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.
Categories: math.AP math.CA
