Narrow Results

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 $s>\frac{1}{2}$. The critical case $s=\frac{1}{2}$ still stands as an interesting open problem.