Abstract
The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in one-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show that nonlocal terms can induce spatial oscillations in the front tails, allowing for the creation of localized structures, that emerge from pinning between two fronts. In parameter space the region where fronts are oscillatory is limited by three transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in which monotonically decaying fronts develop spatial oscillations with a finite wavelength. We show how these transitions are organized by codimension 2 and 3 points and illustrate how by changing the parameters of the nonlocal coupling it is possible to bring the system into the region where localized structures can be formed.
Highlights
Classical evolution equations describing the dynamics of a field in space and time are partial differential equations (PDEs), like the heat and diffusion equations
OF THE OVERALL SCENARIO AND NONLOCAL KERNEL EFFECTS. Spatial dynamics makes it possible to determine the parameter regions in which fronts emerging from a homogeneous steady states (HSSs) have oscillatory tails and where localized structures (LSs) can exist
The presence of oscillatory tails is associated with the fact that the spatial dynamics is led by a quartet of complex eigenvalues or by the combination of a real doublet and a complex quartet
Summary
Classical evolution equations describing the dynamics of a field in space and time are partial differential equations (PDEs), like the heat and diffusion equations. The goal of the present paper is to provide a general framework to understand the effect of nonlocal spatial coupling on the shape of the fronts connecting stable steady states This makes it possible to determine the parameter regions in which fronts have oscillatory tails and LSs can exist. The transition from monotonic to oscillatory fronts that limits the parameter region where LSs can exist is given by the crossover manifold All these codim-2 bifurcations unfold from a codim-3 local bifurcation point characterized by being a sextuple zero of the dispersion relation. By playing with the parameters of the nonlocal interaction it is possible to bring the system into the parameter region where fronts have spatially oscillatory tails, allowing for the existence of LSs. Nonlocal interaction terms can induce the opposite effect, namely to preclude the formation of LSs in systems in which they are present. XI we illustrate how by changing the nonlocal interaction parameters it is possible to bring the system into the parameter regions in which fronts have an oscillatory profile
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