American Institute of Mathematical Sciences

Advanced
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

[1]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & 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 & 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 & 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 & 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 & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021157
[6]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[18]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[19]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[20]

Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences