Phase space analysis of higher-order dispersive equations with point interactions

TL;DR

This paper studies complex higher-order dispersive equations with irregular potentials and initial data, using phase space analysis and Gabor wave packets to understand their fundamental solutions and regularity properties.

Contribution

It introduces a distributional approach employing phase space analysis to examine the regularity of fundamental solutions for dispersive equations with measure potentials.

Findings

Established phase space regularity of generalized Fresnel oscillatory functions.

Developed a harmonic analysis framework for low-regularity dispersive equations.

Provided insights into the modulation and amalgam space properties of solutions.

Abstract

We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution - in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.