Global well-posedness of quadratic and subquadratic half wave Schr{ö}dinger equations

We consider the following $p$ order nonlinear half wave Schr{ö}dinger equations$$(i\mathrm{\partial }\mathrm{\_}t+\mathrm{\partial }\mathrm{\_}{x}^2-\mid D\mathrm{\_}y\mid )u=\pm \mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$$on the plane ${\mathbb{R}}^2$ with $1<p\le 2$. 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 $L\mathrm{\_}{x}^{2}H\mathrm{\_}{y}^s({\mathbb{R}}^2)\cap H\mathrm{\_}{x}^{1}L\mathrm{\_}{y}^2({\mathbb{R}}^2)$($\frac{1}{2}\le s\le 1$), which is the first global well-posedness result of nonlinear half wave Schr{ö}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{ö}dinger equations on $\mathbb{R}\mathrm{\_}x\times \mathbb{T}\mathrm{\_}y$, which can also be called the wave guide Schr{ö}dinger equations on $\mathbb{R}\mathrm{\_}x\times \mathbb{T}\mathrm{\_}y$. Using a similar approach in the analysis of the Cauchy problem of half wave Schr{ö}dinger equations on ${\mathbb{R}}^2$, we can also deduce the global well-posedness of $p$ ($1<p\le 2$) order wave guide Schr{ö}dinger equations in $L\mathrm{\_}{x}^{2}H\mathrm{\_}{y}^s(\mathbb{R}\times \mathbb{T})\cap H\mathrm{\_}{x}^{1}L\mathrm{\_}{y}^2(\mathbb{R}\times \mathbb{T})$ with $\frac{1}{2}\le s\le 1$. With the global well-posedness in the energy space for the focusing wave guide Schr{ö}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.