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Journal ArticleDOI

Standing localized modes in nonlinear lattices.

01 Oct 1994-Physical Review E (Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics)-Vol. 50, Iss: 4, pp 3161-3170
TL;DR: It is shown that spatially localized modes exist in the frequency--wave number domain where the lattice displays modulational instability; two families of localized modes are found for this case as separatrix solutions of the effective equations for the wave envelopes.
Abstract: The theory of standing localized modes in discrete nonlinear lattices is presented. We start from a rather general model describing a chain of particles subjected to an external (on-site) potential with cubic and quartic nonlinearities (the so-called Klein-Gordon model), and, using the approximation based on the discrete nonlinear Schro$iuml---dinger equation, derive a system of two coupled nonlinear equations for slowly varying envelopes of two counterpropagating waves of the same frequency. We show that spatially localized modes exist in the frequency--wave number domain where the lattice displays modulational instability; two families of localized modes are found for this case as separatrix solutions of the effective equations for the wave envelopes. When the nonlinear plane wave in the lattice is stable to small modulations of its amplitude, nonlinear localized modes appear as dark solitons associated with the so-called extended modulational instability. These localized modes may be treated as domain walls or kinks connecting two standing plane-wave modes of the similar structure. We investigate analytically and numerically the special family of such localized solutions that, in the vicinity of the zero-dispersion point, cover exactly the case of the so-called self-induced gap solitons recently introduced by Kivshar [Phys. Rev. Lett. 70, 3055 (1993)]. Application of the theory to the case of parametrically driven damped lattices is also briefly discussed, and it is mentioned that some of the solutions considered in the present paper may be extended to include the case of localized modes in driven damped lattices, provided the mode frequency and amplitude are fixed by the parameters of the external parameters of the external parametric ac force.
Citations
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Journal ArticleDOI
TL;DR: A detailed overview of the physics and applications of optical dark solitons can be found in this article, where the authors discuss the instability-induced dynamics of dark-solitons in the models of generalized (i.e., non-Kerr) optical nonlinearities.

1,076 citations


Cites background or methods from "Standing localized modes in nonline..."

  • ...…and Skinner (1991a,b), Miranda et al. (1992) and Y. Chen and Atai (1992)], dark solitons and dark-profile modes in discrete lattices (Kivshar, 1993a; Kivshar et al., 1994a,c), dark gap solitons in the systems with periodically varying parameters such as optical waveguides with grating [e.g., Feng…...

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  • ...We follow the original paper by Kivshar et al. (1994a)....

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  • ...Applying the same reasoning as in the case of the bright soliton, we obtain the following result (Kivshar et al., 1994a) SdX2T"2D u2 0 P `= ~= Ka(/) v z !b(/) v tK 2 dt , (3.38) where, as above, we have assumed that the field dv is a stochastic complex noise with the only nonzero…...

    [...]

  • ...(3.40) Validity of this result was discussed by Kivshar et al. (1994a)....

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Journal ArticleDOI
TL;DR: In this article, the existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators are considered.

76 citations

Book ChapterDOI
TL;DR: In this article, the modulational instability (MI) of electromagnetic waves in inhomogeneous and in discrete media is discussed, and the origin of the random fluctuations of parameters in optical fibers and other nonlinear optical media is described.
Abstract: Publisher Summary This chapter discusses the modulational instability (MI) of electromagnetic waves in inhomogeneous and in discrete media. MI exists because of the interplay between the nonlinearity and dispersion/diffraction effects. Important models for investigating MI of electromagnetic waves in nonlinear media represent the scalar and vectorial nonlinear Schrodinger (NLS) equations, the system describing evolution of the envelopes of fundamental and second harmonics waves in quadratically nonlinear media, and sine-Gordon equation. The methods such as periodic solutions of the NLS equation and the coupled-mode theory with three modes are discussed. The chapter discusses the MI of electromagnetic waves in optical media with periodic inhomogeneities. The origin of the random fluctuations of parameters in optical fibers and other nonlinear optical media is described. MI in fibers with random amplification and dispersion and MI in randomly birefringent fibers are discussed. The chapter discusses the MI of electromagnetic waves in nonlinear discrete optical systems such as an array of planar waveguides and fibers. Particular cases of MI in discrete media with cubic nonlinearity and quadratic nonlinearity are investigated.

42 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of nontrivial breather solutions of the discrete nonlinear Schrodinger equation with the unbounded potential at infinity was investigated, and a discrete version of compact embedding theorem was derived.
Abstract: In this paper I investigate the existence of nontrivial breather solutions of the discrete nonlinear Schrodinger equation with the unbounded potential at infinity. First I derive a discrete version of compact embedding theorem. Then combining the Nehari manifold approach and the compact embedding theorem, I show the existence of breather solutions without Palais–Smale condition. The results on the exponential decay of breather solutions are also provided in this paper.

38 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of infinitely many non-trivial standing wave solutions of the discrete non-linear Schrodinger equation with the unbounded potential at infinity was proved by using the linking theorem.
Abstract: In this article, we prove the existence of infinitely many non-trivial standing wave solutions of the discrete non-linear Schrodinger equation with the unbounded potential at infinity by using the linking theorem.

36 citations