Growth of Sobolev norms in the cubic defocusing nonlinear Schr "odinger\n equation

2012/05/23 by Marcel Guàrdia, Guardia, Marcel, V. A. Kaloshin +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1205.5188

openalex publication_date 2012/05/23 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

We consider the cubic defocusing nonlinear Schr "odinger equation in the two\ndimensional torus. Fix s>1. Colliander, Keel, Staffilani, Tao and Takaoka\nproved in citeCollianderKSTT10 the existence of solutions with s-Sobolev\nnorm growing in time.\n We establish the existence of solutions with polynomial time estimates. More\nexactly, there is c>0 such that for any \K\≫ 1 we find a solution\nu and a time T such that \‖ u(T)\‖_H^s\≥\K \‖ u(0)\‖_H^s.\nMoreover, time T satisfies polynomial bound 0<T<\K^c.\n

Citations

• Newton's method and periodic solutions of nonlinear wave equations

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