Spectral analysis and long-time asymptotics of complex mKdV equation

Abstract

In this paper, we obtain the long-time asymptotics of the complex mKdV equation via the Defit–Zhou method (nonlinear steepest descent method). The Cauchy problem of the complex mKdV equation is transformed into the corresponding Riemann–Hilbert problem on the basis of the Lax pair and the scattering matrix. After that, Riemann–Hilbert problems are converted through a decomposition of the matrix-valued spectral function and factorizations of the jump matrix for Riemann–Hilbert problem. Finally, by solving the last model problem, the long-time asymptotics of complex mKdV equation are derived.