Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation

Wadee, M. Ahmer; Coman, Ciprian D.; Bassom, Andrew P.

doi:10.1016/s0167-2789(02)00350-0

Cited by 38 publications

(37 citation statements)

References 21 publications

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“…We have shown that for weak dispersion this structure breaks up into a stack of isolas of slowly drifting localized states and that these isolas shrink and eventually vanish as dispersion increases. Stacks of isolas are characteristic of multipulse homoclinic orbits as well [10,11]. We have also (Fig.…”

Section: Discussionsupporting

confidence: 59%

Swift-Hohenberg equation with broken reflection symmetry

2009

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The bistable Swift-Hohenberg equation possesses a variety of time-independent spatially localized solutions organized in the so-called snakes-and-ladders structure. This structure is a consequence of a phenomenon known as homoclinic snaking, and is in turn a consequence of spatial reversibility of the equation. We examine here the consequences of breaking spatial reversibility on the snakesand-ladders structure. We find that the localized states now drift, and show that the snakes-andladders structure breaks up into a stack of isolas. We explore the evolution of this new structure with increasing reversibility breaking and study the dynamics of the system outside of the snaking region using a combination of numerical and analytical techniques.

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Section: Discussionsupporting

confidence: 59%

Swift-Hohenberg equation with broken reflection symmetry

2009

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“…And this process cannot be repeated ad infinitum, since the 2 pulses would have to meet, and therefore the 2-homoclinic orbit cannot follow the full snaking curve. A similar behaviour has been found for different examples in [5,17]. Indeed, so far 2-homoclinic orbits near EP cycles have only been found to lie on isolas and not snaking curves.…”

Section: Discussionsupporting

confidence: 82%

“…This global character of bifurcations near an EP cycle makes a rigorous analytical treatment difficult. Nevertheless, numerical results for N -homoclinic orbits near such a cycle have been described in [5,17] Below we present a corresponding result for the generalized Swift-Hohenberg Equation (1). Similar to Section 2 we view the equation as depending on r, this time setting q c = 0.5, v = 0.41 and g = 1.…”

Section: Discussionmentioning

confidence: 96%

Snaking of Multiple Homoclinic Orbits in Reversible Systems

Knobloch¹,

Wagenknecht²

2008

SIAM J. Appl. Dyn. Syst.

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We study N -homoclinic orbits near a heteroclinic cycle in a reversible system. The cycle is assumed to connect two equilibria of saddle-focus type. Using Lin's method we establish the existence of infinitely many N -homoclinic orbits for each N near the cycle. In particular, these orbits exist along snaking curves, thus mirroring the behaviour one-homoclinic orbits. The general analysis is illustrated by numerical studies for a Swift-Hohenberg system.

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“…In addition to these single-hump pulses, we expect to find multi-pulses that follow the heteroclinic cycle N -times for each N ≥ 2, and the question is then whether these multi-pulses snake as well. The numerical computations in [41,43], and more recently in [6,20], suggest that multi-pulses actually lie on isolas. The techniques employed in this paper can also be used to study multi-pulses, and we currently carry this analysis out.…”

Section: And If Hypothesis 5 Holdsmentioning

confidence: 84%

Snakes, Ladders, and Isolas of Localized Patterns

Beck¹,

Knobloch²,

Lloyd³

et al. 2009

SIAM J. Math. Anal.

163

260

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Abstract. Stable localized roll structures have been observed in many physical problems and model equations, notably in the 1D Swift-Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches, so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.

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Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation

Cited by 38 publications

References 21 publications

Swift-Hohenberg equation with broken reflection symmetry

Swift-Hohenberg equation with broken reflection symmetry

Snaking of Multiple Homoclinic Orbits in Reversible Systems

Snakes, Ladders, and Isolas of Localized Patterns

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