On scattering for the defocusing nonlinear Schrödinger equation on waveguide ${\mathbb{R}}^m$ $\times$ $\mathbb{T}$ (when $m=2,3$)

Abstract

In the article, we prove the large data scattering for two problems, i.e. the defocusing quintic nonlinear Schr{ö}dinger equation on ${\mathbb{R}}^2$ $\times$ $\mathbb{T}$ and the defocusing cubic nonlinear Schr{ö}dinger equation on ${\mathbb{R}}^3$ $\times$ $\mathbb{T}$. Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [36]) on large data scattering for the defocusing critical NLS with integer index nonlinearity on low dimensional waveguides. At this point, that type of problems are almost solved except for two remaining resonant system conjectures and the quintic NLS problem on $\mathbb{R}\times \mathbb{T}$.

27 pages. The author used a compact template which may be better with no big changes. The paper is currently under review. arXiv admin note: substantial text overlap with arXiv:1710.09702; text overlap with arXiv:1205.6132, arXiv:1101.4527 by other authors

References in corpus (5)

• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case
• Small data scattering for the nonlinear Schrödinger equation on product spaces
• The defocusing quintic NLS in four space dimensions
• Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide ${\mathbb{R}}^2$ $\times$ ${\mathbb{T}}^2$

Cited by in corpus (2)

• Global Well-Posedness of the Energy-Critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• On the global well-posedness for the periodic quintic nonlinear Schrödinger equation