The method characteristics is a way to reduce a PDE to a system of ODEs, then it is said that solution $u$ of PDE has to be constant along characteristic curves. Basing on this idea we build a solution as a arbitrary function of first integrals. But why we have decided that the solution $u$ is constant along characteristic curves?
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It is an ansatz that works e.g. like to seperation of variables for parabolic PDEs. – Dan Doe Aug 11 '22 at 15:04