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 Lx2H ys(ℝ2), with s > 1 2. The critical case s = 1 2 still stands as an interesting open problem.

中文翻译：

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半波薛定谔方程的孤立波和柯西问题的备注

在本文中，我们研究了平面上半波薛定谔方程的柯西问题的孤波解。首先，我们展示了基态的存在和轨道稳定性。其次，我们证明给定任何速度v, 行波解存在并且随着速度趋向于收敛到零波1. 最后，我们解决了初始数据的柯西问题大号X2H 是的s(ℝ2)， 和s > 1 2. 危急情况s = 1 2仍然是一个有趣的开放问题。