Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1

By A. Ionescu, Gigliola Staffilani. 2008

63 citations recorded by the source corpus.

AbstractWe prove global well-posedness and scattering in H1 for the defocusing nonlinear Schrödinger equations $$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$$on the hyperbolic spaces $${\mathbb{H}^d}$$, d ≥ 2, for exponents $${\sigma \in (0, 2/(d-2))}$$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H1 is not known for any exponent $${\sigma \in (1/d, 2/d]}$$ and is known to fail for $${\sigma \in (0, 1/d]}$$. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/68a271bc8ace668473582354e387b0b17ab8c1d9

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In Applied Mathematics

  1. Gigliola Staffilani
  2. Nicola Visciglia
  3. Nikolay Tzvetkov
  4. Benoit Pausader
  5. A. Ionescu
  6. Zaher Hani
  7. D. Tataru
  8. Nicolas Burq

In Mathematical Physics

  1. Gigliola Staffilani
  2. Nicola Visciglia
  3. Nikolay Tzvetkov
  4. Benoit Pausader
  5. Zaher Hani
  6. A. Ionescu
  7. Carlos E. Kenig
  8. Luc Molinet

In Statistical and Nonlinear Physics

  1. Zaher Hani
  2. Benoit Pausader
  3. A. Ionescu
  4. Gigliola Staffilani
  5. Nikolay Tzvetkov
  6. Nicola Visciglia
  7. Terence Tao
  8. Tadahiro Oh

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