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

Abstract

Consider the hyperbolic nonlinear Schrödinger equation (HNLS) over ${\mathbb{R}}^d$ $$i{u}_t+u_{xx}-{\Delta }_{\text{y}}u+\lambda |u{|}^{\sigma }u=0.$$ 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 \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.

arXiv comment23 pages