Lax pair, rogue-wave and soliton solutions for a variable-coefficient generalized nonlinear Schrödinger equation in an optical fiber, fluid or plasma

In this paper, a variable-coefficient generalized nonlinear Schrödinger equation, which can be used to describe the nonlinear phenomena in the optical fiber, fluid or plasma, is investigated. Lax pair, higher-order rogue-wave and multi-soliton solutions, Darboux transformation and generalized Darboux transformation are obtained. Wave propagation and interaction are analyzed: (1) The Hirota and Lakshmanan–Porsezian–Daniel coefficients affect the propagation velocity and path of each one soliton; three types of soliton interaction have been attained: the bound state, one bell-shape soliton’s catching up with the other and two bell-shape soliton head-on interaction. Multi-soliton interaction is elastic. (2) The Hirota and Lakshmanan–Porsezian–Daniel coefficients affect the propagation direction of the first-step rogue waves and interaction range of the higher-order rogue waves.