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

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 v , traveling wave solutions exist and converge to the zero wave as the velocity tends to 1. Finally, we solve the Cauchy problem for initial data in L x 2 H y s (ℝ 2) , with s > 1 2 . The critical case s = 1 2 still stands as an interesting open problem. [ABSTRACT FROM AUTHOR]