Linearization Procedure of Nonlinear First Order System of Partial Differential Equations by Means of Canonical Variables Related to Lie Groups of Point Transformations

An algorithm is given to linearize nonlinear first order systems of partial differential equations admitting an infinite-parameter Lie group of point transformations. The associated infinitesimal operator must be a linear combination of commuting operators which individually are not necessarily admitted by the basic system, whereas the coefficients of the combination are given by arbitrary solutions of a suitable linear system. The procedure is based on the introduction of the canonical variables corresponding to the commuting operators. Within such a framework we reformulate a theorem already proved by Kumei and Bluman [SIAM J. Appl. Math.42 (1982)]. The paper concludes with some illustrative examples of the proposed algorithm.