On The Painleve Property For A Class Of Quasilinear Partial Differential Equations

TL;DR

This paper extends the Painleve property analysis to broad classes of high-order quasilinear nonlinear partial differential equations, advancing the understanding of their integrability and singularity structure.

Contribution

It transfers recent classification results from ordinary to partial differential equations, focusing on broad classes of quasilinear PDEs of arbitrary high order.

Findings

Completed Painleve classification for certain broad classes of nonlinear PDEs.

Developed methodology for analyzing singularities in high-order quasilinear PDEs.

Highlighted potential for systematic classification of complex nonlinear PDEs.

Abstract

The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free from moveable critical singularities. The problem of Painleve classification of ordinary and partial nonlinear differential equations lasting since the end of XIX century saw significant advances for the equation of lower (mainly up to fourth with rare exceptions) order, however not that much for the equations of higher orders. Recent works of the author have completed the Painleve classification for several broad classes of ordinary differential equations of arbitrary order, advancing the methodology of the Panleve analysis. This paper transfers one of those results on a broad class of nonlinear partial differential equations - quasilinear equations of an arbitrary order three or higher, algebraic in the dependent variable and including only the highest order derivatives of it. Being a first advance in Painleve classification of broad classes of arbitrary order nonlinear PDE's known to the author, this work highlights the potential in building classifications of that kind going beyond specific equations of a limited order, as mainly considered so far.