Blow up phenomenon and sharp ill-posedness for the short-pulse equation

Abstract

The short-pulse (SP) equation serves as a fundamental model for describing the propagation of ultra-short optical pulses in silica fibers. In this paper, we rigorously establish the finite-time blow-up phenomenon for a class of large initial data in the Sobolev space \(H^{s}(\mathbb {R})\cap \dot{H}^{-1}(\mathbb {R}),~s>\frac{3}{2}.\) In addition, we prove a sharp ill-posedness result for the SP equation with initial data in the critical space \(H^{\frac{3}{2}}(\mathbb {R}).\) Combined with the local well-posedness for initial data in \(H^{s}(\mathbb {R})\) with \(s>\frac{3}{2},\) these results provide a complete characterization of the critical Sobolev regularity for the SP equation, yielding a comprehensive understanding of its well-posedness and ill-posedness dynamics.