Lie Theorem via Rank 2 Distributions (Integration of PDE of Class ω = 1)

Abstract

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed form, as in the method of Darboux, and discuss nonlinear Laplace transformations and symmetric PDE models.

Laplace Transformation of Lie Class ω = 1 Overdetermined Systems

“…Lie demonstrated that a compatible (=formally integrable) class ω = 1 overdetermined system is integrable by reduction to ODEs. Modern proof and applications of this result will be discussed in [16].…”

“…This distinguishes ω = 1 linear systems, but does not extend to nonlinear ω = 1 class systems, which will be discussed in [16].…”

Generalized Lie-Bäcklund theorem for Lie class $$\omega =1$$ ω = 1 overdetermined systems

“…The proof given in [11] uses the following 3 In terminology of Elie Cartan ω is the Cartan integer s 1 (provided the Cartan character is 1: s 2 = 0).…”

“…The PDE system E k has solution space Sol(E k ) that is parametrized by 1 function of 1 argument and dim M − 2 = k(k+1) 2 constants (these are the so-called Lie class ω = 1 systems [15,10,11]). …”

“…Lie class ω = 1 PDE systems are integrable by ODE methods [15,11], and they often arise as symmetry reductions of more complicated PDEs (a Darboux integrable or semi-integrable equation coupled with one intermediate integral is a particular case of class ω = 1 system). Lie-Bäcklund theorem clearly fails for Lie class ω = 1 overdetermined PDE systems, since the internal symmetry group is always infinitedimensional while the external group is usually not.…”