Speed of propagation of fractional dispersive waves

TL;DR

This paper proves that solutions to a broad class of nonlinear dispersive equations cannot stay localized in space over time unless the dispersion relation is analytically extendable, highlighting the influence of fractional dispersion and memory effects.

Contribution

It extends previous results on wave support properties to fractional and more general dispersion relations using complex analysis techniques.

Findings

Non-trivial solutions cannot remain compactly supported over time.

Analytic extension of the dispersion relation is necessary if solutions are initially localized.

Application to fractional Schrödinger equations shows the impact of memory effects.

Abstract

In this paper, we show that all non-trivial solutions of a broad class of nonlinear dispersive equations, whose linear evolution is governed by a dispersion relation under minimal regularity assumptions, cannot remain compactly supported for any non-trivial time interval. Our approach, based on complex-analytic arguments and the Paley-Wiener-Schwartz theorem, yields a stronger result: if linear solutions are compactly supported at two distinct times, then the dispersion relation must admit an analytic extension. This extends previous results beyond polynomial dispersion relations and applies to more general settings, including fractional-order systems. As an application, we examine the generalized space-time fractional Schr\"{o}dinger equation, illustrating the role of memory effects in wave propagation.