Primary branch solutions of first order autonomous scalar partial differential equations

TL;DR

This paper introduces primary branch solutions (PBS) for first order autonomous scalar PDEs, using Lie symmetry groups to explicitly construct solutions and recursion operators, enabling the derivation of diverse implicit exact solutions.

Contribution

It provides a systematic method to determine PBSs for first order autonomous scalar PDEs via Lie symmetry groups, including explicit recursion operators and symmetry sets.

Findings

Explicit recursion operator for (1+1)-dimensional PDEs

Set of high order symmetries for arbitrary PDEs

Method to generate diverse implicit solutions

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.