Modified Strichartz Estimate of the Periodic Fourth Order NLS

TL;DR

This paper establishes modified Strichartz estimates for a fourth-order nonlinear Schrödinger equation on the torus, leading to results on global well-posedness and solution behavior with respect to quantum parameters.

Contribution

It introduces new Strichartz estimates adapted to fourth-order dispersion on the torus, extending low regularity existence theory for related quantum systems.

Findings

Proved global well-posedness of the nonlinear fourth-order Schrödinger equation.

Showed solutions depend continuously on the quantum parameter in compact time intervals.

Demonstrated that this dependence is not uniform over infinite time horizons.

Abstract

We prove modified Strichartz estimates on the one-dimensional torus, that are adapted to a fourth-order dispersion relation, and use them to show global well-posedness of nonlinear fourth-order Schr\"odinger equations. This extends the (low regularity) existence theory of the adiabatic transition of the Quantum Zakharov system to NLS. We show that the solutions behave continuously with respect to the quantum parameter in every compact time interval. Globally in time, however, we also show that such continuous dependence is generally not uniform.