In this work, we examine the solitary wave solutions of the mKdV equation with small singular perturbations. Our analysis is a combination of geometric singular perturbation theory and Melnikov’s method. Our result shows that two families of solitary wave solutions of mKdV equation, having arbitrary positive wave speeds and infinite boundary limits, persist for selected wave speeds after small singular perturbations. More importantly, a new type of solitary wave solution possessing both valley and peak, named as breather in physics, which corresponds to a big homoclinic loop of the associated dynamical system is observed. It reveals an exotic phenomenon and exhibits rich dynamics of the perturbed nonlinear wave equation. Numerical simulations are performed to further detect the wave speeds of the persistent solitary waves and the nontrivial one with both valley and peak.

中文翻译：

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奇异摄动下mKdV方程的一种新型孤波解

在这项工作中，我们研究了具有小奇异扰动的 mKdV 方程的孤立波解。我们的分析是几何奇异摄动理论和梅尔尼科夫方法的结合。我们的结果表明，mKdV 方程的两个孤立波解族，具有任意正波速和无限边界限制，在小的奇异扰动后对于选定的波速仍然存在。更重要的是，我们观察到了一种新的兼有波谷和波峰的孤立波解，在物理学中被称为呼吸器，它对应于相关动力系统的一个大同宿循环。它揭示了一种奇异的现象，并展示了受扰动非线性波动方程的丰富动力学。