Partial and complete linearization of PDEs based on conservation laws

TL;DR

The paper develops a method to factor out linear differential operators from nonlinear PDEs (and their differential consequences) by exploiting infinite-parameter conservation laws, enabling linearization to an equivalent linear PDE or -system when possible. The approach builds conservation-law identities with arbitrary functions, computes the currents and characteristics by solving overdetermined linear PDEs for $P^i$ and $Q^alpha$, and applies a four-step algorithm (Steps 1–4) to transform the identity into a linearized form, with adjoint operators and potential reformulations guiding the process and implemented in the Reduce ConLaw package. It provides sufficient conditions for complete or partial linearization, discusses non-local linearization via potentials, and illustrates the method on the Liouville equation, including handling inhomogeneous linearizations and triangular forms; computational aspects rely on differential Gröbner bases and tools like the Crack package. The work offers a systematic, algorithmic framework for deriving linear or triangular representations of nonlinear PDEs, enabling analytic insight and potential automation across a broad class of problems.

Abstract

A method based on infinite parameter conservation laws is described to factor linear differential operators out of nonlinear partial differential equations (PDEs) or out of differential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (-system) if that is possible. Infinite parameter conservation laws can be computed, for example, with the computer algebra package ConLaw.