Non-uniform continuity of periodic Holm-Staley b-family of equations

TL;DR

This paper demonstrates that the solution map for the Holm-Staley b-family of equations, including well-known integrable cases, is not uniformly continuous, using smooth periodic traveling waves with small amplitude.

Contribution

It establishes non-uniform continuity of the solution map for a broad family of Holm-Staley b-equations, including notable integrable cases.

Findings

Solution map is not uniformly continuous.

Construction of smooth periodic traveling waves with small amplitude.

Applicable to Holm-Staley b-family including Camassa-Holm and Degasperis-Procesi equations.

Abstract

We consider a family of non-evolutionary partial differential equations known as Holm - Staley b - family which includes the integrable Camassa-Holm and Degasperis-Procesi equations. We show that the solution map is not uniformly continuous. The proof relies on a construction of smooth periodic travelling waves with small amplitude.