Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations

Abstract

In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed [Formula: see text], traveling wave solutions exist and converge to the zero wave as the velocity tends to [Formula: see text]. Finally, we solve the Cauchy problem for initial data in [Formula: see text], with [Formula: see text]. The critical case [Formula: see text] still stands as an interesting open problem.