| Near a subcritical pattern-forming instability, there is a special parameter value, called the Maxwell point, where fronts connecting uniform and periodic states are stationnary. In the vicinity of this point, the slow-varying front is pinned to the 'microscopic' periodic structure. However, the size of the resulting periodic domains cannot be predicted with the amplitude equations obtained from the usual weakly nonlinear analysis. We show that what determine this size are exponentially small (but exponentially growing) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. Without loss of genericity, 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. |
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