  We consider the following $p$ order nonlinear half wave Schr{\"o}dinger
equations$$\left(i \partial\_{t}+\partial\_{x }^2-\left|D\_{y}\right|\right)
u=\pm|u|^{p-1} u$$on the plane $\mathbb{R}^2$ with $1<p\leq 2$. This equation
is considered as a toy model motivated by the study of solutions to weakly
dispersive equations. In particular, the global well-posedness of this equation
is a difficult problem due to the anisotropic property of the equation, with
one direction corresponding to the half-wave operator, which is not dispersive.
In this paper, we prove the global well-posedness of this equation in $L\_x^2
H\_y^s(\mathbb{R}^2) \cap H\_x^1 L\_y^2(\mathbb{R}^2)$($\frac{1}{2}\leq s \leq
1$), which is the first global well-posedness result of nonlinear half wave
Schr{\"o}dinger equations. With the global well-posedness in the energy space
for the focusing equation and the study on the solitary wave in [1], we
complete the proof of the stability of the set of ground states. Moreover, we
consider the half wave Schr{\"o}dinger equations on
$\mathbb{R}\_{x}\times\mathbb{T}\_{y}$, which can also be called the wave guide
Schr{\"o}dinger equations on $\mathbb{R}\_{x}\times\mathbb{T}\_{y}$. Using a
similar approach in the analysis of the Cauchy problem of half wave
Schr{\"o}dinger equations on $\mathbb{R}^2$, we can also deduce the global
well-posedness of $p$ ($1<p\leq2$) order wave guide Schr{\"o}dinger 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 $\frac{1}{2}\leq s \leq 1$. With the
global well-posedness in the energy space for the focusing wave guide
Schr{\"o}dinger equations and the study on the ground states in [2], we
complete the proof of the orbital stability of the ground states with small
frequencies.
