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I am looking for examples of the non-autonomous linear wave equation that have some relevant applications.

What articles and sources (maybe, some catalogue like Handbook of Nonlinear Partial Differential Equations) may I refer to?

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  • $\begingroup$ I would think that you need nonlinearity for interesting/nontrivial dynamics. $\endgroup$ – Carlo Beenakker Sep 20 '17 at 9:53
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    $\begingroup$ The wave equation on any non-stationary Lorentzian spacetime (that is, $\square_g u = \frac{1}{\sqrt{|\det g|}} \partial_i g^{ij} \sqrt{|\det g|} \partial_j u = 0$ for a Lorentzian metric $g_{ij}$) is an example. You can read about the (abstract) theory of such equations, for instance, in this book. If you are looking for a list of interesting Lorentzian manifolds, you may want to take a look at this book. $\endgroup$ – Igor Khavkine Sep 20 '17 at 10:42
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If the time dependence of the potentials is periodic in time, then one enters the field of Floquet wave equations, which is a very active field of study with many real-world and even practical applications. (A textbook is Floquet Theory for Partial Differential Equations.)

The recent interest focuses on topological aspects of these driven systems, under the name of "Floquet Topological Insulators". Here is a review article, and here is an online talk. A specific application to the periodically driven Dirac equation is here.

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