Crum transformation and rational solutions of the non-focusing nonlinear Schrödinger equation

Abstract

A Crum transformation for the linear problem of the non-focusing nonlinear Schrödinger (NLS) equation is used to construct the sequence of rational solutions, given in terms of tau-functions expressed as double Wronskian determinants. The rational solutions of the Adler - Kaup - Newell - Segur (AKNS) hierarchy have already been obtained in the work of Sachs, using the equivalence with the classical Boussinesq hierarchy. However, we find that Sachs' description of the restrictions that must be placed on the AKNS tau-functions in order to obtain NLS rational solutions is not entirely correct. We briefly comment on the constrained Calogero - Moser systems associated with the NLS rational solutions, and indicate how the Calogero - Moser equations can be obtained from a trilinear equation for the tau-function which arises by reduction of the Kadomtsev - Petviashvili (KP) hierarchy.