Envelope solitons versus solitons

A theory involving a correspondence between envelope solitonlike solutions of the generalized nonlinear Schrödinger equation (GNLSE) and solitonlike solutions of the generalized Korteweg-de Vries equation (GKVdE) is developed within the context of the Madelung's fluid description (fluid counterpart description of the GNLSE). This correspondence, which, under suitable constraints, can be made invertible, seems to be very helpful for finding one family of solutions (whether envelope solitonlike solutions of the GNLSE or solitonlike solutions of the GKdVE) starting from the knowledge of the other family of solution (whether solitonlike solutions of the GKdVE or envelope solitonlike solutions of the GNLSE). The theory is successfully applied to wide classes of both modified nonlinear Schrödinger equation (MNLSE) and modified Korteweg-de Vries equation (MKVdE), for which bright and gray/dark solitonlike solutions are found. In particular, bright and gray/dark solitary waves are determined for the MNLSE with a quartic nonlinear potential in the modulus of the wavefunction (i.e. q1|Ψ|^2 + q2|Ψ|^4) as well as for the associated MKdVE. Furthermore, the well known bright and gray/dark envelope solitons of the cubic NLSE and the corresponding solitons of the associated standard KdVE are easily recovered from the present theory. Remarkably, this approach opens up the possibility to transfer all the know how concerning the instability criteria for solitonlike solutions of the MKdVE to the instability theory of envelope solitonlike solutions of the MNLSE.