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.

短脉冲方程能够描述超短脉冲，在光纤传播领域中起着至关重要的作用。在本文中，我们研究了一个广义复短脉冲方程及其二分量推广。我们首先证明它们是 Lax 可积的。随后，我们通过全息图变换得到它们新的 Lax 对，分别进行 Darboux 变换。对于广义复短脉冲方程，我们提供了一个不同的Darboux矩阵并验证了它是可行的，然后我们通过广义Darboux变换来关注高阶半有理孤子解。对于耦合的广义复短脉冲方程，我们应用Darboux变换通过选择不同的种子解来讨论精确解。