Coherent structures in nonlocal dispersive active-dissipative systems
journal contributionposted on 2015-06-08, 09:13 authored by Te-Sheng Lin, Marc Pradas, Serafim Kalliadasis, Demetrios T. Papageorgiou, Dmitri TseluikoDmitri Tseluiko
We analyze coherent structures in nonlocal dispersive active-dissipative nonlinear systems, using as a prototype the Kuramoto-Sivashinsky (KS) equation with an additional nonlocal term that contains stabilizing/destabilizing and dispersive parts. As for the local generalized Kuramoto-Sivashinsky (gKS) equation (see, e.g., [T. Kawahara and S. Toh, Phys. Fluids, 31 (1988), pp. 2103-2111]), we show that sufficiently strong dispersion regularizes the chaotic dynamics of the KS equation, and the solutions evolve into arrays of interacting pulses that can form bound states. We analyze the asymptotic characteristics of such pulses and show that their tails tend to zero algebraically but not exponentially, as for the local gKS equation. Since the Shilnikov-type approach is not applicable for analyzing bound states in nonlocal equations, we develop a weak-interaction theory and show that the standard first-neighbor approximation is no longer applicable. It is then essential to take into account long-range interactions due to the algebraic decay of the tails of the pulses. In addition, we find that the number of possible bound states for fixed parameter values is always finite, and we determine when there is long-range attractive or repulsive force between the pulses. Finally, we explain the regularizing effect of dispersion by showing that, as dispersion is increased, the pulses generally undergo a transition from absolute to convective instability. We also find that for some nonlocal operators, increasing the strength of the stabilizing/destabilizing term can have a regularizing/deregularizing effect on the dynamics.
This work was supported by the EPSRC [grant numbers EP/J001740/1 and EP/K041134/1].
- Mathematical Sciences
Published inSIAM Journal on Applied Mathematics
Pages538 - 563
CitationLIN, T.-S. ... et al, 2015. Coherent structures in nonlocal dispersive active-dissipative systems. SIAM Journal on Applied Mathematics, 75 (2), pp. 538 - 563
Publisher© Society for Industrial and Applied Mathematics Publications
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NotesThis is an Open Access Article published by SIAM and distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed.