American Institute of Mathematical Sciences

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

Energy cascades for NLS on the torus

Rémi Carles 1, and  Erwan Faou 2,

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.
Keywords: resonant system, Nonlinear Schrödinger equation, energy cascade..
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35C20, 37K5.
Citation: Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

_____, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2), 148 (1998), 363-439.  Google Scholar

[5]

R. Carles, Cascade of phase shifts for nonlinear Schrödinger equations, J. Hyperbolic Differ. Equ., 4 (2007), 207-231. doi: 10.1142/S0219891607001112.  Google Scholar

[6]

_____, "Semi-Classical Analysis for Nonlinear Schrödinger Equations," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.  Google Scholar

[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.  Google Scholar

[8]

C. Cheverry, Cascade of phases in turbulent flows, Bull. Soc. Math. France, 134 (2006), 33-82.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. Faou, Geometric integration of Hamiltonian PDEs and applications to computational quantum mechanics, European Math. Soc., 2011, to appear. Google Scholar

[13]

E. Faou and B. Grébert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the torus, preprint, arXiv:1003.4845, 2010. Google Scholar

[14]

_____, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, Found. Comput. Math. 11 (2011), no. 4, 381-415.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

W.-M. Wang, Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities, preprint, arXiv:0907.3409, 2009. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

_____, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2), 148 (1998), 363-439.  Google Scholar

[5]

R. Carles, Cascade of phase shifts for nonlinear Schrödinger equations, J. Hyperbolic Differ. Equ., 4 (2007), 207-231. doi: 10.1142/S0219891607001112.  Google Scholar

[6]

_____, "Semi-Classical Analysis for Nonlinear Schrödinger Equations," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.  Google Scholar

[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.  Google Scholar

[8]

C. Cheverry, Cascade of phases in turbulent flows, Bull. Soc. Math. France, 134 (2006), 33-82.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. Faou, Geometric integration of Hamiltonian PDEs and applications to computational quantum mechanics, European Math. Soc., 2011, to appear. Google Scholar

[13]

E. Faou and B. Grébert, A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the torus, preprint, arXiv:1003.4845, 2010. Google Scholar

[14]

_____, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, Found. Comput. Math. 11 (2011), no. 4, 381-415.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal., 7 (1997), 338-363. doi: 10.1007/PL00001622.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

W.-M. Wang, Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities, preprint, arXiv:0907.3409, 2009. Google Scholar

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