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This is a Quantitative Study study.

October 1, 2025Open Access

Hyperbolic nonlinear Schr\"odinger equations on R T

Key Points

The study finds sharp local well-posedness up to critical regularity for cubic and higher odd nonlinearities.
Global existence and scattering are established for small initial data in critical Sobolev spaces, indicating robust solution behavior.
The approach utilizes sharp local/global-in-time Strichartz estimates to establish these findings effectively.
The implications of this research highlight significant insights into the solution dynamics of hyperbolic nonlinear Schrödinger equations.

Abstract

In this paper, we consider the hyperbolic nonlinear Schr\"odinger equations (HNLS) on R. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for higher odd nonlinearities. Moreover, when the initial data is small, we prove the global existence and scattering for the solutions to HNLS with higher nonlinearities (except the cubic one) in critical Sobolev spaces. The main ingredient of the proof is the sharp up to the endpoint local/global-in-time Strichartz estimates.

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