Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations

By Zaher Hani. 2012

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We study the energy cascade problematic for some nonlinear Schrödinger equations on $${\mathbb{T}^2}$$T2 in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one $${|u|^2u}$$|u|2u. Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).

Subfields: Mathematical Physics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ba66adfaa0f59917dd2ae7700b8eded6b5040554