Comptes Rendus
no. 5-6
Partial Differential Equations
Riemann–Hilbert formulation for the KdV equation on a finite interval
[L'équation KdV sur un intervalle borné par la méthode de Riemann–Hilbert]

Présenté par : Bernard Malgrange

Iasonas Hitzazis1 ; Dimitri Tsoubelis1
1 Department of Mathematics, University of Patras, 26500 Patras, Greece
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 261-266.

Nous étudions un problème aux limites pour l'équation KdV sur un intervalle borné : l'étude est faite en termes d'un problème de Riemann–Hilbert singulier dans le k-plan complexe pour une fonction matricielle qui dépend de façon explicite de variables d'espace–temps. Pour un ensemble particulier de données de Cauchy ainsi que de valeurs aux limites, nous donnons les « fonctions spectrales » qui rendent la solution du problème de Riemann–Hilbert unique. A partir de cette solution on obtient une expression intégrale de la solution du problème aux limites pour l'équation KdV.

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.01.012
Iasonas Hitzazis 1 ; Dimitri Tsoubelis 1

1 Department of Mathematics, University of Patras, 26500 Patras, Greece 
@article{CRMATH_2009__347_5-6_261_0,
     author = {Iasonas Hitzazis and Dimitri Tsoubelis},
     title = {Riemann{\textendash}Hilbert formulation for the {KdV} equation on a finite interval},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {261--266},
     publisher = {Elsevier},
     volume = {347},
     number = {5-6},
     year = {2009},
     doi = {10.1016/j.crma.2009.01.012},
     language = {en},
}
TY  - JOUR
AU  - Iasonas Hitzazis
AU  - Dimitri Tsoubelis
TI  - Riemann–Hilbert formulation for the KdV equation on a finite interval
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 261
EP  - 266
VL  - 347
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2009.01.012
LA  - en
ID  - CRMATH_2009__347_5-6_261_0
ER  - 
%0 Journal Article
%A Iasonas Hitzazis
%A Dimitri Tsoubelis
%T Riemann–Hilbert formulation for the KdV equation on a finite interval
%J Comptes Rendus. Mathématique
%D 2009
%P 261-266
%V 347
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2009.01.012
%G en
%F CRMATH_2009__347_5-6_261_0
Iasonas Hitzazis; Dimitri Tsoubelis. Riemann–Hilbert formulation for the KdV equation on a finite interval. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 261-266. doi : 10.1016/j.crma.2009.01.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.012/

[1] J.L. Bona; S.M. Sun; B.-Y. Zhang A nonhomogeneous boundary value problem for the Korteweg–de Vries equation posed on a finite domain, Comm. Partial Differential Equations, Volume 28 (2003), pp. 1391-1436

[2] A. Boutet de Monvel; D. Shepelsky The modified KdV equation on a finite interval, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 517-522

[3] A. Boutet de Monvel; D. Shepelsky Initial boundary value problem for the mKdV equation on a finite interval, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 5, pp. 1477-1495

[4] A. Boutet de Monvel; A.S. Fokas; D. Shepelsky Integrable nonlinear evolution equations on a finite interval, Commun. Math. Phys., Volume 263 (2006), pp. 133-172

[5] A.S. Fokas A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. London, Ser. A, Volume 53 (1997), pp. 1411-1443

[6] A.S. Fokas Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., Volume 230 (2002), pp. 1-39

[7] A.S. Fokas; A.R. Its An initial-boundary value problem for the Korteweg–de Vries equation, Math. Comput. Simul., Volume 37 (1994), pp. 293-321

[8] A.S. Fokas; A.R. Its The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Gen., Volume 37 (2004), pp. 6091-6114

[9] A.S. Fokas; A.R. Its; L.-Y. Sung The nonlinear Schrödinger equation on the half-line, Nonlinearity, Volume 18 (2005), pp. 1771-1822

[10] A.S. Fokas; L.Y. Sung; D. Tsoubelis The inverse spectral method for colliding gravitational waves, Math. Phys. Anal. Geom., Volume 1 (1999), pp. 313-330 (A.S. Fokas, D. Tsoubelis, The inverse spectral method and the initial value problem for colliding gravitational waves, Preprint, Loughborough Univ., 1995)

[11] I. Hitzazis, D. Tsoubelis, The KdV equation on a finite interval, preprint

[12] X. Zhou Inverse scattering transform for systems with rational spectral dependence, J. Differential Equations, Volume 115 (1995), pp. 277-303

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

The modified KdV equation on a finite interval

Anne Boutet de Monvel; Dmitry Shepelsky

C. R. Math (2003)

The Camassa–Holm equation on the half-line with linearizable boundary condition

Anne Boutet de Monvel; Dmitry Shepelsky

C. R. Math (2010)