DARBOUX TRANSFORMATIONS FOR THE NONLINEAR SCHRODINGER-EQUATIONS

Darboux transformations for the AKNS/ZS system are constructed in term s of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reductio n of the Darboux transformation for the nonlinear Schrodinger equation s with standard and anomalous dispersion is presented. Two different f amilies of new solutions for a given seed solution of the nonlinear Sc hrodinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matri ces we present topological solutions, with different asymptotic argume nt for the amplitude and nonzero background. For the anomalous dispers ion case they represent continuous deformations of the bright n-solito n solution, which is recovered for zero background. In particular thes e solutions contain the combination of multiple homoclinic orbits of t he focusing nonlinear Schrodinger equation. Associated with Jordan blo cks we find rational deformations of the just described solutions as w ell as pure rational solutions. The second family contains not only th e solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the d ark soliton solutions.