Multidimensional hierarchies of (1+1)-dimensional integrable partial differential equations. Nonsymmetric ∂̄-dressing

Abstract

In this paper the ∂̄-problem has been constructed for a class of multidimensional integrable partial differential equations (PDE) which can be classified as the multidimensional hierarchies of the (1+1)-dimensional systems of integrable PDE. We introduce the nonsymmetric dressing procedure for this purpose. Among the examples we consider the (n+1)-dimensional (n>1) hierarchies of nonlinear Schrödinger, modified Korteveg–de Vries, and Camassa–Holm equations.