On global solutions of defocusing mKdV equation with specific initial data of critical regularity

We study the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlev\'e equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the matrix-valued function satisfying the associated Riemann-Hilbert problem can be represented by means of a local parametrix around the origin, whose existence can be proved by a vanishing lemma. The aim of this paper is to construct the explicit form of this parametrix and apply it to obtain improved asymptotic relations for the real and purely imaginary Ablowitz-Segur solutions of the inhomogeneous Painlev\'e II equation.