Degasperis2018_Article_IntegrabilityAndLinearStabilit.pdf (2.12 MB)

Integrability and linear stability of nonlinear waves

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journal contribution
posted on 18.05.2018, 10:51 by Antonio Degasperis, Sara Lombardo, Matteo Sommacal
It 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 Science

Volume

28

Issue

4

Pages

1251 - 1291

Citation

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

10/02/2018

Publication date

2018-03-15

Notes

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

0938-8974

eISSN

1432-1467

Language

en

Licence

Exports