We consider the Cauchy problems associated with semi-relativistic NLS (sNLS) and half wave (HW). In particular we focus on the following two main questions: local/global Cauchy theory; existence and stability/instability of ground states. In between other results, we prove the existence and stability of ground states for sNLS in the L2 supercritical regime. This is in sharp contrast with the instability of ground states for the corresponding HW, which is also established along the paper, by showing an inflation of norms phenomenon. Concerning the Cauchy theory we show, under radial symmetry assumption the following results: a local existence result in H1 for energy subcritical nonlinearity and a global existence result in the L2 subcritical regime.
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