Generalized invariant manifolds for integrable equations and their applications

In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature, the method of the differential constraints is well known as a tool for constructing particular solutions for the nonlinear partial differential equations. Its essence is in adding to a given nonlinear PDE, another much simpler, as a rule ordinary, differential equation, consistent with the given one. Then any solution of the ODE is a particular solution of the PDE as well. However the main problem is to find this consistent ODE. Our generalization is that we look for an ordinary differential equation that is consistent not with the nonlinear partial differential equation itself, but with its linearization. Such generalized invariant manifold is effectively sought. Moreover, it allows one to construct such important attributes of integrability theory as Lax pairs and recursion operators for integrable nonlinear equations. In this paper, we show that they provide a way to construct particular solutions to the equation as well.