arXiv subjects

Constance M. Schober

Publications and source records attributed to Constance M. Schober.

3 recordsLinked to original sources

arXiv2020

On the stabilization of breather-type solutions of the damped higher order nonlinear Schrödinger equation

Spatially periodic breather solutions (SPBs) of the nonlinear Schrö\-dinger (NLS) equation are frequently used to model rogue waves and are typically unstable. In this paper we study the effects of dissipation and higher order nonlinearities on the stabilization of both single and multi-mode SPBs in the framework of a damped higher order NLS (HONLS) equation. We observe the onset of novel instabilities associated with the development of critical states which result from symmetry breaking in the damped HONLS system. We broaden the Floquet characterization of instabilities of solutions of the NLS equation, using an even 3-phase solution of the NLS as an example, to show instabilities are associated with degenerate complex elements of both the periodic and continuous Floquet spectrum. As a result the Floquet criteria for the stabilization of a solution of the damped HONLS centers around the elimination of all complex degenerate elements of the spectrum. For an initial SPB with a given mode structure, a perturbation analysis shows that for short time only the complex double points associated with resonant modes split under the damped HONLS while those associated with nonresonant modes remain effectively closed. The corresponding damped HONLS numerical experiments corroborate that instabilities associated with nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes.

nlin.PS↗

arXiv2018

Linear Instability of the Peregrine Breather: Numerical and Analytical Investigations

We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schrödinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov-Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.

nlin.SI↗

arXiv2004

Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrodinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.

physics.comp-ph↗