In this pages, we consider the p order nonlinear half wave Schrödinger equations \(\begin{aligned} \left( i \partial _{t}+\partial _{x }^2-\left| D_{y}\right| \right) u=\pm |u|^{p-1} u \end{aligned}\) on the plane \(\mathbb {R}^2\) with \(1<p\le 2\) . We prove the global well-posedness of this equation in \(L_x^2 H_y^s(\mathbb {R}^2) \cap H_x^1 L_y^2(\mathbb {R}^2)\) ( \(\frac{1}{2}\le s \le 1\) ), which is the first global well-posedness result of nonlinear half wave Schrödinger equations. With the global well-posedness in the energy space for the focusing equation and the study on the solitary wave in Bahri et al. (Commun Contemp Math 23(05), 2020), we complete the proof of the stability of the set of ground states. Moreover, we consider the half wave Schrödinger equations on \(\mathbb {R}_{x}\times \mathbb {T}_{y}\) , which can also be called the wave guide Schrödinger equations on \(\mathbb {R}_{x}\times \mathbb {T}_{y}\) . Using a similar approach in the analysis of the Cauchy problem of half wave Schrödinger equations on \(\mathbb {R}^2\) , we can also deduce the global well-posedness of p ( \(1<p\le 2\) ) order wave guide Schrödinger equations in \(L_x^2 H_y^s(\mathbb {R}\times \mathbb {T}) \cap H_x^1 L_y^2(\mathbb {R}\times \mathbb {T})\) with \(\frac{1}{2}\le s \le 1\) . With the global well-posedness in the energy space for the focusing wave guide Schrödinger equations and the study on the ground states in Bahri et al. J Dyn Differ Equ 1–43, 2021), we complete the proof of the orbital stability of the ground states with small frequencies.
