Forced Snaking

We study spatial localization in the real subcritical Ginzburg-Landau equation $u_t=m_0 u+ m_1\cos\left(\frac{2\pi}{\ell}x\right) u+u_{xx}+d|u|^2u-|u|^4u$ with spatially periodic forcing. When $d>0$ and $m_1 =0$ this equation exhibits bistability between the trivial state $u=0$ and a homogeneous nontrivial state $u=u_0$ with stationary localized structures which accumulate at the Maxwell point $m_0=-3d^2/16$. When spatial forcing is included its wavelength is imprinted on $u_0$ 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 \textit{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 \textit{foliated snaking}.