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

Yakine Bahri; Slim Ibrahim; Hiroaki Kikuchi

paper

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

arXiv:1810.01385

Abstract

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schrödinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero 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 $s > \frac{1}{2}$ .