Energy cascades for NLS on the torus

Previous Article
A minimal approach to the theory of global attractors
DCDS Home
This Issue
Next Article
Absolute and delay-dependent stability of equations with a distributed delay

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

Abstract
Related Papers

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 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.

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.

[1]	Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[2]	Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
[3]	J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665
[4]	Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[5]	Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157
[6]	Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
[7]	Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323
[8]	Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[9]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm. Communications on Pure and Applied Analysis, 2003, 2 (1) : 33-50. doi: 10.3934/cpaa.2003.2.33
[10]	D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]	Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
[12]	Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286
[13]	Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565
[14]	Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[15]	Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030
[16]	Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[17]	Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[18]	Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022039
[19]	Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[20]	Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Rémi Carles Erwan Faou
on Google Scholar
Rémi Carles Erwan Faou

[Back to Top]