UV completions, fixing the equations and nonlinearities in $k$-essence

Scalar-tensor theories with first-derivative self interactions, known as $k$-essence, may provide interesting phenomenology on cosmological scales. On smaller scales, however, initial value evolutions (which are crucial for predicting the behavior of astrophysical systems, such as binaries of compact objects) may run into instabilities related to the Cauchy problem becoming potentially ill-posed. Moreover, on local scales the dynamics may enter in the nonlinear regime, which may lie beyond the range of validity of the infrared theory. Completions of $k$-essence in the ultraviolet, when they are known to exist, mitigate these problems, as they both render Cauchy evolutions well-posed at all times, and allow for checking the relation between nonlinearities and the low energy theory's range of validity. Here, we explore these issues explicitly by considering an ultraviolet completion to $k$-essence and performing vacuum 1+1 dynamical evolutions within it. The results are compared to those obtained with the low-energy theory, and with the low-energy theory suitably deformed with a phenomenological "fixing the equations" approach. We confirm that the ultraviolet completion does not incur in any breakdown of the Cauchy problem's well-posedness, and we find that evolutions agree with the results of the low-energy theory, when the system is within the regime of validity of the latter. However, we also find that the nonlinear behavior of $k$-essence lies (for the most part) outside this regime.