A generalized complex mKdV equation: Darboux transformations and explicit solutions

Abstract

A new generalized complex modified Korteweg–de Vries equation associated with a 3 × 3 matrix spectral problem is proposed by resorting to the zero-curvature equation. Based on the gauge transformations between the Lax pairs, a Darboux transformation for the generalized complex modified Korteweg–de Vries equation is constructed, from which the corresponding N-fold Darboux transformations are derived in terms of both iterative technique and determinants. As an application of the resulting Darboux transformations, explicit solutions, like one-soliton, two-soliton, and three-soliton solutions, first-order breather and second-order breather solutions, for the generalized complex modified Korteweg–de Vries equation are obtained.