Some $L^{\mathrm{\infty }}$ solutions of the hyperbolic nonlinear Schr\"odinger equation and their stability

Consider the hyperbolic nonlinear Schr"odinger equation (HNLS) over ${\mathbb{R}}^d$ $$i{u}_t+u_{xx}-{\mathrm{\Delta }}_{\mathbf{\text{y}}}u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u=0.$$ We deduce the conservation laws associated with (HNLS) and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including $\mathit{\text{spatial}}\mathit{\text{plane}}\mathit{\text{waves}}$ and $\mathit{\text{spatial}}\mathit{\text{standing}}\mathit{\text{waves}}$, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a stability result for both spatial plane waves and spatial standing waves with respect to small $H^1$ perturbations.