On scattering for the defocusing quintic nonlinear Schrödinger equation on the two-dimensional cylinder

2018/09/05 by Xing Cheng, Zihua Guo, Cheng, Xing +3 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1809.01527

openalex publication_date 2018/09/05 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28

Abstract

In this article, we prove the scattering for the quintic defocusing nonlinear Schrödinger equation on cylinder ℝ × \mathbbT in H¹. We establish an abstract linear profile decomposition in L²_x h^α, 0 < α≤ 1, motivated by the linear profile decomposition of the mass-critical Schrödinger equation in L²(ℝ^d ), d≥ 1. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrödinger system, whose scattering can be proved by using the techniques in 1d mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in H¹ by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrödinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schrödinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].

Citations

• Normal forms and quadratic nonlinear Klein‐Gordon equations

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