Abstract

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.

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