Local well-posedness of the nonlinear Schr\"odinger equations on the sphere for data in modulation spaces

TL;DR

This paper establishes local well-posedness for the cubic nonlinear Schrödinger equation on the sphere $S^2$ with initial data in modulation spaces of regularity $s=1/4$, using energy-based a priori estimates.

Contribution

It introduces new a priori estimates enabling the proof of existence of solutions with low regularity initial data on the sphere.

Findings

Existence of solutions for data with regularity $s=1/4$

Development of energy method-based a priori estimates

Extension of well-posedness results to modulation space data

Abstract

In this paper we discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schr\"odinger on the sphere $S^2$. Exploring suitable a priori estimates, we prove the existence of solution for data whose regularity is $s=1/4$.