The Cauchy problem of vector complex modified Korteweg–de Vries equation: Large-time asymptotics with decaying initial data

The nonlinear steepest descent method is applied to investigate the long-time asymptotic behavior of the solution to the Cauchy problem of vector complex modified Korteweg–de Vries (VCMKdV) equation with a (m + 1) × (m + 1) matrix Lax pair. Based on the spectral analysis, the solution of the VCMKdV equation is transformed into the solution of a (m + 1) × (m + 1) matrix Riemann–Hilbert (RH) problem. Then, a nonlinear steepest descent analysis for the associated RH problem demonstrates that, asymptotically in time, the (x, t)-plane primarily divides into three distinct regions: a left slow-decaying region where the asymptotics has the form of Zakharov–Manakov type, a central Painlevé region where the asymptotics is characterized by the solution to a system of coupled Painlevé II equations, which is connected to a (m + 1) × (m + 1) matrix RH problem and appears in a variety of random matrix models, and a right fast-decaying region.