Exponential asymptotics of homoclinic snaking

Abstract

We study homoclinic snaking in the cubic-quintic Swift-Hohenberg equation (SHE) close to the onset of a subcritical pattern-forming instability. Application of the usual multiple-scales method produces a leading-order stationary front solution, connecting the trivial solution to the patterned state. A localized pattern may therefore be constructed by matching between two distant fronts placed back-to-back. However, the asymptotic expansion of the front is divergent, and hence should be truncated. By truncating optimally, such that the resultant remainder is exponentially small, an exponentially small parameter range is derived within which stationary fronts exist. This is shown to be a direct result of the 'locking' between the phase of the underlying pattern and its slowly varying envelope. The locking mechanism remains unobservable at any algebraic order, and can only be derived by explicitly considering beyond-allorders effects in the tail of the asymptotic expansion, following the method of Kozyreff and Chapman as applied to the quadratic-cubic SHE Kozyreff 2009 Physica D 238 319-54, Kozyreff and Chapman 2006 Phys. Rev. Lett. 97 44502). Exponentially small, but exponentially growing, contributions appear in the tail of the expansion, which must be included when constructing localized patterns in order to reproduce the full snaking diagram. Implicit within the bifurcation equations is an analytical formula for the width of the snaking region. Due to the linear nature of the beyond-all-orders calculation, the bifurcation equations contain an analytically indeterminable constant, estimated in the previous work by Chapman and Kozyreff using a best fit approximation. A more accurate estimate of the equivalent constant in the cubic-quintic case is calculated from the iteration of a recurrence relation, and the subsequent analytical bifurcation diagram compared with numerical simulations, with good agreement.

Localized Turing patterns in nonlinear optical cavities

“…Such a refinement may appear superfluous at first sight, but close to the codimension-2 point, the width of the pinning range becomes so thin that the leading order calculation becomes insufficiently accurate. This was noted before in the two instances where a beyond-all-order analysis was carried out [37,40,44] and will be illustrated by comparing our results with the pinning range numerically computed by Gomilà, Scroggie, Firth in [31].…”

“…As a result, a large number of spatial discretization points are necessary and the equations must be integrated for a long time before reaching a stable state. Moreover, the domain of existence of localized patterns in the parameter space is exponentially thin [37,40,44] and can therefore easily be missed. The aim of this paper, therefore, is to locate the region in parameter space where localized patterns can be found for two important classes of optical models.…”

Subcritical Turing bifurcation and the morphogenesis of localized patterns

“…For small q 1 q 3 > 0, then, provided q 1 q 5 < 0, an unfolding of the normal form shows that there is a heteroclinic connection from a background state to a nontrivial periodic orbit. Taking account of beyond-all-orders terms in the normal form [26,27] enables an analysis to be undertaken in which we find infinitely many homoclinic orbits arranged on two closed curves; see Fig. 3(b) below.…”

Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model

“…For the 3-5 Swift-Hohenberg equation, the additional up-down symmetry (x → x, u → −u) admits two additional snaking branches of odd solutions (cf. [12,13]). In contrast to the 2-3 case, traversal through four saddle-nodes on one snaking branch is required to add two wavelengths at the edges of the spatially periodic region in the 3-5 Swift-Hohenberg equation (cf.…”