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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.

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