Homoclinic crossings and pattern selection

Moon, Hie Tae

doi:10.1103/physrevlett.64.412

Cited by 91 publications

(45 citation statements)

References 11 publications

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“…In this sense the physics differs from other integrable models exhibiting a complex nonlinear dynamics of MI already in the undriven regime (e.g. focusing NLSE [26,[30][31][32][33][34][35][36]), around which chaos can develop under weak periodic perturbations [31,37,38].We consider the following periodic NLSEreferring, without loss of generality, to the notation used in optical fibers in suitable scaled units. The dispersion is β(z) = β av + β m f Λ (z), with positive average β av > 0 (equivalent to the defocusing regime); f Λ (z) has period Λ = 2π/k g , zero mean and minimum −1.…”

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confidence: 99%

Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation

et al. 2016

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We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis. [12,13]. While the concept of PR originates in the linear world [14], PRs deeply impact also the behavior of nonlinear conservative systems. However, the full nonlinear dynamics of PRs is relatively well understood only for lowdimensional Hamiltonian systems [2,15,16]. Conversely, the analysis of extended systems described by PDEs with periodicity in the evolution variable [17] is essentially limited to determine the region of parametric instability (Arnold tongues) via Floquet analysis [18][19][20][21], while the nonlinear stage of PR past the linearized growth of the unstable modes remains mostly unexplored.In this letter, taking the periodic defocusing nonlinear Schrödinger equation (NLSE) as a widespread example describing, e.g. periodic management of atom condensates [12,19,20,22], optical beam propagation in layered media [21], or optical fibers with periodic dispersion [18,23,24], we show that the PR gives rise to quasi-periodic evolutions which exhibit on average FermiPasta-Ulam (FPU) recurrence [25] with a remarkably complex (but ordered) underlying phase-plane structure. Such structure describes the continuation into the nonlinear regime of the modulational instability (MI) of a background solution, uniquely due to the parametric forcing (with zero forcing the defocusing NLSE is stable). A byproduct of this structure is the existence of breatherlike solutions [26]. This fact suggests the intriguing possibility of observing rogue waves [27], which are commonly associated with breathers [28], in the defocusing NLSE [29]. On the other hand, the richness of such structure allows us to remarkably predict that optimal parametric amplification occurs at a critical frequency where the system lies off-resonance (outside the PR bandwidth). Our approach retains its validity in the most interesting regime of strong parametric driving, where the system is found to exhibit a remarkably ordered structure despite its broken translational symmetry and integrability. In this sense the physics differs from other integrable models exhibiting a complex nonlinear dynamics of MI already in the undriven regime (e.g. focusing NLSE [26,[30][31][32][33][34][35][36]), around which chaos can develop under weak periodic perturbations [31,37,38].We consider the following periodic NLSEreferring, without loss of genera...

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Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation

et al. 2016

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“…The transition follows an initial inverse pitchfork bifurcation which sets the system close to a heteroclinic trajectory involving one stable fixed point and two distinct unstable fixed points. States with larger number of unstable points are also found to exist as the low-frequency wave vector is varied [11,12].…”

Section: Introductionmentioning

confidence: 95%

Length scale, quasiperiodicity, resonances, separatrix crossings, and chaos in the weakly relativistic Zakharov equations

1995

Phys. Rev. E

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Nonlinear saturation of unstable solutions to the weakly relativistic, one-dimensional Zakharov equations is considered in this paper. In order to perform the analysis, two quantities are introduced.One of them, p" is proportional to the initial energy of the high-frequency 6eld, and the other is the basic wave vector of the low-frequency perturbing mode k = 2'/L, with L as the length scale. With these quantities it becomes possible to identify a number of regions on a p versus k parametric plane. For very small values of p, steady-state solutions become unstable when A: is also very small.In this case ion-acoustic dynamics is found to be unimportant and the system is numerically shown to be approximately integrable, even if k is below a critical value where the solutions are not simply periodic. For larger values of p the unstable wave vectors also become larger and the ion-acoustic Huctuations turn into active dynamical modes of the system, driving a transition to chaos, which follows initial inverse pitchfork bifurcations. The transition includes resonant and quasiperiodic features; separatrix crossing phenomena are also found. The in6uence of relativistic terms on the chaotic dynamics is studied in the context of the Zakharov equations; it is shown that relativistic terms generally enhance the instabilities of the system, therefore anticipating the transition. PACS number(s): 52.35.Ra, 47.20.Ky

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“…2(f) and 2(j) demonstrate that the KAM tori are completely broken down and irregular HMO crossings gradually appear with the off-axis evolution of laser field. As we have known, irregular HMO crossings are associated with spatiotemporal chaos 26,27 and pattern competition. 23 Different from pattern competition that experiences a phase jump from θ = 45 • to θ = 225 • , these two complicated patterns shown in Figs.…”

Section: Numerical Simulations and Nonlinear Dynamicsmentioning

confidence: 99%

Propagation of femtosecond terawatt laser pulses in N2 gas including higher-order Kerr effects

Huang

Zhou

2012

AIP Advances

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Propagation characteristic of femtosecond terawatt laser pulses in N2 gas with higher-order Kerr effect (HOKE) is investigated. Theoretical analysis shows that HOKE acting as Hamiltonian perturbation can destroy the coherent structure of a laser field and result in the appearance of incoherent patterns. Numerical simulations show that in this case two different types of complex structures can appear. It is found that the high-order focusing terms in HOKE can cause continuous phase shift and off-axis evolution of the laser fields when irregular homoclinic orbit crossings of the field in phase space take place. As the laser propagates, small-scale spatial structures rapidly appear and the evolution of the laser field becomes chaotic. The two complex patterns can switch between each other quasi-periodically. Numerical results show that the two complex patterns are associated with the stochastic evolution of the energy contained in the higher-order shorter-wavelength Fourier modes. Such complex patterns, associated with small-scale filaments, may be typical for laser propagation in a HOKE medium

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Homoclinic crossings and pattern selection

Cited by 91 publications

References 11 publications

Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation

Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation

Length scale, quasiperiodicity, resonances, separatrix crossings, and chaos in the weakly relativistic Zakharov equations

Propagation of femtosecond terawatt laser pulses in N2 gas including higher-order Kerr effects

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