The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation

Faou, Erwan; Germain, Pierre; Hani, Zaher

doi:10.1090/jams/845

The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation
HTML articles powered by AMS MathViewer

by Erwan Faou, Pierre Germain and Zaher Hani

J. Amer. Math. Soc. 29 (2016), 915-982

DOI: https://doi.org/10.1090/jams/845

Published electronically: October 20, 2015

PDF | Request permission

Abstract:

We consider the cubic nonlinear Schrödinger (NLS) equation set on a two-dimensional box of size $L$ with periodic boundary conditions. By taking the large-box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on $\mathbb {R}^2$ that approximates the dynamics of the frequency modes. The large-box limit and the weak nonlinearity limit are also performed in weak (or wave) turbulence theory, to which this work is related. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under the Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows one to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.

References

V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295, DOI 10.1007/978-1-4757-2063-1
Jonathan Bennett, Neal Bez, Anthony Carbery, and Dirk Hundertmark, Heat-flow monotonicity of Strichartz norms, Anal. PDE 2 (2009), no. 2, 147–158. MR 2547132, DOI 10.2140/apde.2009.2.147
Jonathan Bennett, Anthony Carbery, and James Wright, A non-linear generalisation of the Loomis-Whitney inequality and applications, Math. Res. Lett. 12 (2005), no. 4, 443–457. MR 2155223, DOI 10.4310/MRL.2005.v12.n4.a1
Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008), no. 5, 1343–1415. MR 2377493, DOI 10.1007/s00039-007-0619-6
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
Jean Bourgain, Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445. MR 1374420
J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2) 148 (1998), no. 2, 363–439. MR 1668547, DOI 10.2307/121001
J. Bourgain, On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 1–20. MR 2331676
Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Math. 20 (1976), no. 2, 151–173. MR 412366, DOI 10.1016/0001-8708(76)90184-5
David Cai, Andrew J. Majda, David W. McLaughlin, and Esteban G. Tabak, Dispersive wave turbulence in one dimension, Phys. D 152/153 (2001), 551–572. Advances in nonlinear mathematics and science. MR 1837929, DOI 10.1016/S0167-2789(01)00193-2
E. A. Carlen, E. H. Lieb, and M. Loss, A sharp analog of Young’s inequality on $S^N$ and related entropy inequalities, J. Geom. Anal. 14 (2004), no. 3, 487–520. MR 2077162, DOI 10.1007/BF02922101
Rémi Carles and Erwan Faou, Energy cascades for NLS on the torus, Discrete Contin. Dyn. Syst. 32 (2012), no. 6, 2063–2077. MR 2885798, DOI 10.3934/dcds.2012.32.2063
Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. MR 1055532, DOI 10.1016/0362-546X(90)90023-A
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
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
C. Connaughton, S. Nazarenko, and A. Pushkarev, Discreteness and quasiresonances in weak turbulence of capillary waves, Phys. Rev. E 63 (2001), 046306.
P. Denissenko, S. Lukaschuk, and S. Nazarenko, Gravity wave turbulence in a laboratory flume, Phys. Rev. Lett. 99 (2007), 014501.
Benjamin Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d\geq 3$, J. Amer. Math. Soc. 25 (2012), no. 2, 429–463. MR 2869023, DOI 10.1090/S0894-0347-2011-00727-3
B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, available at arXiv:1104.1114.
L. Hakan Eliasson and Sergei B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math. (2) 172 (2010), no. 1, 371–435. MR 2680422, DOI 10.4007/annals.2010.172.371
Erwan Faou, Ludwig Gauckler, and Christian Lubich, Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Comm. Partial Differential Equations 38 (2013), no. 7, 1123–1140. MR 3169740, DOI 10.1080/03605302.2013.785562
Damiano Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 739–774. MR 2341830, DOI 10.4171/JEMS/95
Jiansheng Geng, Xindong Xu, and Jiangong You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math. 226 (2011), no. 6, 5361–5402. MR 2775905, DOI 10.1016/j.aim.2011.01.013
Patrick Gérard and Sandrine Grellier, The cubic Szegő equation, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 5, 761–810 (English, with English and French summaries). MR 2721876, DOI 10.24033/asens.2133
Patrick Gérard and Sandrine Grellier, Invariant tori for the cubic Szegö equation, Invent. Math. 187 (2012), no. 3, 707–754. MR 2944951, DOI 10.1007/s00222-011-0342-7
Patrick Gérard and Sandrine Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE 5 (2012), no. 5, 1139–1155. MR 3022852, DOI 10.2140/apde.2012.5.1139
P. Germain, Z. Hani, and L. Thomann, On the continuous resonant equation for NLS. I. Deterministic analysis, available at arXiv:1501.03760. To appear in J. Math. Pures Appl. (9).
P. Germain, Z. Hani, and L. Thomann, On the continuous resonant equation for NLS. II. Statistical study, available at arXiv:1502.05643. To appear in Anal. PDE.
M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 1, 71–149. MR 3312404, DOI 10.4171/JEMS/499
Zaher Hani, Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 211 (2014), no. 3, 929–964. MR 3158811, DOI 10.1007/s00205-013-0689-6
Zaher Hani and Benoit Pausader, On scattering for the quintic defocusing nonlinear Schrödinger equation on $\Bbb R\times \Bbb T^2$, Comm. Pure Appl. Math. 67 (2014), no. 9, 1466–1542. MR 3245101, DOI 10.1002/cpa.21481
Z. Hani and L. Thomann, Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, available at arXiv:1408.6213. To appear in Comm. Pure Appl. Math.
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
K. Hasselmann, On the non-linear energy transfer in a gravity-wave spectrum. I. General theory, J. Fluid Mech. 12 (1962), 481–500. MR 136205, DOI 10.1017/S0022112062000373
K. Hasselmann, On the non-linear energy transfer in a gravity wave spectrum. II. Conservation theorems; wave-particle analogy; irreversibility, J. Fluid Mech. 15 (1963), 273–281. MR 151308, DOI 10.1017/S0022112063000239
Dirk Hundertmark and Vadim Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not. , posted on (2006), Art. ID 34080, 18. MR 2219206, DOI 10.1155/IMRN/2006/34080
Vojtěch Jarník, Über die Gitterpunkte auf konvexen Kurven, Math. Z. 24 (1926), no. 1, 500–518 (German). MR 1544776, DOI 10.1007/BF01216795
E. Kartashova, Wave resonances in systems with discrete spectra, Nonlinear waves and weak turbulence, Amer. Math. Soc. Transl. Ser. 2, vol. 182, Amer. Math. Soc., Providence, RI, 1998, pp. 95–129. MR 1618507, DOI 10.1090/trans2/182/04
E. Kartashova, Exact and quasiresonances in discrete water wave turbulence, Phys. Rev. Lett. 98 (2007), 214502.
Elena Kartashova, Sergey Nazarenko, and Oleksii Rudenko, Resonant interactions of nonlinear water waves in a finite basin, Phys. Rev. E (3) 78 (2008), no. 1, 016304, 9. MR 2496187, DOI 10.1103/PhysRevE.78.016304
E. Kartashova, Discrete wave turbulence, Europhys. Lett. 87 (2009), 44001.
Elena Kartashova, Nonlinear resonance analysis, Cambridge University Press, Cambridge, 2011. Theory, computation, applications. MR 2752246
Rowan Killip, Terence Tao, and Monica Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203–1258. MR 2557134, DOI 10.4171/JEMS/180
Nobu Kishimoto, Remark on the periodic mass critical nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2649–2660. MR 3209321, DOI 10.1090/S0002-9939-2014-12024-7
Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179–208. MR 1069246, DOI 10.1007/BF01233426
Jani Lukkarinen and Herbert Spohn, Weakly nonlinear Schrödinger equation with random initial data, Invent. Math. 183 (2011), no. 1, 79–188. MR 2755061, DOI 10.1007/s00222-010-0276-5
V. S. L’vov and S. Nazarenko, Discrete and mesoscopic regimes of finite-size wave turbulence, Phys. Rev. E 82 (2010), 056322.
A. J. Majda, D. W. McLaughlin, and E. G. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci. 7 (1997), no. 1, 9–44. MR 1431687, DOI 10.1007/BF02679124
F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices 8 (1998), 399–425. MR 1628235, DOI 10.1155/S1073792898000270
François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
Sergey Nazarenko, Wave turbulence, Lecture Notes in Physics, vol. 825, Springer, Heidelberg, 2011. MR 3014432, DOI 10.1007/978-3-642-15942-8
S. Nazarenko, Sandpile behaviour in discrete water-wave turbulence, J. Stat. Mech. Theory Exp. 2006 (2006), L02002.
R. Peierls, The kinetic theory of thermal conduction in crystals, Ann. Phys. 3 (1929), 1055–1101.
Oana Pocovnicu, Traveling waves for the cubic Szegő equation on the real line, Anal. PDE 4 (2011), no. 3, 379–404. MR 2872121, DOI 10.2140/apde.2011.4.379
Oana Pocovnicu, Explicit formula for the solution of the Szegö equation on the real line and applications, Discrete Contin. Dyn. Syst. 31 (2011), no. 3, 607–649. MR 2825631, DOI 10.3934/dcds.2011.31.607
C. Procesi and M. Procesi, A KAM algorithm for the resonant non-linear Schrödinger equation, Adv. Math. 272 (2015), 399–470. MR 3303238, DOI 10.1016/j.aim.2014.12.004
Herbert Spohn, The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics, J. Stat. Phys. 124 (2006), no. 2-4, 1041–1104. MR 2264633, DOI 10.1007/s10955-005-8088-5
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
Terence Tao, Poincaré’s legacies, pages from year two of a mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. MR 2523047, DOI 10.1090/mbk/066
Mitsuhiro Tanaka and Naoto Yokoyama, Effects of discretization of the spectrum in water-wave turbulence, Fluid Dynam. Res. 34 (2004), no. 3, 199–216. MR 2034402, DOI 10.1016/j.fluiddyn.2003.12.001
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
V. E. Zakharov and N. N. Filonenko, Energy spectrum for the stochastic oscillations of a fluid surface, Kohl. Acad. Nauk SSSR 170 (1966), 1292–1295. (translation: Sov. Phys. Dokl. 11, 881–884 (1967)).
V. E. Zakharov and N. N. Filonenko, Weak turbulence of capillary waves, J. Applied Mech. Tech. Phys. 8 (1967), 37–40.
V. E. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence 1. Wave Turbulence, Springer, Berlin, 1992.
Vladimir Zakharov, Frédéric Dias, and Andrei Pushkarev, One-dimensional wave turbulence, Phys. Rep. 398 (2004), no. 1, 1–65. MR 2073490, DOI 10.1016/j.physrep.2004.04.002
V. E. Zakharov, A. C. Korotkevich, and A. N. Pushkarev, Mesoscopic wave turbulence, JETP Lett. 82 (2005), 487–491.