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Journées équations aux dérivées partielles

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Victor Vilaça Da Rocha
Emphasizing nonlinear behaviors for cubic coupled Schrödinger systems
Journées équations aux dérivées partielles (2017), Exp. No. 9, 13 p., doi: 10.5802/jedp.659
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Résumé - Abstract

The purpose of this note is to propose a study of various nonlinear behaviors for a system of two coupled cubic Schrödinger equations with small initial data. Depending on the choice of the spatial domain, we highlight different examples of nonlinear behaviors. The goal is to mix the approaches of the study on the torus (with a truly nonlinear behavior) and of the study on the real line (with an infinite behavior) in order to obtain on the product space $\mathbb{R}\times \mathbb{T}$ a truly nonlinear behavior in infinite time.


[1] J. Bourgain, “Problems in Hamiltonian PDE’s”, Geom. Funct. Anal. (2000) no. Special Volume, Part I, p. 32-56, GAFA 2000 (Tel Aviv, 1999) Article
[2] Jean Bourgain, “On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE”, Internat. Math. Res. Notices (1996) no. 6, p. 277-304 Article
[3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka & T. Tao, “Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation”, Invent. Math. 181 (2010) no. 1, p. 39-113 Article
[4] J. Ginibre & T. Ozawa, “Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n\ge 2$”, Comm. Math. Phys. 151 (1993) no. 3, p. 619-645
[5] Benoît Grébert & Thomas Kappeler, The defocusing NLS equation and its normal form, EMS Series of Lectures in Math., European Mathematical Society, Zürich, 2014 Article
[6] Benoît Grébert, Éric Paturel & Laurent Thomann, “Beating effects in cubic Schrödinger systems and growth of Sobolev norms”, Nonlinearity 26 (2013) no. 5, p. 1361-1376 Article
[7] Zaher Hani, Benoit Pausader, Nikolay Tzvetkov & Nicola Visciglia, “Modified scattering for the cubic Schrödinger equation on product spaces and applications”, Forum Math. Pi 3 (2015) Article
[8] Nakao Hayashi & Pavel I. Naumkin, “Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations”, Amer. J. Math. 120 (1998) no. 2, p. 369-389
[9] Jun Kato & Fabio Pusateri, “A new proof of long-range scattering for critical nonlinear Schrödinger equations”, Differential Integral Equations 24 (2011) no. 9-10, p. 923-940
[10] Tohru Ozawa, “Long range scattering for nonlinear Schrödinger equations in one space dimension”, Comm. Math. Phys. 139 (1991) no. 3, p. 479-493
[11] Gigliola Staffilani, “On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations”, Duke Math. J. 86 (1997) no. 1, p. 109-142 Article
[12] V. E. Zakharov & A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Ž. Èksper. Teoret. Fiz. 61 (1971) no. 1, p. 118-134
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