Boutet de Monvel, Anne; Shepelsky, Dmitry
Initial boundary value problem for the mKdV equation on a finite interval  [ Problème aux limites pour l'équation de Korteweg de Vries modifiée sur un intervalle borné ]
Annales de l'institut Fourier, Tome 54 (2004) no. 5 , p. 1477-1495
Zbl 02162431 | MR 2127855 | 1 citation dans Numdam
doi : 10.5802/aif.2056
URL stable : http://www.numdam.org/item?id=AIF_2004__54_5_1477_0

Classification:  35Q53,  37K15,  35Q15,  34A55,  34L25
On analyse l’équation de «Korteweg-de Vries modifiée» sur un intervalle borné (0,L), avec conditions aux limites en t=0 et en x=0,L, en exprimant sa solution q(x,t) en termes de la solution d’un problème de Riemann-Hilbert associé. Ce problème est défini par des fonctions spectrales déterminées par les conditions aux limites. Nous explicitons la relation globale qui reflète en termes de ces fonctions spectrales la compatibilité des conditions aux limites.
We analyse an initial-boundary value problem for the mKdV equation on a finite interval (0,L) by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k-plane. This RH problem is determined by certain spectral functions which are defined in terms of the initial-boundary values at t=0 and x=0,L. We show that the spectral functions satisfy an algebraic “global relation” which express the implicit relation between all boundary values in terms of spectral data.

Bibliographie

[1] A. Boutet De Monvel, A.S. Fokas & D. Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu 3 (2004) p. 139-164 MR 2055707 | Zbl 1057.35050

[2] A. Boutet De Monvel, A.S. Fokas & D. Shepelsky, Analysis of the global relation for the nonlinear Schrödinger equation on the half-line, Lett. Math. Phys 65 (2003) p. 199-212 MR 2033706 | Zbl 1055.35107

[3] A. Boutet De Monvel & D. Shepelsky, The modified KdV equation on a finite interval, C. R. Math. Acad. Sci. Paris 337 (2003) p. 517-522 MR 2017129 | Zbl 1044.35080

[4] A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London, Ser. A 453 (1997) p. 1411-1443 MR 1469927 | Zbl 0876.35102

[5] A.S. Fokas, On the integrability of linear and nonlinear partial differential equations, J. Math. Phys 41 (2000) p. 4188-4237 MR 1768651 | Zbl 0994.37036

[6] A.S. Fokas, Two dimensional linear PDEs in a convex polygon, Proc. Roy. Soc. London, Ser. A 457 (2001) p. 371-393 MR 1848093 | Zbl 0988.35129

[7] A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys 230 (2002) p. 1-39 MR 1930570 | Zbl 1010.35089

[8] A.S. Fokas & A.R. Its, The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal 27 (1996) p. 738-764 MR 1382831 | Zbl 0851.35122

[9] A.S. Fokas & A.R. Its, The nonlinear Schrödinger equation on the interval, Preprint MR 2074625 | Zbl 1057.37063

[10] A.S. Fokas, A.R. Its & L.-Y. Sung, The nonlinear Schrödinger equation on the half-line, Preprint MR 2150354 | Zbl 02201258

[12] X. Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal 20 (1989) p. 966-986 MR 1000732 | Zbl 0685.34021

[13] X. Zhou, Inverse scattering transform for systems with rational spectral dependence, J. Differential Equations 115 (1995) p. 277-303 MR 1310933 | Zbl 0816.35104

[11] V.E. Zakharov & A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl. 8 (1974) p. 226-235 Zbl 0303.35024

[11] V.E. Zakharov & A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering. II, Funct. Anal. Appl. 13 (1979) p. 166-174 Zbl 0448.35090