A coupled complex mKdV equation and its N-soliton solutions via the Riemann-Hilbert approach

This paper concerns the initial value problem of a coupled complex mKdV (CCMKDV) equations u(t) + u(xxx) + 6(|u|(2) + |v|(2))u(x) + 6u(|v|(2))(x) = 0, v(t) + v(xxx) + 6(|u|(2) + |v|(2))v(x) + 6v(|u|(2))(x) = 0, proposed by Yang (Nonlinear Waves in Integrable and Nonintegrable Systems, 2010), which is associated with a 4x4 scattering problem. Based on matrix spectral analysis, a fourth-order matrix Riemann-Hilbert problem is formulated. By solving a specific nonregular Riemann-Hilbert problem with zeros, we present the N-soliton solutions for the CCMKDV system. Moreover, the single-soliton solutions are displayed graphically.