Initial boundary value problem for the mKdV equation on a finite interval

Boutet de Monvel, Anne; Shepelsky, Dmitry

Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1477-1495.

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.

Classification : 35Q53, 37K15, 35Q15, 34A55, 34L25

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

[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

[<L>11</L>] 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