Energy Thresholds for Discrete Breathers in One-, Two-, and Three-Dimensional Lattices

Flach, Sergej; Kladko, K.; MacKay, Robert S.

doi:10.1103/physrevlett.78.1207

Cited by 189 publications

(271 citation statements)

References 11 publications

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“…The value P th (1, m) ≈ 3.2 corresponds to the threshold power of discrete surface solitons in a semi-infinite array [4]. For 1D localized modes there exists no power threshold in the continuum limit [13], but in our system P th (8, m) ≈ 0.4 due to finite-size effects.…”

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confidence: 79%

“…In the nonlinear case, we look for localized stationary solutions of the form u n,m (ξ) = u n,m exp(iλξ), where the amplitudes u n,m are real, and λ is the nonlinear propagation constant. For a given λ, localized solutions are found in a 15 × 15 lattice by using the Newton-Raphson method.We calculate the power threshold P th that characterizes the discrete solitons in 2D lattices [13]. We study three different modes: corner [ Fig.…”

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confidence: 99%

“…We calculate the power threshold P th that characterizes the discrete solitons in 2D lattices [13]. We study three different modes: corner [ Fig.…”

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Discrete surface solitons in two-dimensional anisotropic photonic lattices

et al. 2007

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We study nonlinear surface modes in two-dimensional anisotropic periodic photonic lattices and demonstrate that, in a sharp contrast to one-dimensional discrete surface solitons, the mode threshold power is lower at the surface, and two-dimensional discrete solitons can be generated easier near the lattice corners and edges. We analyze the crossover between effectively one-and two-dimensional regimes of the surface-mediated beam localization in the lattice. PACS numbers:Surface modes have been studied in different branches of physics; in guided wave optics surface states were predicted to exist at interfaces separating periodic and homogeneous dielectric media [1]. The interest in studying surface waves has been renewed recently because the interplay of discreteness and nonlinearity can facilitate the formation of discrete surface solitons [2,3] at the edge of the waveguide array. That can be understood as the localization of a discrete optical soliton near the surface [4] for powers exceeding a certain threshold value, for which the repulsive effect of the surface is balanced. A similar effect of light localization near the edge of the waveguide array and the formation of surface gap solitons have been predicted and observed for defocusing nonlinear media [5,6].It is important to analyze how the properties of nonlinear surface waves are modified by the lattice dimensionality, and the first studies of different types of discrete surface solitons in two-dimensional lattices [7,8,9,10] revealed, in particular, that the presence of a surface increases the stability region for two-dimensional (2D) discrete solitons [10] and the threshold power for the edge surface state is slightly higher than that for the corner soliton [9].In this Letter we consider anisotropic semi-infinite twodimensional photonic lattices and study the crossover between one-and two-dimensional surface solitons emphasizing the crucial effect of the lattice dimensionality on the formation of surface solitons.We consider a semi-infinite 2D lattice [shown schematically in Fig.2(a) below], described by the system of coupled-mode equations for the normalized amplitudes u n,m [11,12],where ξ is the normalized propagation distance. We de- fine the lattice coupling as follows:where α characterizes the lattice anisotropy.Linear lattice waves of the form u n,m (ξ) = u 0 sin(kn) sin(qm) exp(iβξ) satisfy the dispersion relation β kq = 2(cos k + α cos q). In the nonlinear case, we look for localized stationary solutions of the form u n,m (ξ) = u n,m exp(iλξ), where the amplitudes u n,m are real, and λ is the nonlinear propagation constant. For a given λ, localized solutions are found in a 15 × 15 lattice by using the Newton-Raphson method.We calculate the power threshold P th that characterizes the discrete solitons in 2D lattices [13]. We study three different modes: corner [ Fig. 1(a)], edge surface [ Fig. 1(b,c)] and central [ Fig. 1(d)] localized modes. The corner and edge modes represent 2D surface localized

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Discrete surface solitons in two-dimensional anisotropic photonic lattices

et al. 2007

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“…The existence of localized solutions is well known for spatially discrete systems (including quintic and higher order nonlinearities) [25,26,27,28]. Moreover the stability of these solutions at weak coupling is established as well, independent of the integrability of the underlying equations of motion.…”

Section: Introductionmentioning

confidence: 97%

Exact localized solutions of quintic discrete nonlinear Schrödinger equation

2003

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We study a new quintic discrete nonlinear Schrödinger (QDNLS) equation which reduces naturally to an interesting symmetric difference equation of the form φ n+1 + φ n−1 = F (φ n ). Integrability of the symmetric mapping is checked by singularity confinement criteria and growth properties. Some new exact localized solutions for integrable cases are presented for certain sets of parameters. Although these exact localized solutions represent only a small subset of the large variety of possible solutions admitted by the QDNLS equation, those solutions presented here are the first example of exact localized solutions of the QDNLS equation. We also find chaotic behavior for certain parameters of nonintegrable case.

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“…This is, in particular, the case of 2D and 3D lattices [6,8,11]. In order to explore such a possibility in the 1D model (1) we recall that the condition for the modulational [3,9] (also referred to as dynamical [20]) instability now reads M (σ) n χ (σ) n < 0.…”

Section: The Modelmentioning

confidence: 99%

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

Bludov

Brazhnyi

2007

Phys. Rev. A

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It is shown that inhomogeneous nonlinear interactions in a Bose-Einstein condensate loaded in an optical lattice can result in delocalizing transition in one dimension, what sharply contrasts to the known behavior of discrete and periodic systems with homogeneous nonlinearity. The transition can be originated either by decreasing the amplitude of the linear periodic potential or by the change of the mean value of the periodic nonlinearity. The dynamics of the delocalizing transition is studied.

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Energy Thresholds for Discrete Breathers in One-, Two-, and Three-Dimensional Lattices

Cited by 189 publications

References 11 publications

Discrete surface solitons in two-dimensional anisotropic photonic lattices

Discrete surface solitons in two-dimensional anisotropic photonic lattices

Exact localized solutions of quintic discrete nonlinear Schrödinger equation

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

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