2019/03/18 by Carlos E. Kenig, Kenig, Carlos, Dana Mendelson +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1903.07246
openalex publication_date 2019/03/18 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We consider the focusing energy-critical quintic nonlinear wave equation in\nthree dimensional Euclidean space. It is known that this equation admits a\none-parameter family of radial stationary solutions, called solitons, which can\nbe viewed as a curve in Hsx(\ℝ3) \×\nHs-1x(\ℝ3), for any s > 1/2. By randomizing radial initial data\nin Hsx(\ℝ3) \× Hs-1x(\ℝ3) for s > 5/6,\nwhich also satisfy a certain weighted Sobolev condition, we produce with high\nprobability a family of radial perturbations of the soliton which give rise to\nglobal forward-in-time solutions of the focusing nonlinear wave equation that\nscatter after subtracting a dynamically modulated soliton. Our proof relies on\na new randomization procedure using distorted Fourier projections associated to\nthe linearized operator around a fixed soliton. To our knowledge, this is the\nfirst long-time random data existence result for a focusing wave or dispersive\nequation on Euclidean space outside the small data regime.\n