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We consider the Cauchy problems associated with semirelativistc 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 groundstates. In between other results, we prove the existence and stability ofground states for sNLS in the $L^2$ supercritical regime. This is in sharpcontrast with the instability of ground states for the corresponding HW, whichis also established along the paper, by showing an inflation of normsphenomenon. Concerning the Cauchy theory we show, under radial symmetryassumption the following results: a local existence result in $H^1$ for energysubcritical nonlinearity and a global existence result in the $L^2$ subcriticalregime.

Long time dynamics for semirelativistic NLS and Half Wave in arbitrary dimension