PDEs on manifolds