Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability

Consider the hyperbolic nonlinear Schrödinger equation (HNLS) over $\mathbb{R}^d$ $iu_t + u_{xx} - Δ_{\textbf{y}} u + λ|u|^σu=0.$ 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 \textit{spatial plane waves} and \textit{spatial standing 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.

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