We are concerned with the defocusing modified Korteweg–de Vries equation equipped with the particular type of irregular initial conditions that are given as linear combination of the Dirac delta function and Cauchy principal value. For the initial value problem we prove the existence of smooth self-similar solution, whose profile function is the Ablowitz–Segur solution of the second Painlevé equation. Our method is to use the approach based on the Riemann–Hilbert problem to improve asymptotics of these Painlevé transcendents and find desired profile function by constructing its Stokes multipliers.

中文翻译：

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关于具有关键正则性的特定初始数据的散焦mKdV方程的整体解

我们关注的是散焦的改进的Korteweg-de Vries方程，该方程配备了特定类型的不规则初始条件，这些初始条件由Dirac delta函数和Cauchy主值的线性组合给出。对于初始值问题，我们证明了光滑的自相似解的存在，其轮廓函数是第二个Painlevé方程的Ablowitz-Segur解。我们的方法是使用基于Riemann-Hilbert问题的方法来改善这些Painlevé先验者的渐近性，并通过构造其Stokes乘数来找到所需的轮廓函数。