Asymptotics of Large Bound States of Localized Structures

We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods. We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling.

Figure 1

Bifurcation diagram for $\epsilon =0.55$. Thick line: analytical formula. Thin line: numerical simulation.

Figure 2

Amplitude of the snaking oscillations (pinning range) in Fig. 1 as a function of $\epsilon$. Dots: numerical; line: analytical.