Universality of critical behavior in small-dispersion limit of the modified KdV hierarchy

This paper investigates critical behavior of solutions to the modified KdV (mKdV) hierarchy in the small dispersion limit, focusing on the region of the gradient catastrophe point of the corresponding dispersionless system. By applying the Deift–Zhou nonlinear steepest descent method to the associated Riemann–Hilbert problem, we rigorously establish a double scaling asymptotic expansion of the solution. The first correction term beyond the dispersionless approximation is universally governed by the smooth solution of the second member of the Painlevé-I hierarchy. This further confirms Dubrovin's universality conjecture in the mKdV setting, which states that such Painlevé type transcendents universally capture the critical behavior near a gradient catastrophe for Hamiltonian perturbations of hyperbolic equations.