Defect-mediated snaking: A new growth mechanism for localized structures

Ma, Yi-Ping; Burke, John; Knobloch, Edgar

doi:10.1016/j.physd.2010.06.014

Cited by 57 publications

(81 citation statements)

References 32 publications

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“…11. The resulting behavior resembles in all aspects the phenomenon of defect-mediated snaking described in [28] except for the exponential shrinking of the region of existence of these states as the hole broadens. Consequently we refer to this behavior as collapsed at ρ M are starting to separate ( Fig.…”

Section: Dark Solitons For θ >mentioning

confidence: 55%

Dark solitons in the Lugiato-Lefever equation with normal dispersion

et al. 2016

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The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.

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Section: Dark Solitons For θ >mentioning

confidence: 55%

Dark solitons in the Lugiato-Lefever equation with normal dispersion

et al. 2016

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“…These dark solitons are connected by unstable solution branches that serve to add additional spatial oscillations in their profiles, leading to the broadening of the dark states. 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].…”

Section: Modification Of the Bifurcation Structure Of The Solitonsmentioning

confidence: 92%

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

2017

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Using the Lugiato-Lefever model, we analyze the effects of third-order chromatic dispersion on the existence and stability of dark-and bright-soliton Kerr frequency combs in the normal dispersion regime. While in the absence of third-order dispersion only dark solitons exist over an extended parameter range, we find that third-order dispersion allows for stable dark and bright solitons to coexist. Reversibility is broken and the shape of the switching waves connecting the top and bottom homogeneous solutions is modified. Bright solitons come into existence thanks to the generation of oscillations in the switching-wave profiles. Temporal oscillatory instabilities of dark solitons are suppressed in the presence of sufficiently strong third-order dispersion, while bright solitons are never found to oscillate in time. As a result of third-order dispersion both bright and dark solitons are found to move with a velocity that depends on their width.

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“…This asymmetry is greatly amplified by the rapid phase slips triggered by the continued self-compression of the structure (see the time interval 250 t 400 in figure 19b), much as described by Ma, Burke & Knobloch (2010). These set in at t ≈ 400, creating a complex transient that allows the structure to find its preferred wavelength.…”

Section: Effect Of Domain Sizementioning

confidence: 98%

Travelling convectons in binary fluid convection

et al. 2013

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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. With no-slip, fixed-temperature, no-mass-flux boundary conditions at the top and bottom stationary odd-and even-parity convectons fall on a pair of intertwined branches connected by branches of travelling asymmetric states. In appropriate parameter regimes the stationary convectons may be stable. When the boundary condition on the top is changed to Newton's law of cooling the odd-parity convectons start to drift and the branch of odd-parity convectons breaks up and reconnects with the branches of asymmetric states. We explore the dependence of these changes and of the resulting drift speed on the associated Biot number using numerical continuation, and compare and contrast the results with a related study of the Swift-Hohenberg equation by Houghton & Knobloch (Phys. Rev. E, vol. 84, 2011, art. 016204). We use the results to identify stable drifting convectons and employ direct numerical simulations to study collisions between them. The collisions are highly inelastic, and result in convectons whose length exceeds the sum of the lengths of the colliding convectons.

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Defect-mediated snaking: A new growth mechanism for localized structures

Cited by 57 publications

References 32 publications

Dark solitons in the Lugiato-Lefever equation with normal dispersion

Dark solitons in the Lugiato-Lefever equation with normal dispersion

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

Travelling convectons in binary fluid convection

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