Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition

Anne Boutet de Monvel; Alexander Its; Vladimir Kotlyarov

doi:10.1016/j.crma.2007.10.018

Partial Differential Equations/Mathematical Physics

Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition

Presented by: Paul Malliavin

Anne Boutet de Monvel ¹; Alexander Its ²; Vladimir Kotlyarov ³

¹ Institut de mathématiques de Jussieu, case 7012, Université Paris 7, 2, place Jussieu, 75251 Paris cedex 05, France
² Indiana University – Purdue University, Indianapolis, IN 46202-3216, USA
³ Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 615-620.

Abstract
Résumé

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic ( $a e^{2 i ω t}$ ). The goal of this Note is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem. We show that the solution of the IBV problem has different asymptotic behaviors in different regions. In the region $x > 4 b t$ ( $b = \sqrt{(a^{2} - ω) / 2} > 0$ ) the solution has the form of a Zakharov–Manakov vanishing asymptotics. In the region $4 b t - \frac{1}{2 a} N \log t < x < 4 b t$ , where N is an integer, the solution behaves as a finite train of asymptotic solitons. In the region $4 (b - a \sqrt{2}) t < x < 4 b t$ the solution is a modulated elliptic wave. Finally, in the sector $0 < x < 4 (b - a \sqrt{2}) t$ the solution is a plane wave.

Nous considérons l'équation de Schrödinger non linéaire focalisante sur le premier quadrant ( $x > 0$ , $t > 0$ ), avec donnée initiale décroissante à l'infini et donnée au bord t-périodique, de la forme $a e^{2 i ω t}$ . Notre objectif est d'étudier le comportement asymptotique de la solution de ce problème aux limites. Notre méthode consiste à faire l'analyse asymptotique d'un problème de Riemann–Hilbert matriciel associé. Nous montrons que la solution du problème aux limites a des comportements asymptotiques différents suivant la région dans laquelle $(x, t)$ tend vers l'infini. Dans le secteur $x > 4 b t$ , pour $b = \sqrt{(a^{2} - ω) / 2} > 0$ , le comportement asymptotique est décrit par des formules du type de Zakharov–Manakov. Dans la région $4 b t - \frac{1}{2 a} N \log t < x < 4 b t$ , où N est un entier, la solution se comporte comme un train fini de solitons asymptotiques. Dans le secteur $4 (b - a \sqrt{2}) t < x < 4 b t$ la solution est comme une onde elliptique modulée. Enfin, dans le secteur $0 < x < 4 (b - a \sqrt{2}) t$ la solution se comporte comme une onde plane.

Received: 2007-06-14
Accepted: 2007-10-04
Published online: 2007-11-19

DOI: 10.1016/j.crma.2007.10.018

Author's affiliations:

Anne Boutet de Monvel ¹; Alexander Its ²; Vladimir Kotlyarov ³

¹ Institut de mathématiques de Jussieu, case 7012, Université Paris 7, 2, place Jussieu, 75251 Paris cedex 05, France
² Indiana University – Purdue University, Indianapolis, IN 46202-3216, USA
³ Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

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     author = {Anne Boutet de Monvel and Alexander Its and Vladimir Kotlyarov},
     title = {Long-time asymptotics for the focusing {NLS} equation with time-periodic boundary condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {615--620},
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     volume = {345},
     number = {11},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.018},
     language = {en},
}

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Anne Boutet de Monvel; Alexander Its; Vladimir Kotlyarov. Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 615-620. doi : 10.1016/j.crma.2007.10.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.018/

References
Cited by

[1] A. Boutet de Monvel; V. Kotlyarov Generation of asymptotic solitons of the Nonlinear Schrödinger equation by boundary data, J. Math. Phys., Volume 44 (2003) no. 8, pp. 3185-3215

[2] A. Boutet de Monvel; V. Kotlyarov The focusing nonlinear Schrödinger equation on the quarter plane with time-periodic boundary condition: a Riemann–Hilbert approach, J. Inst. Math. Jussieu, Volume 6 (2007) no. 4, pp. 579-611

[3] R. Buckingham; S. Venakides Long-time asymptotics of the nonlinear Schrödinger equation shock problem, Comm. Pure Appl. Math., Volume 60 (2007) no. 9, pp. 1349-1414

[4] P. Deift; X. Zhou A steepest descent method for oscillatory Riemann–Hilbert problems, Bull. Amer. Math. Soc. (N.S.), Volume 26 (1992) no. 1, pp. 119-123

[5] L.D. Faddeev; L.A. Takhtajan Hamiltonian Methods in the Theory of Solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987

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