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Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. (English) Zbl 1397.35262
Summary: 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 \(3\times 3\) matrices via the Fokas method. We write the solution in terms of the solution of a \(3\times 3\) Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions \(s(k)\), \(S(k)\), and \(S_{L}(k)\), which are determined by the initial values, boundary values at \(x = 0\), and at \(x = L\), 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.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35Q15 Riemann-Hilbert problems in context of PDEs
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
35B40 Asymptotic behavior of solutions to PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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