Eur. Phys. J. Special Topics 225, 2549-2561 (2016)
https://doi.org/10.1140/epjst/e2016-60006-8

Regular Article

Forced snaking: Localized structures in the real Ginzburg-Landau equation with spatially periodic parametric forcing

Department of Physics, University of California at Berkeley, Berkeley CA 94720, USA

^a e-mail: bponede1@berkeley.edu

Received: 20 January 2016
Revised: 19 March 2016
Published online: 22 November 2016

Abstract

We study spatial localization in the real subcritical Ginzburg-Landau equation u_t = m₀u + Q(x)u + u_xx + d|u|²u −|u|⁴u with spatially periodic forcing Q(x). When d>0 and Q ≡ 0 this equation exhibits bistability between the trivial state u = 0 and a homogeneous nontrivial state u = u₀ with stationary localized structures which accumulate at the Maxwell point m₀ = −3d²/16. When spatial forcing is included its wavelength is imprinted on u₀ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as forced snaking. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting foliated snaking.