American Institute of Mathematical Sciences

June  2012, 32(6): 2063-2077. doi: 10.3934/dcds.2012.32.2063

Energy cascades for NLS on the torus

 1 CNRS & Univ. Montpellier 2, UMR 5149, CC 051, F-34095 Montpellier, France 2 INRIA & ENS Cachan Bretagne, Avenue Robert Schumann, F-35170 Bruz, France

Received  April 2011 Revised  July 2011 Published  February 2012

We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.
Citation: Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063
References:
 [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285. doi: 10.1007/s00220-002-0774-4. [2] D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135 (2006), 507-567. doi: 10.1215/S0012-7094-06-13534-2. [3] J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., 6 (1996), 201-230. doi: 10.1007/BF02247885. [4] _____, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2), 148 (1998), 363-439. [5] R. Carles, Cascade of phase shifts for nonlinear Schrödinger equations, J. Hyperbolic Differ. Equ., 4 (2007), 207-231. doi: 10.1142/S0219891607001112. [6] _____, "Semi-Classical Analysis for Nonlinear Schrödinger Equations," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. [7] R. Carles, E. Dumas and C. Sparber, Multiphase weakly nonlinear geometric optics for Schrödinger equations, SIAM J. Math. Anal., 42 (2010), 489-518. doi: 10.1137/090750871. [8] C. Cheverry, Cascade of phases in turbulent flows, Bull. Soc. Math. France, 134 (2006), 33-82. [9] 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), 39-113. doi: 10.1007/s00222-010-0242-2. [10] W. Craig and C. E. Wayne, Periodic solutions of nonlinear Schrödinger equations and the Nash-Moser method, in "Hamiltonian Mechanics" (Toruń, 1993), NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, (1994), 103-122. [11] L. H. Eliasson and S. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math. (2), 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [12] E. Faou, Geometric integration of Hamiltonian PDEs and applications to computational quantum mechanics, European Math. Soc., 2011, to appear. [13] E. Faou and B. Grébert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the torus, preprint, arXiv:1003.4845, 2010. [14] _____, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, Found. Comput. Math. 11 (2011), no. 4, 381-415. [15] L. Gauckler and C. Lubich, Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169. doi: 10.1007/s10208-010-9059-z. [16] B. Grébert, Birkhoff normal form and Hamiltonian PDEs, in "Partial Differential Equations and Applications," Sémin. Congr., 15, Soc. Math. France, Paris, 2007, 1-46. [17] S. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622. [18] S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2), 143 (1996), 149-179. doi: 10.2307/2118656. [19] C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153. doi: 10.1090/S0025-5718-08-02101-7. [20] L. van Veen, The quasi-periodic doubling cascade in the transition to weak turbulence, Phys. D, 210 (2005), 249-261. doi: 10.1016/j.physd.2005.07.020. [21] W.-M. Wang, Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities, preprint, arXiv:0907.3409, 2009.

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References:
 [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285. doi: 10.1007/s00220-002-0774-4. [2] D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135 (2006), 507-567. doi: 10.1215/S0012-7094-06-13534-2. [3] J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., 6 (1996), 201-230. doi: 10.1007/BF02247885. [4] _____, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2), 148 (1998), 363-439. [5] R. Carles, Cascade of phase shifts for nonlinear Schrödinger equations, J. Hyperbolic Differ. Equ., 4 (2007), 207-231. doi: 10.1142/S0219891607001112. [6] _____, "Semi-Classical Analysis for Nonlinear Schrödinger Equations," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. [7] R. Carles, E. Dumas and C. Sparber, Multiphase weakly nonlinear geometric optics for Schrödinger equations, SIAM J. Math. Anal., 42 (2010), 489-518. doi: 10.1137/090750871. [8] C. Cheverry, Cascade of phases in turbulent flows, Bull. Soc. Math. France, 134 (2006), 33-82. [9] 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), 39-113. doi: 10.1007/s00222-010-0242-2. [10] W. Craig and C. E. Wayne, Periodic solutions of nonlinear Schrödinger equations and the Nash-Moser method, in "Hamiltonian Mechanics" (Toruń, 1993), NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, (1994), 103-122. [11] L. H. Eliasson and S. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math. (2), 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [12] E. Faou, Geometric integration of Hamiltonian PDEs and applications to computational quantum mechanics, European Math. Soc., 2011, to appear. [13] E. Faou and B. Grébert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the torus, preprint, arXiv:1003.4845, 2010. [14] _____, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, Found. Comput. Math. 11 (2011), no. 4, 381-415. [15] L. Gauckler and C. Lubich, Nonlinear Schrödinger equations and their spectral semi-discretizations over long times, Found. Comput. Math., 10 (2010), 141-169. doi: 10.1007/s10208-010-9059-z. [16] B. Grébert, Birkhoff normal form and Hamiltonian PDEs, in "Partial Differential Equations and Applications," Sémin. Congr., 15, Soc. Math. France, Paris, 2007, 1-46. [17] S. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622. [18] S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2), 143 (1996), 149-179. doi: 10.2307/2118656. [19] C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153. doi: 10.1090/S0025-5718-08-02101-7. [20] L. van Veen, The quasi-periodic doubling cascade in the transition to weak turbulence, Phys. D, 210 (2005), 249-261. doi: 10.1016/j.physd.2005.07.020. [21] W.-M. Wang, Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities, preprint, arXiv:0907.3409, 2009.
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