Abstract
We review the results of recent investigations dealing with a connection between the envelope soliton-like solutions of a wide family of nonlinear Schrödinger equations (NLSEs) and the soliton-like solutions of a wide family of Korteweg-de Vries equations (KdVEs). The investigation is carried out within the context of the Madelung's fluid picture, which plays the role of the fluid counterpart of the NLSE. In two different fluid motion regimes (uniform current velocity and stationary-profile current velocity variation, respectively), bright and gray/dark soliton-like solutions of both modified NLSE and modified KdVE are found. Remarkably, the present approach represents an alternative key of reading for the envelope soliton theory of the NLSE. In particular, the well known envelope soliton solutions and soliton solutions of the cubic NLSE and the standard KdVE, respectively, are recovered from the general approach and in terms of the fluid language presented in this paper.
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