Modified scattering and beating effect for coupled Schrödinger systems on product spaces with small initial data

Vilaça Da Rocha, Victor

doi:10.1090/tran/7396

Modified scattering and beating effect for coupled Schrödinger systems on product spaces with small initial data
HTML articles powered by AMS MathViewer

by Victor Vilaça Da Rocha PDF

Trans. Amer. Math. Soc. 371 (2019), 4743-4768 Request permission

Abstract:

In this paper, we study a coupled nonlinear Schrödinger system with small initial data in a product space. We establish a modified scattering of the solutions of this system and we construct a modified wave operator. The study of the resonant system, which provides the asymptotic dynamics, allows us to highlight a control of the Sobolev norms and interesting dynamics with the beating effect. The proof uses a recent work of Hani, Pausader, Tzvetkov, and Visciglia for the modified scattering, and a recent work of Grébert, Paturel, and Thomann for the study of the resonant system.

References

G. Agrawal and R. Boyd, Contemporary Nonlinear Optics, pp. 60–67, 1992.
Valeria Banica, Rémi Carles, and Thomas Duyckaerts, On scattering for NLS: from Euclidean to hyperbolic space, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1113–1127. MR 2505694, DOI 10.3934/dcds.2009.24.1113
Valeria Banica, Rémi Carles, and Gigliola Staffilani, Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space, Geom. Funct. Anal. 18 (2008), no. 2, 367–399. MR 2421543, DOI 10.1007/s00039-008-0663-x
Rémi Carles, Eric Dumas, and Christof Sparber, Multiphase weakly nonlinear geometric optics for Schrödinger equations, SIAM J. Math. Anal. 42 (2010), no. 1, 489–518. MR 2607351, DOI 10.1137/090750871
Rémi Carles and Thomas Kappeler, Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces, Bull. Soc. Math. France 145 (2017), no. 4, 623–642 (English, with English and French summaries). MR 3770969, DOI 10.24033/bsmf.2749
B. Cassano and M. Tarulli, $H^1$-scattering for systems of $N$-defocusing weakly coupled NLS equations in low space dimensions, J. Math. Anal. Appl. 430 (2015), no. 1, 528–548. MR 3347230, DOI 10.1016/j.jmaa.2015.05.008
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal. 33 (2001), no. 3, 649–669. MR 1871414, DOI 10.1137/S0036141001384387
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113. MR 2651381, DOI 10.1007/s00222-010-0242-2
Luiz Gustavo Farah and Ademir Pastor, Scattering for a 3D coupled nonlinear Schrödinger system, J. Math. Phys. 58 (2017), no. 7, 071502, 33. MR 3671163, DOI 10.1063/1.4993224
Patrick Gérard and Sandrine Grellier, On the growth of Sobolev norms for the cubic Szegő equation, Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2014–2015, Ed. Éc. Polytech., Palaiseau, 2016, pp. Exp No. XI, 20. MR 3560350
J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys. 151 (1993), no. 3, 619–645. MR 1207269, DOI 10.1007/BF02097031
B. Grébert, Du ressort à l’atome, une histoire de résonance..., http://www.lebesgue.fr/fr/ video/5min/grebert, 2016.
Benoît Grébert, Éric Paturel, and Laurent Thomann, Beating effects in cubic Schrödinger systems and growth of Sobolev norms, Nonlinearity 26 (2013), no. 5, 1361–1376. MR 3056129, DOI 10.1088/0951-7715/26/5/1361
Benoît Grébert, Éric Paturel, and Laurent Thomann, Modified scattering for the cubic Schrödinger equation on product spaces: the nonresonant case, Math. Res. Lett. 23 (2016), no. 3, 841–861. MR 3533198, DOI 10.4310/MRL.2016.v23.n3.a13
Benoît Grébert and Laurent Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 3, 455–477. MR 2926244, DOI 10.1016/j.anihpc.2012.01.005
Benoît Grébert and Carlos Villegas-Blas, On the energy exchange between resonant modes in nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 1, 127–134 (English, with English and French summaries). MR 2765514, DOI 10.1016/j.anihpc.2010.11.004
Zaher Hani, Benoit Pausader, Nikolay Tzvetkov, and Nicola Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi 3 (2015), e4, 63. MR 3406826, DOI 10.1017/fmp.2015.5
Zaher Hani and Laurent Thomann, Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, Comm. Pure Appl. Math. 69 (2016), no. 9, 1727–1776. MR 3530362, DOI 10.1002/cpa.21594
Lysianne Hari and Nicola Visciglia, Small data scattering for energy critical NLKG on product spaces ${\bf {R}}^d\times \mathcal M^2$, Commun. Contemp. Math. 20 (2018), no. 2, 1750036, 11. MR 3730754, DOI 10.1142/S0219199717500365
E. Haus and M. Procesi, KAM for beating solutions of the quintic NLS, Comm. Math. Phys. 354 (2017), no. 3, 1101–1132. MR 3668616, DOI 10.1007/s00220-017-2925-7
Nakao Hayashi and Pavel I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math. 120 (1998), no. 2, 369–389. MR 1613646, DOI 10.1353/ajm.1998.0011
Alexandru D. Ionescu, Benoit Pausader, and Gigliola Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE 5 (2012), no. 4, 705–746. MR 3006640, DOI 10.2140/apde.2012.5.705
Jun Kato and Fabio Pusateri, A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations 24 (2011), no. 9-10, 923–940. MR 2850346
Donghyun Kim, A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension, Asymptot. Anal. 98 (2016), no. 1-2, 79–90. MR 3502373, DOI 10.3233/ASY-161362
Chunhua Li and Hideaki Sunagawa, On Schrödinger systems with cubic dissipative nonlinearities of derivative type, Nonlinearity 29 (2016), no. 5, 1537–1563. MR 3481342, DOI 10.1088/0951-7715/29/5/1537
Tohru Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), no. 3, 479–493. MR 1121130, DOI 10.1007/BF02101876
T. Tao. A physical space proof of the bilinear Strichartz and local smoothing estimates for the Schrödinger equation, UTN-FRA, Buenos Aires.
Nikolay Tzvetkov and Nicola Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces, Comm. Partial Differential Equations 37 (2012), no. 1, 125–135. MR 2864809, DOI 10.1080/03605302.2011.574306
Nikolay Tzvetkov and Nicola Visciglia, Well-posedness and scattering for nonlinear Schrödinger equations on $\Bbb {R}^d\times \Bbb {T}$ in the energy space, Rev. Mat. Iberoam. 32 (2016), no. 4, 1163–1188. MR 3593518, DOI 10.4171/RMI/911
V. Vilaça Da Rocha, Emphasising nonlinear behaviors for cubic coupled Schrödinger systems, Theses, Université de Nantes Faculté des sciences et des techniques, June 2017.
Victor V. Da Rocha, Asymptotic behavior of solutions to the cubic coupled Schrödinger systems in one space dimension, Dyn. Partial Differ. Equ. 13 (2016), no. 1, 53–74. MR 3484080, DOI 10.4310/DPDE.2016.v13.n1.a3
Haiyan Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z. 286 (2017), no. 1-2, 443–489. MR 3648505, DOI 10.1007/s00209-016-1768-9
C. Zuily, Éléments de distributions et d’équations aux dérivées partielles : cours et problèmes résolus, Sciences sup. Dunod, 2002.