SOLITON EIGENFUNCTION EQUATIONS: THE IST INTEGRABILITY AND SOME PROPERTIES

Abstract

Eigenfunctions of the linear eigenvalue problems for the soliton equations obey nonlinear differential equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform (IST) method. They have triad operator representations. Eigenfunction equations are the generating equations and possess other interesting properties. Eigenfunction equations form a new wide class of nonlinear integrable equations. Eigenfunction equations for several typical, well-known (1+1)-, (2+1)- and multi-dimensional soliton equations are considered. A general method for constructing the auxiliary linear systems for the eigenfunction equations is proposed. It is shown that the vertical hierarchies of the eigenfunction equations contain only finite numbers of different members in the cases considered. The properties of such hierarchies for soliton equations are closely connected with their Painleve properties. Some “linear” properties of the eigenfunction equations are also discussed.