Even and odd nonautonomous NLSE hierarchy and reversible transformations

Abstract

Under investigation are the nonautonomous Nonlinear Schrödinger equation (NLSE) hierarchies. We have presented the soliton solution of the hierarchy in the most generalized form, namely in terms of the spectral parameter of nonautonomous NLSE spectral problem. We analysed crucial differences between the even order hierarchies and the odd order hierarchies and the corresponding solutions. We have also presented the generalized reversible transformations, which transform the nonautonomous NLSE hierarchy into variable coefficient autonomous NLSE hierarchy. We found that the reversible coordinate transformations are monotonic for odd order hierarchies but not strictly monotonic for even order hierarchies or for a combination of the both. The Jacobian of the transformation is, however, unity in all the transformations. We have identified a set of constraints among the dispersion and the nonlinear coefficients of the general nonautonomous NLSE hierarchy, which are preserved under the transformations. We hope that our analysis provides a universal mathematical platform to study diverse physical systems in nonlinear optics, plasma physics, atomic physics and climate physics.