Rank two prolongations of second-order PDE and geometric singular solutions

TL;DR

This paper investigates the geometric structures of rank two prolongations of second-order PDEs with two independent and one dependent variable, classifies PDE types via fiber topology, and derives explicit geometric singular solutions using contact geometry.

Contribution

It introduces a novel geometric framework for analyzing second-order PDEs through rank two prolongations and explicitly constructs singular solutions using contact geometric methods.

Findings

Classified PDE types by fiber topology of prolongations

Derived explicit geometric singular solutions

Connected PDE analysis with contact geometry

Abstract

In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of these prolongations, we give explicit expressions of geometric singular solutions of second-order PDEs from the point of view of contact geometry of second order.