Madelung fluid description on a generalized mixed nonlinear Schrödinger equation

Abstract

Within the framework of the Madelung fluid description, in the present paper, we will derive bright and dark (including gray- and black-soliton) envelope solutions for a generalized mixed nonlinear Schrodinger model $${\mathrm {i}}\,\dfrac{\partial \varPsi }{\partial t}=\dfrac{\partial ^2 \varPsi }{\partial x^2}+{\mathrm {i}}\,a\,|\varPsi |^{2}\,\dfrac{\partial \varPsi }{\partial x}+{\mathrm {i}}\,b\,\varPsi ^{2}\,\dfrac{\partial \varPsi ^*}{\partial x}+c\,|\varPsi |^{4}\varPsi +d\,|\varPsi |^{2}\varPsi $$ , by virtue of the corresponding solitary wave solutions for the generalized stationary Gardner equations. Via corresponding parametric constraints, our results are achieved under suitable assumptions for the current velocity associated with different boundary conditions of the fluid density $$\rho $$ , while we have only considered the motion with stationary-profile current velocity case and excluded the motion with constant current velocity case. Note that our model is a generalized one with the inclusion of multiple coefficients ( $$a$$ , $$b$$ , $$c$$ and $$d$$ ).