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

2012/10/28 by Zaher Hani, Hani, Zaher · 2 citations
Mathematics · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1210.7509

openalex publication_date 2012/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the energy cascade problematic for some nonlinear Schr "odinger\nequations on the torus T² in terms of the growth of Sobolev norms. We\ndefine the notion of long-time strong instability and establish its connection\nto the existence of unbounded Sobolev orbits. This connection is then explored\nfor a family of cubic Schr "odinger nonlinearities that are equal or closely\nrelated to the standard polynomial one |u|²u. Most notably, we prove the\nexistence of unbounded Sobolev orbits for a family of Hamiltonian cubic\nnonlinearities that includes the resonant cubic NLS equation (a.k.a. the first\nBirkhoff normal form).\n

Citations

• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential
• Inertial Ranges in Two-Dimensional Turbulence
• Global well-posedness of the cubic nonlinear Schrödinger equation on compact manifolds without boundary
• Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
• First and second order approximations for a nonlinear wave equation

Cited by

• Long-time behavior of resonant time-dependent perturbations of periodic transport equations on ℝ
• Plane wave stability of the split-step Fourier method for the nonlinear\n Schr "odinger equation

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