Nonlinear differential operators of first and second order possessing invariant linear spaces of maximal dimension

Abstract

In connection with the approach to the construction of explicit solutions to nonlinear partial differential equations proposed by S. S. Titov and V. A. Galaktionov, the problem of describing nonlinear differential operators F[y(x)] possessing finite-dimensional invariant linear spaces arises. It has previously been proved that for the m-th order operators, the dimension of an invariant space cannot exceed2m+1. In the present paper, we consider the cases where this value is achieved. The first- and second-order operators are studied and it is shown that they are quadratic in y. Full descriptions of the first-order operators and the second-order quadratic operators with constant coefficients are obtained.