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Formation of caustics in k-essence and Horndeski theory

We study propagation of waves and appearance of caustics in k-essence and galileon theories. First we show that previously known solutions for travelling waves in k-essence and galileon models correspond to very specific fine-tuned initial conditions. On the contrary, as we demonstrate by the method of characteristics, generic initial conditions leads to a wave in k-essence which ends up with formation of caustics. Finally, we find that any wave solution in pure k-essence is also a solution for a galileon theory with the same k-essence term. Thus in the Horndeski theory with a k-essence term formation of caustics is generic. We discuss physical consequences of the caustics formation and possible ways to cure the problem.