Attractor for the nonlinear Schrödinger equation with a nonlocal nonlinear term

Zhu, Chaosheng; Mu, Chunlai; Zhang, Pu

doi:10.1007/s10883-010-9108-6

Cited by 6 publications

(5 citation statements)

References 6 publications

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“…Such a behavior was observed experimentally. 18 To find a more rigorous expression for the expected oscillation frequency one have to find Joptimal analyzing the value of the oscillation threshold and then solve equation (28).…”

Section: T Rnomentioning

confidence: 99%

“…One is able to describe the threshold and averaged frequency of the oscillation in the resonator, made of a material with known dispersion and nonlinearity, using equations (18), (19), (28), and (30). This technique allows studying generation of just one dominating pair of sidebands, which can be produced either in the case oflarge enough dispersion when the mode spectrum is significantly non-equidistant,14 or in the case of a significantly nonlinear dispersion curve and operation in the vicinity of zero dispersion point.…”

Section: T Rnomentioning

confidence: 99%

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Whispering Gallery Mode Oscillators and Optical Comb Generators

Matsko

Savchenkov

Liang

et al. 2009

Frequency Standards and Metrology

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We discuss an approach for generating optical combs using four-wave mixing process in a nonlinear whispering gallery mode resonator. We show that pumping the resonator with strong enough continuous wave coherent light results in appearance of optical sidebands around the optical carrier. The pumping threshold of the oscillation can be in a few microWatt range for resonators with an ultra-high quality factor. Natural cascading of the the nonlinear process leads to mode locking of the sidebands and emergence of an optical frequency comb. Demodulation of the comb by means of a fast photodiode produces a high frequency spectrally pure RF signal, the frequency of which is given by the resonator morphology.

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Section: T Rnomentioning

confidence: 99%

Section: T Rnomentioning

confidence: 99%

Whispering Gallery Mode Oscillators and Optical Comb Generators

Matsko

Savchenkov

Liang

et al. 2009

Frequency Standards and Metrology

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“…To obtain an attractor, we have to assume that the equation has a positive damping parameter > 0. The dynamical behavior of the damped Schrödinger equation was widely investigated by many physicists and mathematicians (see, e.g., [3][4][5][6][7][8][9][10][11]) but restricted in the autonomous case; that is, the force is time-independent (only space dependent). This paper deals with dynamics for the nonautonomous Schrödinger equation; that is, the force is time-dependent.…”

Section: Introductionmentioning

confidence: 99%

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

She

Wang

2018

Discrete Dynamics in Nature and Society

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This paper deals with pullback dynamics for the weakly damped Schrödinger equation with time-dependent forcing. An increasing, bounded, and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable. Also, we obtain the forward absorption, which is only used to deduce the backward compact-decay decomposition according to high and low frequencies. Based on a new existence theorem of a backward compact pullback attractor, we show that the nonautonomous Schrödinger equation has a pullback attractor which is compact in the past. The method of energy, high-low frequency decomposition, Sobolev embedding, and interpolation are quite involved in calculating a priori pullback or forward bound.

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“…Later, Ma and Chang [33] studied the semi-dicretized NLS equation (1.1) and proved that for each mesh size, there exist attractors for the discretized system. In the one-dimensional case for Equation (1.1), it is also proved in [47] that the attractor is made of H 2 ([0, 1]) functions if the external force f (x) is H 2 ([0, 1]). In this paper, our main aim is to improve results in [47].…”

Section: Introductionmentioning

confidence: 99%

“…In the one-dimensional case for Equation (1.1), it is also proved in [47] that the attractor is made of H 2 ([0, 1]) functions if the external force f (x) is H 2 ([0, 1]). In this paper, our main aim is to improve results in [47]. To this end, we first prove that the existence of the global attractor A γ in the strong topology of H 1 (R) and the existence of the exponential attractor M which contains the global attractor A γ , are still finite dimensional, and attract the trajectories exponentially fast.…”

Section: Introductionmentioning

confidence: 99%

Global Attractor of Nonlocal Nonlinear Schrödinger Equation on R

Chao-sheng¹

2016

AAN

Self Cite

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This paper is devoted to the large time behavior and especially to the regularity of the global attractor of the dissipative 1D nonlinear Schrödinger equation with nonlocal integral term on R. We first prove that the existence of the global attractor Aγ in the strong topology of H 1 (R) and the existence of the exponential attractor M which contains the global attractor Aγ, are still finite dimensional, and attract the trajectories exponentially fast. We also show that the global attractor Aγ is regular, i.e., Aγ is included, bounded and compact in H 2 (R) assuming that the forcing term f (x) is of class H 2 (R). Furthermore we estimate the number of the determining modes for this equation. Moreover, we show that the solution trajectories and the global attractor of the nonlocal Schrödinger equation converge to those of the usual Schrödinger equation, as the coefficient of the nonlocal integral term goes to zero.

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Attractor for the nonlinear Schrödinger equation with a nonlocal nonlinear term

Cited by 6 publications

References 6 publications

Whispering Gallery Mode Oscillators and Optical Comb Generators

Whispering Gallery Mode Oscillators and Optical Comb Generators

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

Global Attractor of Nonlocal Nonlinear Schrödinger Equation on R

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