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 a truly nonlinear behavior in infinite time.
@article{JEDP_2017____A9_0,
author = {Vila\c{c}a Da Rocha, Victor},
title = {Emphasizing nonlinear behaviors for cubic coupled {Schr\"odinger} systems},
journal = {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 = {http://www.numdam.org/articles/10.5802/jedp.659/}
}
TY - JOUR AU - Vilaça Da Rocha, Victor 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 - http://www.numdam.org/articles/10.5802/jedp.659/ DO - 10.5802/jedp.659 LA - en ID - JEDP_2017____A9_0 ER -
%0 Journal Article %A Vilaça Da Rocha, Victor %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 http://www.numdam.org/articles/10.5802/jedp.659/ %R 10.5802/jedp.659 %G en %F JEDP_2017____A9_0
Vilaça Da Rocha, Victor. Emphasizing nonlinear behaviors for cubic coupled Schrödinger systems. Journées équations aux dérivées partielles (2017), Talk no. 9, 13 p. doi : 10.5802/jedp.659. http://www.numdam.org/articles/10.5802/jedp.659/
[1] 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] 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] 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] Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension , Comm. Math. Phys., Volume 151 (1993) no. 3, pp. 619-645 http://projecteuclid.org/euclid.cmp/1104252243 | MR
[5] 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] Beating effects in cubic Schrödinger systems and growth of Sobolev norms, Nonlinearity, Volume 26 (2013) no. 5, pp. 1361-1376 | DOI | MR
[7] Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, Volume 3 (2015), e4, 63 pages | DOI | MR
[8] 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] 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] 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] 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] 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
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