Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations

Abstract

Using the inverse strong symmetry of the Korteweg–de Vries (KdV) equation on the trivial symmetry and τ0 symmetry, one gets four new sets of symmetries of the KdV equation. These symmetries are expressed explicitly by the multi-integrations of the Jost function of the KdV equation and constitute an infinite dimensional Lie algebra together with two hierarchies of the known symmetries. Contrary to the general belief, the time-independent symmetry groups of the KdV and mKdV equations are non-Abelian and the infinite dimensional Lie algebras of the KdV and mKdV equations are nonisomorphic though two equations are related by the Miura transformation. Starting from these sets of symmetries, four hierarchies of the integrodifferential KdV equations, which can be solved by the Schrödinger inverse scattering transformation method, are obtained. Some of these hierarchies enjoy a common strong symmetry and/or same local conserved densities.