Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios

Knobloch, Jürgen; Lloyd, David J. B.; Sandstede, Björn; Wagenknecht, Thomas

doi:10.1007/s10884-010-9195-9

Cited by 32 publications

(43 citation statements)

References 23 publications

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“…The difference with that case, though is that there the isolas represent travelling solitary structures, whereas here they are stationary. Other relevant results for continuum systems is the analysis in [19,3] in which two-pulse stationary solitons near a homoclinic snake are shown to generically lie on isolas with a similar topology to those encountered here, even in the absence of symmetry-breaking.…”

supporting

confidence: 52%

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Yulin¹,

Champneys²

2010

SIAM J. Appl. Dyn. Syst.

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Abstract. This paper continues an investigation into a one-dimensional lattice equation that models the light field in a system comprised of a periodic array of pumped optical cavities with saturable nonlinearity. The additional effects of a spatial gradient of the phase of the pump field are studied, which in the presence of loss terms is shown to break the spatial reversibility of the steady problem. Unlike for continuum systems, small symmetry-breaking is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons are no-longer arranged in a homoclinic snaking bifurcation diagrams, but instead break up into discrete isolas. For large enough symmetry-breaking, the fold bifurcations of the lowest intensity solitons no longer overlap, which is argued to be the trigger point of moving localised structures. Due to the dissipative nature of the problem, any radiation shed by these structures is damped and so they appear to be true attractors. Careful direct numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking. Introduction.Recently, the present authors [33,34] studied the rich variety of stable localised structures that can occur in a spatially discrete model for an optical cavity with an imposed periodic structure and saturable nonlinearity. In particular, it was shown how infinitely many distinct stable localised structures can occur for a wide range of parameter values due to the so-called homoclinic snaking mechanism that has received a lot of attention in continuum models of pattern formation [1,4,5,6,8,14,20,22,35]. This paper explores the consequences of introducing a spatial gradient to the optical pumping field (the forcing term in the model), which breaks the spatial reversibility of the steady system.The model can be written in dimensionless form as the one-dimensional lattice equation for a complex field A n ∈ C, n ∈ Z:

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supporting

confidence: 52%

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Yulin¹,

Champneys²

2010

SIAM J. Appl. Dyn. Syst.

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“…Numerical continuation reveals that, for a fixed driving amplitude (here, a = 0.2 μm), the DBs and multibreathers appear to be located on a single coiling solution branch. This structure, sometimes referred to as "snaking" in the dynamical systems community [35][36][37][38][39][40][41], has received considerable recent attention in settings such as nematic liquid crystals [42] and classical fluid problems such as Couette flow [43]. To the best of the authors' knowledge, snaking behavior in FPU-like chains (such as a granular crystal chain) has not been reported previously.…”

Section: The Damped-driven Modelmentioning

confidence: 99%

Damped-driven granular chains: An ideal playground for dark breathers and multibreathers

Chong

Yang

et al. 2014

Phys. Rev. E

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By applying an out-of-phase actuation at the boundaries of a uniform chain of granular particles, we demonstrate experimentally that time-periodic and spatially localized structures with a nonzero background (so-called dark breathers) emerge for a wide range of parameter values and initial conditions. We demonstrate a remarkable control over the number of breathers within the multibreather pattern that can be "dialed in" by varying the frequency or amplitude of the actuation. The values of the frequency (or amplitude) where the transition between different multibreather states occurs are predicted accurately by the proposed theoretical model, which is numerically shown to support exact dark breather and multibreather solutions. Moreover, we visualize detailed temporal and spatial profiles of breathers and, especially, of multibreathers using a full-field probing technology and enable a systematic favorable comparison among theory, computation, and experiments. A detailed bifurcation analysis reveals that the dark and multibreather families are connected in a "snaking" pattern, providing a roadmap for the identification of such fundamental states and their bistability in the laboratory.

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“…On the other hand, unequally spaced two-convecton states are expected to lie on nested isolas (Burke & Knobloch 2009;Knobloch et al 2010); on the real line there will be an infinite number of such isolas, corresponding to the infinite number of discrete separations permitted by the locking between the oscillatory tails of neighbouring convectons. Moreover, each set of nested isolas corresponds to a fixed number of rolls within the two convectons forming the two-pulse state.…”

Section: Two-convecton States: Neumann Boundary Conditionsmentioning

confidence: 99%

“…Moreover, each set of nested isolas corresponds to a fixed number of rolls within the two convectons forming the two-pulse state. Thus, on the real line the snaking region is expected to contain an infinite stack of such isolas, each corresponding to a bound state of convectons with different number of rolls (Burke & Knobloch 2009;Knobloch et al 2010). In general, the breakup of the snakes-and-ladders structure of the pinning region (Burke & Knobloch 2007a) that gives rise to these structures becomes exponentially small as soon as the two convectons are substantially far apart.…”

Section: Two-convecton States: Neumann Boundary Conditionsmentioning

confidence: 99%

Convectons, anticonvectons and multiconvectons in binary fluid convection

et al. 2010

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Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. Numerical continuation is used to compute such states in the presence of both Neumann boundary conditions and no-slip no-flux boundary conditions in the horizontal. In addition to the previously identified convectons, new states referred to as anticonvectons with a void in the centre of the domain, and wall-attached convectons attached to one or other wall are identified. Bound states of convectons and anticonvectons called multiconvecton states are also computed. All these states are located in the so-called snaking or pinning region in the Rayleigh number and may be stable. The results are compared with existing results with periodic boundary conditions.

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Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios

Cited by 32 publications

References 23 publications

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Damped-driven granular chains: An ideal playground for dark breathers and multibreathers

Convectons, anticonvectons and multiconvectons in binary fluid convection

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