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.

中文翻译：

---

广义复杂 mKdV 方程：达布变换和显式解

摘要 借助零曲率方程，提出了一个新的广义复修正 Korteweg-de Vries 方程，该方程与 3 × 3 矩阵谱问题相关。基于 Lax 对之间的规范变换，构造了广义复修正 Korteweg-de Vries 方程的 Darboux 变换，从中根据迭代技术和行列式推导出相应的 N 折 Darboux 变换。作为所得 Darboux 变换的应用，获得了广义复修正 Korteweg-de Vries 方程的显式解，如单孤子、双孤子和三孤子解、一阶呼吸器和二阶呼吸器解.