Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion

Parra‐Rivas, Pedro; Gomila, Damià; Gelens, Lendert; Knobloch, Edgar

doi:10.1103/physreve.97.042204

Cited by 69 publications

(93 citation statements)

References 84 publications

(146 reference statements)

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“…This FW bifurcation originates in the codimension-two point X, which appears to organize these connections. Finally, as θ → 2 and k c → 0 the bifurcation structure of patterns transforms into foliated snaking of localized structures [20], as a pattern with infinite wavelength corresponds in effect to a single peak localized structure in a finite size system.…”

Section: Discussionmentioning

confidence: 99%

“…A detailed analysis of how the bifurcation structure of such localized structures changes as one approaches this critical point θ = 2 can be found in Ref. [20]. At this point we can already identify several distinct solution regimes based on the existence of patterns and the stability of A 0 :…”

Section: Linear Stability Analysis Of the Homogeneous Steady Statesmentioning

confidence: 90%

“…While the MI bifurcation corresponds to the point where the HSSs lose stability to temporal perturbations, it is also possible to study this transition in the context of spatial dynamics. Here, the HSS is interpreted as a fixed point in a four-dimensional phase space [20], and the MI corresponds to a Hamiltonian-Hopf (HH) bifurcation with eigenvalues λ = ±ik c of double multiplicity. In this formulation the pattern state corresponds to a periodic orbit, and this orbit bifurcates from HSS at ρ c (for θ < 2) with initial period (wavelength) 2π/k c .…”

Section: Bifurcation Structure Of Patternsmentioning

confidence: 99%

“…Thus a parameter regime is present in which A b 0 and P kc coexist stably. As a result localized structures (LS) are also present and these are organized in a so-called homoclinic snaking structure [20,[25][26][27][28]. These LS are found in the region of bistability between A b 0 and P kc colored in dark gray in Fig.…”

Section: Bifurcation Structure Of Patternsmentioning

confidence: 99%

“…For the parameter values for which the patterns are subcritical, this bifurcation also leads to the formation of localized structures. For a detailed study of the bifurcation structure of such localized states in the LL model, we refer to [20].…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

et al. 2018

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We study the stability and bifurcation structure of spatially extended patterns arising in nonlinear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number kc arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number 2kc. This next branch also connects to a branch of patterns with double wave number, this time 4kc, and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching θ = 2 from below the critical wave number kc approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied.

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Section: Discussionmentioning

confidence: 99%

Section: Linear Stability Analysis Of the Homogeneous Steady Statesmentioning

confidence: 90%

Section: Bifurcation Structure Of Patternsmentioning

confidence: 99%

Section: Bifurcation Structure Of Patternsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

et al. 2018

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Stationary localised patterns without Turing instability

Saadi

Worthy

Msmali

et al. 2022

Math Methods in App Sciences

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Since the pioneering work of Turing, it has been known that diffusion can destablise a homogeneous solution that is stable in the underlying model in the absence of diffusion. The destabilisation of the homogeneous solutions leads to the generation of patterns. In recent years, techniques have been developed to analyse so-called localised spatial structures. These are solutions in which the spatial structure occurs in a localised region. Unlike Turing patterns, they do not spread out across the whole domain. We investigate the existence of localised structures that occur in two predator-prey models. The functionalities chosen have been widely used in the literature. The existence of localised spatial structures has not investigated previously. Indeed, it is easy to show that these models cannot exhibit the Turing instability. This has perhaps led earlier researchers to conclude that interesting spatial solutions can therefore not occur for these models. The novelty of our paper is that we show the existence of stationary localised patterns in systems which do not undergo the Turing instability. The mathematical tools used are a combination of Linear and weakly nonlinear analysis with supporting numerical methods. By combining these methods, we are able to identify conditions for a wide range of increasing exotic behaviour. This includes the Belyckov-Devaney transition, a codimension two spatial instability point and the formation of localised patterns. The combination of spectral computations and numerical simulations reveals the crucial role played by the Hopf bifurcation in mediating the stability of localised spatial solutions. Finally, numerical solutions in two spatial dimensions confirms the onset of intricate spatio-temporal patterns within the parameter regions identified within one spatial dimension.

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Dissipative Systems

Knobloch¹

2020

Emerging Frontiers in Nonlinear Science

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Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion

Cited by 69 publications

References 84 publications

Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

Stationary localised patterns without Turing instability

Dissipative Systems

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