J. Phys. A: Math. Gen. 29 (1996) 7721–7737. Printed in the UK Darboux transformations for the nonlinear Schr¨ odinger equations Manuel Ma˜ nas† Departamento de F´ ısica Te´ orica, Universidad Complutense, E28040-Madrid, Spain Received 14 May 1996 Abstract. Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schr¨ odinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schr¨ odinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schr¨ odinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions. 1. Introduction The nonlinear Schr¨ odinger (NLS) equation is one of the more relevant among the set of integrable equations in (1 + 1)-dimensions and has been extensively studied since the seminal papers [14, 15, 1]. Its role in nonlinear optics is central to the study of solitons in optical fibres [9] and as a partial differential equation is a universal equation describing the propagation of a quasi-monochromatic wave in a weakly dispersive nonlinear one- dimensional media. In [6] one can find a very detailed study of the inverse scattering and the Riemann problem as well as its Hamiltonian structure. Darboux transformations are one of the main tools in the theory of integrable systems [11]. Given a spectral problem defined for some potentials the Darboux transformation acts on the potentials and wavefunctions at the same time giving us solutions to a similar problem. When applied to the Lax pairs associated to integrable systems one obtains new solutions from old ones, an auto-B¨ acklund transformation. For the Ablowitz–Kaup–Newell– Segur/Zakharov–Shabat (AKNS/ZS) these transformations have been analysed for example in [10], see also [11, 13, 5, 4]. Our results regarding the 3-waves resonant interaction equations [7] and the Davey– Stewartson (DS) equations and its Darboux transformation lead us to study the NLS equation from this point of view. We remark here that in [12] the Darboux transformations for the DSI were presented but the reduction to the NLS was just briefly considered. For the † E-mail address: manuel@dromos.fis.ucm.es 0305-4470/96/237721+17$19.50 c  1996 IOP Publishing Ltd 7721