We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schrödinger type and have recently been obtained by Dumas et al. in the context of nonlinear optics. In contrast to the usual nonlinear Schrödinger equation, this new model incorporates the additional effects of self-steepening and partial off-axis variations of the group velocity of the laser pulse. We prove global-in-time existence of the corresponding solution for various choices of parameters. In addition, we present a series of careful numerical simulations concerning the (in-)stability of stationary states and the possibility of finite-time blow-up.
Keywords: Nonlinear Schrödinger equation, derivative nonlinearity, orbital stability, finite-time blow-up, self-steepening, spectral resolution, Runge–Kutta algorithm
Arbunich, Jack 1 ; Klein, Christian 1 ; Sparber, Christof 1
@article{M2AN_2019__53_5_1477_0,
author = {Arbunich, Jack and Klein, Christian and Sparber, Christof},
title = {On a class of derivative {Nonlinear} {Schr\"odinger-type} equations in two spatial dimensions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1477--1505},
year = {2019},
publisher = {EDP Sciences},
volume = {53},
number = {5},
doi = {10.1051/m2an/2019018},
mrnumber = {3989598},
zbl = {1427.65277},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2019018/}
}
TY - JOUR AU - Arbunich, Jack AU - Klein, Christian AU - Sparber, Christof TI - On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 1477 EP - 1505 VL - 53 IS - 5 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2019018/ DO - 10.1051/m2an/2019018 LA - en ID - M2AN_2019__53_5_1477_0 ER -
%0 Journal Article %A Arbunich, Jack %A Klein, Christian %A Sparber, Christof %T On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 1477-1505 %V 53 %N 5 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2019018/ %R 10.1051/m2an/2019018 %G en %F M2AN_2019__53_5_1477_0
Arbunich, Jack; Klein, Christian; Sparber, Christof. On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 5, pp. 1477-1505. doi: 10.1051/m2an/2019018
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