Abstract
In this paper, we study the initial-boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the finite interval with Lax pairs involving \begin{document}$3× 3$\end{document} matrices via the Fokas method. We write the solution in terms of the solution of a \begin{document}$3× 3$\end{document} Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions \begin{document}$s(k)$\end{document} , \begin{document}$S(k)$\end{document} , and \begin{document}$S_{L}(k)$\end{document} , which are determined by the initial values, boundary values at \begin{document}$x = 0$\end{document} , and at \begin{document}$x = L$\end{document} , respectively. Some of the boundary values are known for a well-posed problem, however, the remaining boundary data are unknown. By using the so-called global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation.
Highlights
Several important partial differential equations (PDEs) in mathematics and physics are integrable, which can be rewritten in terms of two linear eigenvalue equations, called a Lax pair [22]
The solution at time t can be recovered by using the solution of an inverse problem
This inverse problem is most conveniently formulated as a Riemann-Hilbert problem (RHP)
Summary
Several important partial differential equations (PDEs) in mathematics and physics are integrable, which can be rewritten in terms of two linear eigenvalue equations, called a Lax pair [22]. The IST method can be often used to study the initial value problems for integrable evolution equations on the line [1, 5, 6, 16]. The Fokas method can be used to study the boundary value problems of several important integrable equations including 2 × 2 Lax pair equations, such as the KdV equation [10, 11], the nonlinear Schrodinger equation [12, 13, 14, 15] and other PDEs [2, 3, 4, 17, 21, 23, 24, 25, 29, 30, 34].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
