Linear stability analysis for travelling waves of second order in time PDE's
In: Nonlinearity, Jg. 25 (2012-08-10), S. 2625-2654
Online
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Zugriff:
We develop a general theory for linear stability of traveling waves of second order in time PDE's. More precisely, we introduce an explicitly computable index $\om^*\in (0, \infty]$ (depending on the self-adjoint part of the linearized operator) so that the wave is stable if and only if $|c|\geq \om^*$. The results are applicable both in the periodic case and in the whole line case. As an application, we consider three classical models - the Boussinesq equation, the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam equation. For the Boussinesq model and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable traveling waves (and for all powers of $p$ in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander-Sachs, which apply to the case $p=2$. For the beam equation, we provide an explicit formula (depending of the function $\|\vp_c'\|_{L^2}$), which works for all $p$ and for both the periodic and the whole line cases. Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves.
Comment: Added some references, proof of Section 3.3 much simplified
Titel: |
Linear stability analysis for travelling waves of second order in time PDE's
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Autor/in / Beteiligte Person: | Stefanov, Atanas ; Stanislavova, Milena |
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Zeitschrift: | Nonlinearity, Jg. 25 (2012-08-10), S. 2625-2654 |
Veröffentlichung: | IOP Publishing, 2012 |
Medientyp: | unknown |
ISSN: | 1361-6544 (print) ; 0951-7715 (print) |
DOI: | 10.1088/0951-7715/25/9/2625 |
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