Nowhere continuity of the flow map of an integrable derivative nonlinear Schrödinger system on the torus

Abstract

We consider a derivative nonlinear Schrödinger system called the Chen-Lee-Liu type system on the torus. This system is known as a completely integrable system. We prove the flow map fails to be continuous at every point in the Sobolev space $H^s(\mathbb{T}) \times H^s(\mathbb{T})$. Moreover, we establish an additional condition required for the flow map to be continuous. For the discontinuity, we take a sequence converging to the initial data for which the corresponding solutions do not exist.