Direct construction method for conservation laws of partial differential equations. I: Examples of conservation law classifications. (English) Zbl 1034.35070

In the authors’ works [Phys. Rev. Lett. 78, 2869–2873 (1997; Zbl 0948.58015), Eur. J. Appl. Math. 9, 245–259 (1998; Zbl 0922.34006)] the authors have presented an algorithmic approach replacing Noether’s theorem allowing to obtain all local conservation laws for any differential equations, especially for ordinary ones, whether or not they have variational principles. Here they concentrate on partial differential equations. This method reduces the computation of conservation laws to solving a system of linear determining equations similar to those for finding symmetries. It gives an explicit construction formula yielding a conservation law for each solution of the determining system. Part 1 presents examples of nonlinear wave equations to exhibit the method and classification results for conservation laws of these equations. For Part 2 see the review Zbl 1034.35071 below.


35L65 Hyperbolic conservation laws
58J70 Invariance and symmetry properties for PDEs on manifolds
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