On the solitary waves for anisotropic nonlinear Schrödinger models on the plane

The focusing anisotropic nonlinear Schrödinger equation iut-∂xxu+(-∂yy)su=|u|p-2uinR×R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {i}\, u_t-\partial _{xx} u + (-\partial _{yy})^s u=|u|^{p-2}u \quad \text{ in }\;\; {{\mathbb {R}}}\,{\times }\, {{\mathbb {R}}}^2 \end{aligned}$$\end{document}is considered for 0<s<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<1$$\end{document} and p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2$$\end{document}. Here the equation is of anisotropy, it means that dispersion of solutions along x-axis and y-axis is different. We show that while localized time-periodic waves, that are solutions in the form u=e-iωtϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=e^{-\text {i} \omega t} \phi $$\end{document}, do not exist in the range [inline-graphic not available: see fulltext], they do exist in the complementary range 2<p<ps\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p<p_s$$\end{document}. We construct them variationally and establish a number of key properties. Importantly, we completely characterize their spectral stability properties. Our consideration is easily extendable to higher dimensional case. We also show uniqueness of these waves under a natural weak non-degeneracy assumption. This assumption is actually removed for s close to 1, implying uniqueness for the waves in the full range of parameters.