Well-posedness of the mixed-Fractional Nonlinear Schr\"odinger Equation on $\mathbb{R}^2$

TL;DR

This paper establishes the well-posedness and scattering properties of a 2-D mixed-fractional nonlinear Schrödinger equation with anisotropic dispersion, relevant for optics and photonics applications.

Contribution

It provides the first well-posedness results for the mixed-fractional NLSE with anisotropic dispersion and analyzes solution continuity with respect to dispersion parameters.

Findings

Dispersive estimates in anisotropic Sobolev spaces are developed.

Small data solutions exhibit scattering in the critical space.

Solution dependence on dispersion parameters is continuous on compact intervals.

Abstract

We investigate the well-posedness theory of the 2-D fractional nonlinear Schr\"odinger equation (NLSE) with a mixed degree of derivatives. Motivated by models in optics and photonics where the light propagation is governed by non-quadratic, fractional, and anisotropic dispersion profile, this paper presents first results in this direction. Dispersive estimates are developed in the context of anisotropic Sobolev spaces defined by inhomogeneous symbols. The main model is shown to exhibit scattering for small data in the scaling-critical space. Furthermore the continuity of solution with respect to the dispersion parameter is shown on a compact time interval.