Exact Solutions and Dynamics in Schrödinger–Hirota Model Having Multiplicative White Noise via Itô Calculus

Abstract

For the Schrödinger–Hirota model having multiplicative white noise described via Itô calculus, to find exact explicit solutions, the corresponding differential system of the amplitude component is formulated, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems to analyze the parameter conditions of the associated system and to find the corresponding phase portraits, the dynamical behavior of the amplitude component can be derived. Under a special parameter condition, exact explicit homoclinic solutions, periodic solution families as well as compacton solutions can be found.