Abstract
This paper summarizes a talk given in the PDE Session at the 2006 International Congress on Mathematical Physics about joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao. We build new smooth solutions of the cubic defocussing nonlinear Schrödinger equation on the two dimensional torus which are weakly turbulent: given any δ≪1,K≫1,s>1, we construct smooth initial data u 0 in the Sobolev space H s with \(\|u_{0}\|_{{H}^{s}}<\delta\) , so that the corresponding time evolution u satisfies \(\|u(T)\|_{{H}^{s}}>K\) at some time T.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)
J. Bourgain, Problems in Hamiltonian PDE’s. Geom. Funct. Anal. (Special Volume, Part I), 32–56 (2000)
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Weakly turbulent solutions for the cubic defocussing nonlinear Schrödinger equation. Preprint (2008), pp. 1–54. http://arxiv.org/abs/0808.1742v1
S. Dyachenko, A.C. Newell, A.C.A. Pushkarev, and V.E. Zakharov, Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D 57(1–2), 96–160 (1992)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media B.V.
About this paper
Cite this paper
Colliander, J. (2009). Weak Turbulence for Periodic NLS. In: Sidoravičius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2810-5_11
Download citation
DOI: https://doi.org/10.1007/978-90-481-2810-5_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2809-9
Online ISBN: 978-90-481-2810-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)