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

Abstract

Abstract This paper concerns the initial value problem of a coupled complex mKdV (CCMKDV) equations u t + u x x x + 6 ( | u | 2 + | v | 2 ) u x + 6 u ( | v | 2 ) x = 0 , v t + v x x x + 6 ( | u | 2 + | v | 2 ) v x + 6 v ( | u | 2 ) x = 0 , $$ \begin{aligned} &u_{t}+u_{xxx}+6 \bigl( \vert u \vert ^{2}+ \vert v \vert ^{2} \bigr)u_{x}+6u\bigl( \vert v \vert ^{2} \bigr)_{x}=0, \\ &v_{t}+v_{xxx}+6\bigl( \vert u \vert ^{2}+ \vert v \vert ^{2}\bigr)v_{x}+6v \bigl( \vert u \vert ^{2}\bigr)_{x}=0, \end{aligned} $$ proposed by Yang (Nonlinear Waves in Integrable and Nonintegrable Systems, 2010), which is associated with a 4 × 4 $4 \times 4$ 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.

Keywords

• Riemann–Hilbert approach
• N-soliton solutions
• Lax pair
• Scattering data