Primary branch solutions of first order autonomous scalar partial differential equations

Abstract

A primary branch solution (PBS) is defined as a solution with $n$ independent $m-1$ dimensional arbitrary functions for an $n$ order $m$ dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach. Especially, one recursion operator and some sets of infinitely many high order symmetries are also explicitly given for arbitrary (1+1)-dimensional first order autonomous PDEs. Because of the intrusion of the arbitrary function, various implicit special exact solutions can be find by fixing the arbitrary functions and selecting different seed solutions.