Volume 53, Number 5, September-October 2019
|Page(s)||1477 - 1505|
|Published online||06 August 2019|
On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions
Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA
2 Institut de Mathématiques de Bourgogne, University of Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21000 Dijon, France
* Corresponding author: firstname.lastname@example.org
Accepted: 8 March 2019
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.
Mathematics Subject Classification: 65M70 / 65L05 / 35Q55
Key words: Nonlinear Schrödinger equation / derivative nonlinearity / orbital stability / finite-time blow-up / self-steepening / spectral resolution / Runge–Kutta algorithm
© EDP Sciences, SMAI 2019
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