Application of the Riemann-Hilbert method to the vector modified Korteweg-de Vries equation

摘要

Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable systems. For the direct scattering problem, the spectral analysis is performed for the equation, from which a Riemann-Hilbert problem is well constructed. For the inverse scattering problem, the Riemann-Hilbert problem corresponding to the reflection-less case is solved. Furthermore, as applications, three types of multi-soliton solutions are found. Finally, some figures are presented to discuss the soliton behaviors of the vmKdV equation.