Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability
Abstract
Consider the hyperbolic nonlinear Schrödinger equation (HNLS) over $\mathbb{R}^d$ $$ iu_t + u_{xx}  \Delta_{\textbf{y}} u + \lambda u^\sigma u=0. $$ 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 wellposedness 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.08745
 Bibcode:
 2015arXiv151008745C
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 23 pages