SYMMETRIES AND LAX INTEGRABILITY OF THE GENERALIZED PROUDMAN-JOHNSON EQUATION - O. I. Morozov (Научный вестник МГТУ ГА, 2017)
We study local symmetries of the generalized Proudman-Johnson equation. Symmetries of a partial differential equation may be used to find its invariant solutions. In particular, if <р is a characteristic of a symmetry for a PDE Н = О then the <р-invariant solution of the PDE is a solution to the compatible over-determined system Н = О, < р = О. We show that the Lie algebra of local symmetries for the generalized Proudman-Johnson equation is infinite-dimensional. Reductions of equation with respect to the local symmetries provide ordinary differential equations that describe invariant solutions. For a certain value of the parameter entering the equation we find some cases when the reduced ODE is integrable by quadratures and thus allows one to construct exact solutions. Differential coverings (or Wahlquist-Estabrook prolongation structures, or zero-curvature representations, or integrable extensions, etc.) are of great importance in geometry of PDEs. The theory of coverings is a natural framework for dealing with inverse scattering constructions for soliton equations, Bäcklund transformations, recursion operators, nonlocal symmetries and nonlocal conservation laws, Darboux transformations, and deformations of nonlinear PDEs. In the last section we show that in the case of a certain value of the parameter entering the equation it has a differential covering. This property is referred to as a Lax integrability.