Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg-Landau equation

by Gelens, L., Matías, M. A., Gomila, D., Dorissen, T. and Colet, P.
Abstract:
We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-dimensional real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.
Reference:
Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg-Landau equation Gelens, L., Matías, M. A., Gomila, D., Dorissen, T. and Colet, P., Physical Review E, volume 89, pp. 012915, 2014.
Bibtex Entry:
@article{gelens_formation_2014,
	title = {Formation of localized structures in bistable systems through nonlocal spatial coupling. {II}. {The} nonlocal {Ginzburg}-{Landau} equation},
	volume = {89},
	url = {http://link.aps.org/doi/10.1103/PhysRevE.89.012915},
	doi = {10.1103/PhysRevE.89.012915},
	abstract = {We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-dimensional real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.},
	number = {1},
	urldate = {2016-11-07},
	journal = {Physical Review E},
	author = {Gelens, L. and Matías, M. A. and Gomila, D. and Dorissen, T. and Colet, P.},
	month = jan,
	year = {2014},
	pages = {012915},
	file = {APS Snapshot:/home/jan/.zotero/zotero/djuw86a6.default/zotero/storage/WQIKNUEC/PhysRevE.89.html:text/html}
}