Linearization of the nonlinear Schrödinger equation and the isotropic Heisenberg spin chain

Abstract

A new description in terms of one and the same linear inhomogeneous integral equation is proposed for the nonlinear Schrödinger equation (NLS), as well as for the equation of motion for the classical isotropic Heisenberg spin chain in the continuum limit (IHSC). From the integral equation which contains a two-fold integration over an arbitrary contour in the complex plane with an arbitrary measure one can obtain the various solutions of the NLS as well as of the IHSC in a direct way without going through the details of the inverse scattering formalism. Well-known properties such as the Miura transformation, the Gel'fand-Levitan equation and the Lax representations for NLS and IHSC can be derived as a corollary from the integral equation. The treatment leads also to a few more general (integrable) partial differential equations which contain the NLS and the IHSC as special cases.