In this paper, we consider a perturbed Korteweg–de Vries (KdV) equation with weak dissipation and Marangoni effects. Main attention is focused on the existence conditions of periodic and solitary wave solutions of the perturbed KdV equation. Based on bifurcation theory of dynamic system and geometric singular perturbation method, the parameter conditions and wave speed conditions for the existence of one periodic solution, one solitary solution and the coexistence of a solitary solution and infinite number of periodic solutions are given. By using Chebyshev criterion to analyze the ratio of Abelian integrals, the monotonicity of wave speed is proved, and the upper and lower bounds of wave speed are obtained.

中文翻译：

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扰动KdV方程中的孤波和周期波

在本文中，我们考虑具有弱耗散和Marangoni效应的摄动Korteweg-de Vries（KdV）方程。主要的注意力集中在被扰动的KdV方程的周期和孤波解的存在条件上。基于动力系统的分岔理论和几何奇异摄动法，给出了一个周期解，一个孤立解以及一个孤立解与无限个周期解并存的参数条件和波速条件。利用Chebyshev准则分析阿贝尔积分的比率，证明了波速的单调性，并获得了波速的上下界。