MNLS and Generalized Fourier Transforms for Constant Boundary Conditions

The Lax pair for the MNLS describing spin 1 Bose–Einstein condensate proposed by Wadati et al. is associated to the symmetric C.I space (Formula presented.). We construct the fundamental analytic solutions of the Lax operator (Formula presented.) with constant boundary conditions (CBC) and determine two minimal sets (Formula presented.) of scattering data of (Formula presented.). We derive Wronskian relations, which allow us to introduce “squared solutions” of (Formula presented.), which map the potential (Formula presented.) and its variation onto (Formula presented.) and (Formula presented.), respectively. Using the Green functions (Formula presented.), we prove the completeness relations for the “squared solutions” and expand (Formula presented.) and (Formula presented.) over the squared solutions, that is, we derive generalized Fourier transforms. The recursion operator (Formula presented.), for which the squared solutions are eigenfunctions allows us to describe the hierarchy of MNLS with CBC.