Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy

Abstract

The traveling wave reduction of the Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found. The Lax pair is used to obtain the first integrals of the traveling wave reduction for the Korteweg-de Vries hierarchy. Exact formulas for the first integrals of the hierarchy are given. The first three members of the hierarchy are considered in more detail. These first integrals are the examples of the integrable nonlinear differential equations with the Painlevé property. Exact solutions in the form of the soliton for the traveling wave reduction of the Korteweg-de Vries hierarchy and its first integrals are presented.