The nonlinear Schrödinger equation on the hyperbolic space

Valeria Banica

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The nonlinear Schrödinger equation on the hyperbolic space

Valeria Banica

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Abstract

In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space $\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties : we prove that the dispersion inequality is valid, in a stronger form than the one on $\mathbb{R}^n$. However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.

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Valeria Banica. 2007-11-29. The nonlinear Schrödinger equation on the hyperbolic space. https://arxiv.org/abs/math/0406058

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