Symmetries of Kadomtsev–Petviashvili equation, isomonodromic deformations, and nonlinear generalizations of the special functions of wave catastrophes

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.