2026/07/28 by Zhenhua Shi, Mingyue Guo
#math-ph #math.MP
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 \(g22\) diverges to \(+∞\); when \(μ≠0\), the mixed component \(g12\) 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.