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.

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