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Localised structures appear in a wide variety of systems, arising from apinning mechanism due to the presence of a small-scale pattern or an imposedgrid. When there is a separation of lengthscales, the width of the pinningregion is exponentially small and beyond the reach of standard asymptoticmethods. We show how this behaviour can be obtained using a variational method,for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation,this gives results that are in agreement with recent work using exponentialasymptotics. Secondly, the method is applied to a discrete system withcubic-quintic nonlinearity, giving results that agree well with numericalsimulations.

Variational approximations to homoclinic snaking in continuous and discrete systems