Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function

TL;DR

This paper investigates an integrable multi-component peakon system involving arbitrary polynomial functions, establishing existence, uniqueness, continuity, and unique continuation properties within Gevrey-Sobolev spaces using advanced analytical methods.

Contribution

It extends the analysis of peakon systems by proving well-posedness and unique continuation for a broad class involving arbitrary polynomial nonlinearities.

Findings

Existence and uniqueness of solutions in Gevrey-Sobolev spaces.

Continuity of the data-to-solution map.

Demonstration of unique continuation property.

Abstract

In this paper, we study the Cauchy problem for an integrable multi-component (2N-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence and uniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system. Furthermore, by introducing a family of continuous diffeomorphisms of a line and utilizing the fine structure of the system, we demonstrate the system exhibits unique continuation.