A linearization of PDEs based on conservation laws

A method based on innnite parameter conservation laws is described to factor linear differential operators out of nonlinear partial diierential equations (PDEs) or out of diierential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (system) if that is possible. Innnite parameter conservation laws can be computed, for example, with a computer algebra package ConLaw. The method is compared with the symmetry based linearization method of Kumei and Bluman and extended functionality is explained. 1 Conservation laws with arbitrary functions With the availability of computer algebra programs for the automatic computation of all conservation laws up to a given diierential order of the integrating factors (as described in 2], 3]) conservation laws have been found that involve arbitrary functions, i.e. innnitely many parameters. In this paper we show how based on such conservation laws a linear diierential operator can be factored out of a combination of the nonlinear partial diierential equations (PDEs) and their diierential consequences. Possible outcomes include the complete linearization into an equivalent linear system, the derivation of at least one linear equation from a nonlinear system (with the possibility of deriving further linear equations for the new mixed linear-nonlinear system), the lowering of the diierential order with respect to at least one variable such that, for example, ordinary diierential equations (ODEs) result. For this to work we do not need a form of the conservation law in which the arbitrary functions occur explicitly. It is enough to have the determining conditions for the integrating factors solved up to the solution of consistent PDEs which are necessarily linear.