Darboux transformation of two novel two-component generalized complex short pulse equations

Abstract

The short pulse equation is able to describe ultra short pulse, which plays a crucial part in the field of optical fiber propagation. In this paper, we investigate a generalized complex short pulse equation and its two-component generalization. We first prove that they are Lax integrable. Subsequently, we obtain their new Lax pairs through hodograph transformation to carry out Darboux transformation, respectively. For the generalized complex short pulse equation, we provide a different Darboux matrix and verify that it is feasible, then we focus on higher-order semi-rational soliton solutions by means of generalized Darboux transformation. For the coupled generalized complex short pulse equations, we apply Darboux transformation to discuss exact solutions by choosing different seed solutions.