Localized states in the generalized Swift-Hohenberg equation

Abstract

The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time.

A mechanism for streamwise localisation of nonlinear waves in shear flows

“…This will most probably be the case for localised plane Couette and plane Poiseuille solutions too Zammert & Eckhardt 2014). The non-snaking localisation mechanism (Burke & Knobloch 2006) found in double-diffusive convection (Beaume et al 2013b) cannot apply to plane Couette, plane Poiseuille or pipe flows, as the required condition that a nontrivial streamwise-independent state exists is not fulfilled by any of these flows. It is clear from k-continuation that MTSWs connect both with upper and lower branch TSWs, and how this comes about when smoothly varying Re at constant k is vital to putting the snaking hypothesis to test.…”

“…Regular snaking results from stationary front pinning to the wave train underlying the localised solution (Burke & Knobloch 2006). Homogeneous extended solutions give rise to non-snaking behaviour as the fronts can pin to the underlying solution regardless of its spatial extent but only so for values of the governing parameter corresponding to Maxwell points (Burke & Knobloch 2006;Beaume et al 2013b).…”

“…It is precisely the collision of these two saddle-nodes upon smoothly varying k that disrupts the snaking mechanism. This is reminiscent of the blending of separate regions of regular and semi-infinite snaking into a unique region of semi-infinite snaking observed for some parameter values of the Swift-Hohenberg equation (Burke & Knobloch 2006). Furthermore, the existence of a conserved quantity, which in our case is mass-flux, accounts for the slanted arrangement of the leftmost saddle-nodes (Dawes 2008;Beaume et al 2013a), thus allowing for localised solutions to exist at lower and higher values of the parameter than is possible in regular snaking.…”

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

“…This type of bifurcation structure is called collapsed snaking [16,17,41,42], which is significantly different from the homoclinic snaking appearing for dissipative solitons associated with subcritical patterns. Such homoclinic snaking, where many solutions coexist over a fixed parameter range around the Maxwell point, is probably better known and has been widely studied in physics [43,44] and optics [13,[45][46][47]. When TOD is taken into account, S u gradually develops oscillations when increasing d 3 , which allows bright solitons to come into existence.…”

Transitions between patterned states in vegetation models for semiarid ecosystems

“…In a study of the generalized Klausmeier model by van der Stelt et al [20], a patterned vegetated state collapses directly to desert via long-wavelength, sideband, and Hopf instabilities. Additionally, homoclinic snaking [21,22] has been proposed as a mechanism for the stabilization and motion of localized patterned states that emerge en route to desertification [23]. These approaches, as well as the approach we presented in this paper, can help to form a catalog of scenarios for transition between vegetated and desert states in semi-arid ecosystems, which may provide crucial insight into how semi-arid ecosystems will respond to the change in precipitation conditions that will accompany global climate change in the coming decades.…”