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

Anne Boutet de Monvel; Dmitry Shepelsky

doi:10.1016/j.crma.2010.05.002

Partial Differential Equations/Mathematical Physics

The Camassa–Holm equation on the half-line with linearizable boundary condition
[L'équation de Camassa–Holm sur la demi-droite avec condition aux limites linéarisable]

Présenté par : Gilles Lebeau

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

¹ Institut de mathématiques de Jussieu, université Paris Diderot Paris 7, 175, rue du Chevaleret, 75013 Paris, France
² Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 775-780.

Résumé
Abstract

Nous considérons un problème aux limites pour l'équation de Camassa–Holm sur la demi-droite $x > 0$ avec condition de Dirichlet homogène au bord $x = 0$ . Nous montrons que, comme dans le cas du problème de Cauchy sur la droite, la solution $u (x, t)$ s'exprime, sous forme paramétrique, en termes de la solution d'un problème de Riemann–Hilbert auxiliaire, entièrement déterminé par des fonctions spectrales associées aux seules données initiales. Cela permet d'appliquer la méthode de plus grande descente non linéaire et d'obtenir ainsi le comportement asymptotique de la solution pour les grandes valeurs du temps. Cette analyse met en évidence trois secteurs du quadrant $x > 0$ , $t > 0$ où la solution a des comportements asymptotiques de types différents.

We present a Riemann–Hilbert problem formalism for the initial boundary value problem for the Camassa–Holm equation on the half-line $x > 0$ with homogeneous Dirichlet boundary condition at $x = 0$ . We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann–Hilbert problem determined only by the initial data via associated spectral functions. This allows us to apply the nonlinear steepest descent method and to describe the large-time asymptotics of the solution. There are three sectors of the quarter plane $x > 0$ , $t > 0$ where the asymptotic behavior is qualitatively different.

Reçu le : 2010-01-19
Accepté le : 2010-05-27
Publié le : 2010-06-17

DOI : 10.1016/j.crma.2010.05.002

Affiliations des auteurs :

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

¹ Institut de mathématiques de Jussieu, université Paris Diderot Paris 7, 175, rue du Chevaleret, 75013 Paris, France
² Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

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     author = {Anne Boutet de Monvel and Dmitry Shepelsky},
     title = {The {Camassa{\textendash}Holm} equation on the half-line with linearizable boundary condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {775--780},
     publisher = {Elsevier},
     volume = {348},
     number = {13-14},
     year = {2010},
     doi = {10.1016/j.crma.2010.05.002},
     language = {en},
}

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Anne Boutet de Monvel; Dmitry Shepelsky. The Camassa–Holm equation on the half-line with linearizable boundary condition. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 775-780. doi : 10.1016/j.crma.2010.05.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.05.002/

Bibliographie
Cité par

[1] A. Boutet de Monvel; A.S. Fokas; D. Shepelsky The analysis of the global relation for the nonlinear Schrödinger equation on the half-line, Lett. Math. Phys., Volume 65 (2003), pp. 199-212

[2] A. Boutet de Monvel; A.S. Fokas; D. Shepelsky The modified KdV equation on the half-line, J. Inst. Math. Jussieu, Volume 3 (2004), pp. 139-164

[3] A. Boutet de Monvel; A.S. Fokas; D. Shepelsky Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys., Volume 263 (2006) no. 1, pp. 133-172 (See also C. R. Math. Acad. Sci. Paris, 337, 8, 2003, pp. 517-522)

[4] A. Boutet de Monvel; A. Kostenko; D. Shepelsky; G. Teschl Long-time asymptotics for the Camassa–Holm equation, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1559-1588

[5] A. Boutet de Monvel; D. Shepelsky Riemann–Hilbert problem in the inverse scattering for the Camassa–Holm equation on the line, C. R. Math. Acad. Sci. Paris (Math. Sci. Res. Inst. Publ.), Volume vol. 55 (2008) no. 10, pp. 53-75 (See also, 343, 2006, pp. 627-632)

[6] A. Boutet de Monvel; D. Shepelsky Long-time asymptotics of the Camassa–Holm equation on the line, Contemp. Math., Volume 458 (2008), pp. 99-116

[7] A. Boutet de Monvel; D. Shepelsky The Camassa–Holm equation on the half-line: A Riemann–Hilbert approach, J. Geom. Anal., Volume 18 (2008) no. 2, pp. 285-323

[8] R. Camassa; D.D. Holm An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Volume 71 (1993) no. 11, pp. 1661-1664

[9] R. Camassa; J. Huang; L. Lee On a completely integrable numerical scheme for a nonlinear shallow-water wave equation, J. Nonlinear Math. Phys., Volume 12 (2005) no. suppl. 1, pp. 146-162

[10] A. Constantin On the scattering problem for the Camassa–Holm equation, R. Soc. Lond. Proc. Ser. A, Volume 457 (2001) no. 2008, pp. 953-970

[11] A. Constantin; J. Escher Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 26 (1998) no. 2, pp. 303-328

[12] P. Deift; X. Zhou A steepest descent method for oscillatory Riemann–Hilbert problem. Asymptotics for the MKdV equation, Ann. Math., Volume 137 (1993) no. 2, pp. 295-368

[13] J. Escher; Zh. Yin Initial boundary value problem for nonlinear dispersive wave equations, J. Funct. Anal., Volume 256 (2009) no. 2, pp. 479-508

[14] A.S. Fokas A unified transform method for solving linear and certain nonlinear PDE's, Proc. R. Soc. Lond. A, Volume 453 (1997), pp. 1411-1443

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

[16] A.S. Fokas A generalised Dirichlet to Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math., Volume LVIII (2005), pp. 639-670

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

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

[19] A.S. Fokas; J. Lenells Explicit soliton asymptotics for the Korteweg–de Vries equation on the half-line, Nonlinearity, Volume 23 (2010), pp. 937-976

[20] J. Lenells; A.S. Fokas An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons, Inverse Problems, Volume 25 (2009) no. 11, p. 115006

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