Local well-posedness and blow-up for a family of $U(1)$-invariant peakon equations

TL;DR

This paper investigates the initial value problem for a family of integrable peakon equations with $U(1)$ symmetry, establishing local well-posedness in Besov spaces and demonstrating finite-time blow-up using conservation laws.

Contribution

It provides the first rigorous analysis of well-posedness and blow-up phenomena for this specific family of $U(1)$-invariant peakon equations from the NLS hierarchy.

Findings

Proved local well-posedness in Besov spaces.

Established finite-time blow-up using an $L^1$ conservation law.

Analyzed the dynamics of solutions within the integrable peakon family.

Abstract

The Cauchy problem for a unified family of integrable $U(1)$-invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an $L^1$ conservation law.