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Absolute and delay-dependent stability of equations with a distributed delay
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 |
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. |
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. |
[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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