New symmetry solution techniques for first-order non-linear PDEs

Abstract

Using recent extensions of work of S. Lie (Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, Teubner, 1891) and É. Cartan (Leçons sur les invariants intégreaux, Hermann, Paris, 1922; Les systèms differentials extérieures et leurs applications géométrique, Hermann, Paris, 1945) for integrating Frobenius integrable vector field distributions via symmetry, we examine some symmetry techniques for finding local solutions of first-order non-linear partial differential equations (PDEs). In the language of exterior differential systems, we develop a technique for solving first-order quasilinear PDEs that is then applied to general, first-order non-linear partial differential equations. Our results are significant inasmuch as we give algorithms for solving first-order PDEs in the presence of symmetry that are essentially mechanical in nature and done in the original coordinate system.