The existence of solitary wave solutions for a perturbed generalized mKdV equation with a cubic evolution term is investigated. All possible solitary waves for the corresponding unperturbed equation are firstly explored by dynamical system analysis. Then by using geometric singular perturbation theory and Melnikov’s method, we prove that two solitary wave solutions of the unperturbed generalized mKdV equation with particularly chosen wave speeds will persist under small singular perturbation. The results of numerical simulations are consistent with our theoretical analysis.

中文翻译：

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奇异扰动广义 mKdV 方程的孤立波解的持久性

研究了具有三次演化项的扰动广义 mKdV 方程的孤立波解的存在性。首先通过动力系统分析探索了对应未扰动方程的所有可能的孤立波。然后利用几何奇异摄动理论和Melnikov方法，证明了在小奇异摄动下，具有特定波速的未摄动广义mKdV方程的两个孤立波解将保持不变。数值模拟结果与我们的理论分析一致。