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We study the asymptotic behavior of the Ablowitz-Segur solutions for thesecond Painlev\'e equation using the Riemann-Hilbert approach and methods basedon asymptotic expansions of classical special functions. Recent results showthat the matrix-valued function satisfying the associated Riemann-Hilbertproblem 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 toconstruct the explicit form of this parametrix and apply it to obtain improvedasymptotic relations for the real and purely imaginary Ablowitz-Segur solutionsof the inhomogeneous Painlev\'e II equation.

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