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

Abstract

Consider the hyperbolic nonlinear Schrödinger equation $\mathrm {(HNLS)}$ over $\mathbb R^d$ $$ iu_t + u_{xx} - \Delta_{\textbf{y}} u + \lambda |u|^\sigma u=0. $$ We deduce the conservation laws associated with $\mathrm {(HNLS)}$ and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including hyperbolically symmetric solutions, spatial plane waves and spatial plane 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.