Long-time asymptotic behavior for the matrix modified Korteweg–de Vries equation

Abstract

The long-time asymptotics of solution to the integrable matrix modified Korteweg–de Vries equation with a 4 × 4 Lax pair on the line in the case of decaying initial data is established. The study makes crucial use of the inverse scattering transform in the form of an associated 4 × 4 matrix Riemann–Hilbert problem, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. It was shown that the (x,t)-plane mainly decomposes asymptotically in time into three types of regions: a left oscillating region, where to leading order the solution exhibits decaying (of the order O(t−1/2)) modulated oscillations, a central Painlevé region where the asymptotic behavior is expressed in terms of the solution of a coupled Painlevé II equation which is related to a 4 × 4 matrix Riemann–Hilbert problem, and a right fast decaying region.