A Rigorous Proof of Metric Blow-Up for Pseudospherical Metrics Associated with the Degasperis--Procesi Equation

Abstract

This paper studies finite-time blow-up of pseudospherical metrics induced by solutions of the Cauchy problem for the Degasperis--Procesi equation. For initial momentum profiles satisfying suitable left--right sign conditions, we use the method of characteristics, the momentum-transport formula, the Green's-function representation, and Riccati-type differential inequalities to analyze the metric along a distinguished characteristic. We prove that the associated coframe remains non-degenerate before the critical time, so that the induced pseudospherical metric is well defined in the precritical region. Moreover, as wave breaking is approached, the metric component \(g_{22}\) diverges to \(+\infty\); when \(μ\neq0\), the mixed component \(g_{12}\) also blows up in absolute value. Thus, finite-time wave breaking for the Degasperis--Procesi equation is shown to induce blow-up of certain components of the corresponding pseudospherical metric.