On the high-low method for NLS on the hyperbolic space

Abstract

In this paper, we first prove that the cubic, defocusing nonlinear Schrödinger equation on the two dimensional hyperbolic space with radial initial data in $H^s({\mathbb{H}}^2)$ is globally well-posed and scatters when $s>\frac{3}{4}$. Then we extend the result to nonlineraities of order $p>3$. The result is proved by extending the high-low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by the first author and Ionescu.

arXiv commentThe result is extended to general nonlineraities

References in corpus (6)

• Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d=2$
• The energy-critical defocusing NLS on T^3
• Nonlinear Schrödinger equation on real hyperbolic spaces
• Global well-posedness and scattering for nonlinear Schr{ö}dinger equations with algebraic nonlinearity when $d=2,3$, $u_0$ radial
• On the high-low method for NLS on the hyperbolic space
• Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane Under the Schrödinger Maps Evolution

Cited by in corpus (2)

• Global well-posedness for the cubic fractional NLS on the unit disk
• On the high-low method for NLS on the hyperbolic space