Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schr\"odinger equations

V.E. Zakharov, Weak turbulence in media with decay spectrum. Zh. Priklad. Tech. Fix. 4, 35–39 (1965) [J. Appl. Mech. Tech. Phys. 4, 22–24 (1965)]

1965

Earlier work this paper cites.

R. H. Kraichman, Inertial ranges in two-dimensional turbulence , Phys. Fluids 10 (7) 1417–1423 (1967)

1967

Earlier work this paper cites.

V. E. Zakharov, A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media , Soviet Physics JETP 34 (1972), no. 1, 62–69.; translated from Z ˇ \check{Z} . Éksper. Teoret. Fiz. 61 (1971), no. 1, 118–134

1971

Earlier work this paper cites.

S. Dyachenko, A.C. Newell, A.C.A. Pusharev, V. E. Zakharov, Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation , Phys. D 57 (1992), no. 1-2, 96–160

1992

Earlier work this paper cites.

V.E. Zakharov, V.S. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence I , Springer Series in Nonlinear Dynamics, Springer-Verlag, 1992

1992

Earlier work this paper cites.

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I: Schrödinger Equations , GAFA 3 (1993), 107–156

1993

Earlier work this paper cites.

J. Bourgain, Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations , Geom. Funct. Anal. 5 (1995), no. 2, 105–140

1995

Earlier work this paper cites.

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE , Internat. Math. Res. Notices 1996, no. 6, 277–304

1996

Earlier work this paper cites.

S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations , Geom. Funct. Anal. 7 (1997), no. 2, 338–363

1997

Earlier work this paper cites.

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations , Duke Math. J. 86 (1997), no. 1, 109–142

1997

Earlier work this paper cites.

J. Bourgain, Problems in Hamiltonian PDE’s , Geom. Funct. Anal. 2000, Special Volume, Part I, 32–56

2000

Cited alongside, same era.

J. E. Colliander, J.-M. Delort, C. E. Kenig, G. Staffilani, Bilinear estimates and applications to 2D NLS , Trans. Amer. Math. Soc. 353 (2001), no. 8, 3307–3325

2001

Cited alongside, same era.

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Polynomial growth and orbital instability bounds for L 2 L^{2} -subcritical NLS below the energy norm , Comm. Pure Appl. Anal. 2 (2003), 33–50

2003

Cited alongside, same era.

J. Bourgain, A remark on normal forms and the I-method for periodic NLS , J. Anal. Math., 94 (2004), 125–157

2004

Cited alongside, same era.

J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations , Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1331–1357

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation , Invent. Math. 181 (2010), no. 1, 39–113

2010

Later among the works it cites.

P. Gerard, S. Grellier, The cubic Szegö equation , Ann. Sci. Ec. Norm. Supér.(4), 43(5):761–810, 2010

2010

Later among the works it cites.

P. Gerard, S. Grellier, Effective integrable dynamics for some nonlinear wave equation , preprint arXiv:1110.5719 (2011)

Original

2011

Later among the works it cites.

O. Pocovnicu, Explicit formula for the solution of the Szegö equation on the real line and applications , Discrete Contin. Dyn. Syst., 31(3):607–649, 2011

2011

Later among the works it cites.

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger Equations on S 1 S^{1} , Differential Integral Equations 24 (2011), no. 7-8, 653–718