On the solitary waves for anisotropic nonlinear Schr\"odinger models on the plane

The focussing anisotropic nonlinear Schr\"odinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0<s<1$ and $p>2$. 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 $u=e^{-\mathrm{i} \omega t} \phi$, do not exist in the regime $p\geq p_s:=\frac{2(1+s)}{1-s}$, they do exist in the complementary regime $2<p<p_s$. In fact, we construct them variationally and we establish a number of key properties. Importantly, we completely characterize their spectral stability properties. Our consideration are easily extendable to the higher dimensional situation. 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.