Emphasizing nonlinear behaviors for cubic coupled Schrödinger systems
Victor Vilaça Da Rocha1
1 Laboratoire de Mathématiques Jean Leray Université de Nantes UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France
Journées équations aux dérivées partielles (2017), Exposé no. 9, 13 p.

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 ℝ×𝕋 a truly nonlinear behavior in infinite time.

Publié le :
DOI : 10.5802/jedp.659
Victor Vilaça Da Rocha 1

1 Laboratoire de Mathématiques Jean Leray Université de Nantes UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France 
@incollection{JEDP_2017____A9_0,
     author = {Victor Vila\c{c}a Da Rocha},
     title = {Emphasizing nonlinear behaviors for cubic coupled {Schr\"odinger} systems},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:9},
     pages = {1--13},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2017},
     doi = {10.5802/jedp.659},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.659/}
}
TY  - JOUR
AU  - Victor Vilaça Da Rocha
TI  - Emphasizing nonlinear behaviors for cubic coupled Schrödinger systems
JO  - Journées équations aux dérivées partielles
N1  - talk:9
PY  - 2017
SP  - 1
EP  - 13
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.659/
DO  - 10.5802/jedp.659
LA  - en
ID  - JEDP_2017____A9_0
ER  - 
%0 Journal Article
%A Victor Vilaça Da Rocha
%T Emphasizing nonlinear behaviors for cubic coupled Schrödinger systems
%J Journées équations aux dérivées partielles
%Z talk:9
%D 2017
%P 1-13
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.659/
%R 10.5802/jedp.659
%G en
%F JEDP_2017____A9_0
Victor Vilaça Da Rocha. Emphasizing nonlinear behaviors for cubic coupled Schrödinger systems. Journées équations aux dérivées partielles (2017), Exposé no. 9, 13 p. doi : 10.5802/jedp.659. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.659/

[1] J. Bourgain Problems in Hamiltonian PDE’s, Geom. Funct. Anal. (2000) no. Special Volume, Part I, pp. 32-56 GAFA 2000 (Tel Aviv, 1999) | DOI | MR

[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, pp. 277-304 | DOI | MR

[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., Volume 181 (2010) no. 1, pp. 39-113 | DOI | MR

[4] J. Ginibre; T. Ozawa Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n≥2, Comm. Math. Phys., Volume 151 (1993) no. 3, pp. 619-645 http://projecteuclid.org/euclid.cmp/1104252243 | MR

[5] Benoît Grébert; Thomas Kappeler The defocusing NLS equation and its normal form, EMS Series of Lectures in Math., European Mathematical Society, Zürich, 2014, x+166 pages | DOI | MR

[6] Benoît Grébert; Éric Paturel; Laurent Thomann Beating effects in cubic Schrödinger systems and growth of Sobolev norms, Nonlinearity, Volume 26 (2013) no. 5, pp. 1361-1376 | DOI | MR

[7] Zaher Hani; Benoit Pausader; Nikolay Tzvetkov; Nicola Visciglia Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, Volume 3 (2015), e4, 63 pages | DOI | MR

[8] Nakao Hayashi; Pavel I. Naumkin Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., Volume 120 (1998) no. 2, pp. 369-389 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.2hayashi.pdf | MR

[9] Jun Kato; Fabio Pusateri A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations, Volume 24 (2011) no. 9-10, pp. 923-940 | MR

[10] Tohru Ozawa Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., Volume 139 (1991) no. 3, pp. 479-493 http://projecteuclid.org/euclid.cmp/1104203467 | MR

[11] Gigliola Staffilani On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., Volume 86 (1997) no. 1, pp. 109-142 | DOI | MR

[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., Volume 61 (1971) no. 1, pp. 118-134 | MR

Cité par Sources :