In this paper, we construct the generalized perturbation ([Formula: see text], [Formula: see text])-fold Darboux transformation of the fifth-order modified Korteweg-de Vries (KdV) equation by the Taylor expansion. We use this transformation to derive the higher-order rational soliton solutions of the fifth-order modified KdV equation. We find that these higher-order rational solitons admit abundant interaction structures. We graphically present the dynamics behaviors from the first- to fourth-order rational solitons. Furthermore, by the Miura transformation, we obtain the complex rational soliton solutions of the fifth-order KdV equation.

中文翻译：

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五阶修正 KdV 和 KdV 方程的高阶有理孤子解

在本文中，我们通过泰勒展开构造了五阶修正 Korteweg-de Vries (KdV) 方程的广义扰动（[公式：见正文]，[公式：见正文]）-折叠 Darboux 变换。我们使用这种变换来推导五阶修正 KdV 方程的高阶有理孤子解。我们发现这些高阶有理孤子具有丰富的相互作用结构。我们以图形方式展示了从一阶到四阶有理孤子的动力学行为。此外，通过 Miura 变换，我们得到了五阶 KdV 方程的复有理孤子解。