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

TL;DR

This paper investigates specific solutions of the hyperbolic nonlinear Schrödinger equation that are not in $H^1$, introduces new functional spaces to include these solutions, and proves their stability under small perturbations.

Contribution

It constructs new classes of solutions outside $H^1$, develops functional spaces accommodating these solutions, and establishes their stability, advancing understanding of HNLS dynamics.

Findings

Constructed spatial plane and standing wave solutions outside $H^1$

Developed new functional spaces including these solutions

Proved stability of these solutions under small $H^1$ perturbations

Abstract

Consider the hyperbolic nonlinear Schr\"odinger equation (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 (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.