On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation

Alfimov, G. L.; Brazhnyi, V. A.; Konotop, V. V.

doi:10.1016/j.physd.2004.02.001

Cited by 83 publications

(95 citation statements)

References 25 publications

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“…They were found to exist below a maximum value of , which is max = 1.086, 1.789, 2.584 and 4.426, for σ = 1, 1.5, 2, and σ = 3, respectively (the presence of the upper limit for their existence is natural, as they, obviously, have no counterparts in the continuum limit, which corresponds to → ∞). For more details on the nature and bifurcation structure of the termination of such branches, the interested reader is directed to the detailed study of [31]. On the other hand, these branches may become unstable for > (1) cr , where the critical values are…”

Section: Two-humped Statesmentioning

confidence: 99%

Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

Cuevas¹,

Kevrekidis

Frantzeskakis

et al. 2009

Physica D: Nonlinear Phenomena

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We study the discrete nonlinear Schrödinger lattice model with the onsite nonlinearity of the general form, |u| 2σ u. We systematically verify the conditions for the existence and stability of discrete solitons in the one-dimensional version of the model predicted by means of the variational approximation (VA), and demonstrate the following: monostability of fundamental solitons (FSs) in the case of the weak nonlinearity, 2σ + 1 < 3.68; bistability, in a finite range of values of the soliton's power, for 3.68 < 2σ + 1 < 5, and the presence of a threshold (minimum norm of the FS), for 2σ + 1 ≥ 5. We also perform systematic numerical simulations to study higher-order solitons in the same general model, i.e., bound states of the FSs. While all in-phase bound states are unstable, stability regions are identified for antisymmetric double solitons, and their triple counterparts. These numerical findings are supplemented by an analytical treatment of the stability problem, which allows quantitively accurate predictions for the stability features of such multipulses. When these waveforms are found to be unstable, we show, by means of direct simulations, that they self-trap into a persistent lattice breather, or relax into a stable FS, or sometimes decay completely.

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Section: Two-humped Statesmentioning

confidence: 99%

Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

Cuevas¹,

Kevrekidis

Frantzeskakis

et al. 2009

Physica D: Nonlinear Phenomena

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show abstract

“…Such a classification is always possible for stationary states of high symmetry. Moreover, it can be used for each stationary solution, providing a complete description of the discrete spectrum, up to ω ≈ −5.45 [32]. What appears as a middle branch of the double-peaked (DP) solutions in Fig.…”

Section: Frequency Spectrummentioning

confidence: 99%

“…As it is shown in the Appendix, if the separation of the two peaks in the 2D lattice is given by S x and S y , then Fig. 9 presents numerical results in the case of χ < 0 regarding the magnitude of instability of symmetric DP states with different interpeak separations S x , S y (points), as well as a comparison with the power-paw (32).…”

Section: Double-peaked Solutionsmentioning

confidence: 99%

Multi-peaked localized states of DNLS in one and two dimensions

Kalosakas¹

2006

Physica D: Nonlinear Phenomena

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Multi-peaked localized stationary solutions of the discrete nonlinear Schrödinger (DNLS) equation are presented in one (1D) and two (2D) dimensions. These are excited states of the discrete spectrum and correspond to multi-breather solutions. A simple, very fast, and efficient numerical method, suggested by Aubry, has been used for their calculation. The method involves no diagonalization, but just iterations of a map, starting from trivial solutions of the anti-continuous limit. Approximate analytical expressions are presented and compared with the numerical results. The linear stability of the calculated stationary states is discussed and the structure of the linear stability spectrum is analytically obtained for relatively large values of nonlinearity.

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“…In the limit of ε → ∞, for C 1 = −1 or C 1 = 1 (in respective vicinities of the Brillouin zone), one can devise a long wavelength limit of the system; for C 1 = 1 and ∆β = 0, we fall back on the DNLS limit of waveguides of the same type. The key limit that we will utilize herein is the AC-limit of ε → 0 [30,31].…”

Section: Physical Model and Setupmentioning

confidence: 99%

Existence, stability and dynamics of discrete solitary waves in a binary waveguide array

Shen¹,

Kevrekidis²,

Srinivasan³

et al. 2016

J. Phys. A: Math. Theor.

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Recent work has explored binary waveguide arrays in the long-wavelength, near-continuum limit, here we examine the opposite limit, namely the vicinity of the so-called anti-continuum limit. We provide a systematic discussion of states involving one, two and three excited waveguides, and provide comparisons that illustrate how the stability of these states differ from the monoatomic limit of a single type of waveguide. We do so by developing a general theory which systematically tracks down the key eigenvalues of the linearized system. When we find the states to be unstable, we explore their dynamical evolution through direct numerical simulations. The latter typically illustrate, for the parameter values considered herein, the persistence of localized dynamics and the emergence for the duration of our simulations of robust quasi-periodic states for two excited sites. As the number of excited nodes increase, the unstable dynamics feature less regular oscillations of the solution's amplitude.

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On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation

Cited by 83 publications

References 25 publications

Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

Multi-peaked localized states of DNLS in one and two dimensions

Existence, stability and dynamics of discrete solitary waves in a binary waveguide array

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