Semilinear Schrödinger flows on hyperbolic spaces: scattering in \(H^{1}\). (English) Zbl 1203.35262
The authors prove the well-posedness and scattering in \(H^1\) of the initial-value problem for a nonlinear Schrödinger equation of the form \((i\partial_t+\Delta_g)u=N(u)\), \(u(0)=\phi\) on the hyperbolic spaces \(\mathbb{H}^d\), \(d\geq 2\), where the nonlinearity is of the form \(N(u)=u|u|^{2\sigma}\), for appropriate values of \(\sigma\in (0,\infty)\). Such a problem on Euclidean spaces has been studied extensively, see the books [T. Cazenave, Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics 10. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences. (2003; Zbl 1055.35003)] and [T. Tao, Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics 106. Providence, RI: American Mathematical Society (AMS). (2006; Zbl 1106.35001)] and the references therein. On Euclidean spaces, the global well-posedness is known, and scattering to linear solutions has been shown for \(\sigma\in (2/d,2/(d-2))\) while it is known to fail for \(\sigma\in (0,1/d)\), which makes the results of the present paper surprising.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35P25 | Scattering theory for PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 47A40 | Scattering theory of linear operators |
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