In this paper, the general triple-pole multi-soliton solutions are proposed for the focusing modified Korteweg–de Vries (mKdV) equation with both nonzero boundary conditions (NZBCs) and triple zeros of analytical scattering coefficients by means of the inverse scattering transform. Furthermore, we also give the corresponding trace formulae and theta conditions. Particularly, we analyze some representative reflectionless potentials containing the triple-pole multi-dark-anti-dark solitons and breathers. The idea can also be extended to the whole mKdV hierarchy (e.g. the fifth-order mKdV equation, and third-fifth-order mKdV equation) with NZBCs and triple zeros of analytical scattering coefficients. Moreover, these obtained triple-pole solutions can also be degenerated to the triple-pole soliton solutions with zero boundary conditions.

中文翻译：

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具有非零边界条件的聚焦 mKdV 层次的多三极孤子

在本文中，通过逆散射变换，提出了具有非零边界条件（NZBCs）和解析散射系数三重零的聚焦修正 Korteweg-de Vries（mKdV）方程的一般三极多孤子解。此外，我们还给出了相应的迹公式和theta条件。特别是，我们分析了一些包含三极多暗反暗孤子和呼吸器的具有代表性的无反射势。这个想法也可以扩展到整个 mKdV 层次结构（例如，五阶 mKdV 方程和三五阶 mKdV 方程），具有 NZBC 和解析散射系数的三重零。此外，这些获得的三极子解也可以退化为具有零边界条件的三极子孤子解。