Degasperis2018_Article_IntegrabilityAndLinearStabilit.pdf (2.12 MB)
Integrability and linear stability of nonlinear waves
journal contribution
posted on 2018-05-18, 10:51 authored by Antonio Degasperis, Sara Lombardo, Matteo SommacalIt is well known that the linear stability of solutions of (Formula presented.) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general (Formula presented.) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for (Formula presented.) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.
History
School
- Science
Department
- Mathematical Sciences
Published in
Journal of Nonlinear ScienceVolume
28Issue
4Pages
1251 - 1291Citation
DEGASPERIS, A., LOMBARDO, S. and SOMMACAL, M., 2018. Integrability and linear stability of nonlinear waves. Journal of Nonlinear Science, 28(4), pp. 1251–1291.Publisher
© The Author(s). Published by Springer.Version
- VoR (Version of Record)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0) licence. Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/Acceptance date
2018-02-10Publication date
2018-03-15Notes
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.ISSN
0938-8974eISSN
1432-1467Publisher version
Language
- en