Initial-boundary value problems for the two-component complex modified Korteweg-de Vries equation on the interval

Abstract

<p style='text-indent:20px;'>We apply the Fokas unified method to study initial-boundary value (IBV) problems for the two-component complex modified Korteweg-de Vries (mKdV) equation with a <inline-formula><tex-math id="M1">\begin{document}$ 4\times 4 $\end{document}</tex-math></inline-formula> Lax pair on the interval. The solution can be written by the solution of a <inline-formula><tex-math id="M2">\begin{document}$ 4\times 4 $\end{document}</tex-math></inline-formula> Riemann-Hilbert (RH) problem constructed in the complex <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-plane. The relevant jump matrices are explicitly expressed in terms of three matrix-valued spectral functions related to the initial values, and the Dirichlet-Neumann boundary values, respectively. Moreover, we get that these spectral functions satisfy a global relation and also study the asymptotic analysis of the spectral functions. By considering the global relation, we express the unknown boundary values in terms of the known initial and boundary values via a Gelfand-Levitan-Marchenko (GLM) representation.</p>