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

Abstract

This paper concerns the initial value problem of a coupled complex mKdV (CCMKDV) equations ut+uxxx+6(|u|2+|v|2)ux+6u(|v|2)x=0,vt+vxxx+6(|u|2+|v|2)vx+6v(|u|2)x=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{aligned} &u_{t}+u_{xxx}+6 \\bigl( \\vert u \\vert ^{2}+ \\vert v \\vert ^{2} \\bigr)u_{x}+6u\\bigl( \\vert v \\vert ^{2} \\bigr)_{x}=0, \\\\ &v_{t}+v_{xxx}+6\\bigl( \\vert u \\vert ^{2}+ \\vert v \\vert ^{2}\\bigr)v_{x}+6v \\bigl( \\vert u \\vert ^{2}\\bigr)_{x}=0, \\end{aligned} $$\\end{document} proposed by Yang (Nonlinear Waves in Integrable and Nonintegrable Systems, 2010), which is associated with a 4 times 4 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.