Boutet de Monvel, Anne; Shepelsky, Dmitry The modified KdV equation on a finite interval. (English. Abridged French version) Zbl 1044.35080 C. R., Math., Acad. Sci. Paris 337, No. 8, 517-522 (2003). Summary: We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex \(k\)-plane. This Riemann-Hilbert problem has explicit \((x,t)\)-dependence and it involves certain functions of \(k\) referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition \(q(x,0)=q_0(x)\), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms. Cited in 15 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 35Q15 Riemann-Hilbert problems in context of PDEs 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Keywords:initial-boundary value problem; mKdV equation; matrix Riemann-Hilbert problem; spectral functions PDFBibTeX XMLCite \textit{A. Boutet de Monvel} and \textit{D. Shepelsky}, C. R., Math., Acad. Sci. Paris 337, No. 8, 517--522 (2003; Zbl 1044.35080) Full Text: DOI References: [1] A. Boutet de Monvel, A.S. Fokas, D. Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu, in press; A. Boutet de Monvel, A.S. Fokas, D. Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu, in press · Zbl 1057.35050 [2] A. Boutet de Monvel, D. Shepelsky, Initial boundary value problem for the modified KdV equation on a finite interval, Preprint; A. Boutet de Monvel, D. Shepelsky, Initial boundary value problem for the modified KdV equation on a finite interval, Preprint · Zbl 1137.35419 [3] Fokas, A. S., A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A, 453, 1411-1443 (1997) · Zbl 0876.35102 [4] Fokas, A. S., On the integrability of linear and nonlinear partial differential equations, J. Math. Phys., 41, 4188-4237 (2000) · Zbl 0994.37036 [5] Fokas, A. S., Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys., 230, 1-39 (2002) · Zbl 1010.35089 [6] A.S. Fokas, A.R. Its, The nonlinear Schrödinger equation on the interval, Preprint; A.S. Fokas, A.R. Its, The nonlinear Schrödinger equation on the interval, Preprint This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.