Localized periodic patterns for the non-symmetric generalized Swift-Hohenberg equation

Abstract

A new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized Swift-Hohenberg equation. An analysis of this envelope equation results in parametric conditions for localized patterns. Furthermore, it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case. The analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing Hamiltonian and Lagrangian for periodic solutions. One striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an O(1) distance from the bifurcation, thus providing a novel approach for studying these localized patterns.

Variational approximations to homoclinic snaking

“…which are the parameter values of the front at the Maxwell point, in exact agreement with the multiplescales asymptotic method [1,2,12]. Note that L ef f depends linearly on L, but this dependence vanishes at the Maxwell point.…”

“…We obtain the symmetric snaking solutions and the asymmetric 'ladder' states, and also predict the stability of the localised states. The resulting approximate formulas for the width of the snaking region show good agreement with numerical results.There has been much recent interest in the phenomenon of spatially localised patterns, [2,4,7,11], extending our understanding of earlier work on this topic [1,13]. As discussed in the review article by Dawes [9], this work is motivated by wide-ranging applications in many different areas of physics, including buckling of struts and cylinders [6,10,15], nonlinear optics [7], convection patterns [1,12], gas discharge systems and granular media.…”

“…The interesting case leading to snaking bifurcations is b 3 > 0, b 5 > 0, in which case localised solutions can exist for r < 0. It is possible to rescale u to set b 5 = 1 but we will keep b 5 as a parameter for consistency with [2,4]. The stationary solutions of (1) can be derived from a Lagrangian…”

“…This same conclusion was reported in [15] after a lengthy 'beyond all orders' calculation. It is expected that the ansatz (4) only resembles exact [2,4,14]. For localised states in the snaking regime, the envelope solution has a plateau, which becomes longer as the norm N increases.…”

A mechanism for streamwise localisation of nonlinear waves in shear flows

“…In simple models featuring snaking, localised solutions can be understood as homoclinic orbits of the spatial ordinary differential equation that results from dropping the time derivative (Kirchgässner 1982;Budd & Kuske 2005). These homoclinic orbits take from the base flow (a spatial fixed point) back to the base flow, with a natural number of intervening quasi-orbits around the nontrivial patterned flow (a spatial periodic orbit), hence the qualifier homoclinic with which the word snaking is commonly prefixed.…”