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Abstract
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in nonlocal equations, we develop a weak-interaction theory and show that the standard first-neighbor approximation is no longer applicable.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.
Highlights
Nonlocal models arise in a wide variety of natural phenomena and technological applications, such as optical systems [43], cell biology [4], plastic materials [59], and in many problems involving nonequilibrium interfaces, such as flame propagation and thin-film growth [37]
Some well-known examples of nonlocal partial differential equations (PDEs) are the Benjamin–Ono (BO) equation describing propagation of internal waves in a deep stratified fluid [8]; the Smith equation governing certain types of continental-shelf waves [48]; a nonlocal Korteweg–de Vries (KdV) equation describing shallow-water waves when a viscous boundary layer is taken into account [23, 24, 35]; the same nonlocal KdV equation but with an additional nonlocal term to account for Marangoni effects on the free sur
Η(x, t) = eλitAiφi(x) + eσta(σ)φ(x, σ) dσ, i where the summation is over all the isolated eigenvalues λi with corresponding eigenfunctions φi(x), Ai are constants, Σ is the essential spectrum of L given by (4.5), and φ(x, σ) are the eigenfunctions of L, i.e., the functions belonging to the null space of σI − L
Summary
Nonlocal models arise in a wide variety of natural phenomena and technological applications, such as optical systems [43], cell biology [4], plastic materials [59], and in many problems involving nonequilibrium interfaces, such as flame propagation and thin-film growth [37]. For a single-pulse solution of Case 2 in section 3.3 and under the condition that p2 ∈ (0, 1), the right tail is negative and the left tail is positive, which gives P (l 1) > 0 and P (−l 1) < 0, and the following result can be concluded: If δ = 0 or p2 < p1 and p2 ∈ (0, 1), we have P (−l) − P (l) < 0 for |l| 1, which implies that there exists a long-range attractive force so that if the separation distance between two pulses is sufficiently large, they attract each other and form a bound state with finite separation distance.
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