Exponential asymptotics and homoclinic snaking - Dean, Andrew David (2012)
There is much current interest in systems exhibiting homoclinic snaking, in which solution curves of localised patterns snake back and forth within a narrow region of parameter space. Such solutions comprise superimposed, back-to-back stationary fronts, each front connecting a homogeneous and a patterned state. These fronts are pinned to the underlying pattern within the snaking region; elsewhere, they become travelling waves and cannot form localised solutions. Application of standard asymptotic techniques near bifurcation can only produce a stationary front at the centre of the snaking region; this is the Maxwell point, where patterned and homogeneous states are equally energetically favourable. Such methods fail to capture the pinning mechanism because it is an exponentially small effect, and must be studied using exponential asymptotics. Deriving the late terms in the asymptotic expansion and observing that it is divergent, we truncate optimally after the least term. The resultant remainder is exponentially small and governed by an inhomogeneous differential equation. Rescaling this equation near Stokes lines---lines in the complex plane at which forcing is maximal---we observe a smooth but rapid increase from zero to exponentially small in the coefficient of an exponentially growing complementary function as Stokes lines are crossed. Requiring that unbounded terms vanish fixes the phase of the underlying pattern relative to the leading-order front. Furthermore, matching two fronts together produces a set of formulae describing the snaking bifurcation diagram. We successfully apply this method to continuous and discrete systems. In the former, we also show how symmetric solutions comprising two localised patches form figure-of-eight isolas in the bifurcation diagram. In the latter, we investigate snaking behaviour of a one-dimensional localised solution rotated into a square lattice, and find that the snaking region vanishes when the tangent of the angle of orientation is irrational.