The algebraic structures of isospectral Lax operators and applications to integrable equations

Abstract

For a general spectral operator, the author establishes types of algebraic structures of the spaces of the corresponding isospectral Lax operators, which essentially form the theoretical basis of the Lax operator method. Furthermore the author introduces the concepts of tau -algebras and master algebras to describe time-polynomial-dependent symmetries of nonlinear integrable equations. Finally the author applies the theory of Lax operators to the KP hierarchy of integrable equations as an illustrative example, and thus obtain the master symmetry algebra of the KP hierarchy.