Coherent Structures in Nonlocal Dispersive Active-Dissipative Systems

Lin, Te-Sheng; Pradas, Marc; Kalliadasis, Serafim; Papageorgiou, Demetrios T. and Tseluiko, Dmitri (2015). Coherent Structures in Nonlocal Dispersive Active-Dissipative Systems. SIAM Journal on Applied Mathematics, 75(2) pp. 538–563.



We analyze coherent structures in non-local dispersive active-dissipative nonlinear systems, using as a prototype the Kuramoto-Sivashinsky (KS) equation with an additional non-local term that contains stabilizing/ destabilizing and dispersive parts. As for the local generalized Kuramoto-Sivashinsky (gKS) equation (see, e.g., T. Kawara and S. Toh, Phys. Fluids, 31, 2103, 1988), 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 non-local equations, we develop a weak-interaction theory and show that the standard first-neighbor approximation is not applicable anymore. 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 find that for some nonlocal operators, increasing the strength of the stabilizing/destabilizing term can have a regularizing/de-regularizing effect on the dynamics.

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