On description of generalized invariant manifolds for nonlinear equations

In the paper we discuss the problem on constructing generalized invariant manifolds for nonlinear partial differential equations. A generalized invariant manifold for a given nonlinear equation is a differential connection that is compatible with the linearization of this equation. In fact, this concept generalizes symmetry. Examples of generalized invariant manifolds obtained from symmetries are given in the paper. However, there exist generalized invariant manifolds irreducible to symmetries, exactly they are of the greatest interest. Such generalized invariant manifolds allow one to construct effectively Lax pairs, recursion operators, and particular solutions to integrable equations. In the work we present the algorithm for constructing a generalized invariant manifold for a given equation. A complete description of generalized invariant manifolds of order $(2,2)$ is given for the Korteweg–de Vries equation. We describe briefly a method for constructing a Lax pair and a recursion operator by means of the generalized invariant manifolds. As an example, the Korteweg–de Vries equation is considered.