Families of particular solutions to multidimensional partial differential equations

Abstract

Dressing methods are known as productive tools for construction of the particular solutions to the big class of nonlinear partial differential equations (PDEs) which are integrable by the inverse scattering technique. Recently the modification of the dressing method based on the system of algebraic equations has been suggested which allows us to find the families of particular solutions to certain types of nonintegrable (in classical sense) nonlinear PDEs. This modification represents PDEs as closure reductions of an appropriate differential-difference system. In this article we study the dressing procedure in more detail. Particularly, we consider different families of particular solutions available through the dressing method based on the algebraic system. We give two examples of the differential-difference systems and related PDEs and point to other possible generalizations of the dressing method.