Spatial properties of integrable and nonintegrable discrete nonlinear Schrödinger equations

Hennig, Dirk; Sun, Ning; Gabriel, H.; Tsironis, G. P.

doi:10.1103/physreve.52.255

Cited by 38 publications

(24 citation statements)

References 51 publications

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“…2. For the doubly periodic wave solution u 2,n expressed by (24), in the long wave limit m → 1, the black soliton solution can be derived in the following…”

Section: Sd ξ N CD ξ N and Nd ξ N Expansionmentioning

confidence: 99%

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Doubly Periodic Wave Solutions and Soliton Solutions of Ablowitz–Ladik Lattice System

Huang

Liu

2007

Int J Theor Phys

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The general Jacobi elliptic function expansion method is developed and extended to construct doubly periodic wave solutions for discrete nonlinear equations. Applying this method, many exact elliptic function doubly periodic wave solutions are obtained for Ablowitz-Ladik lattice system. When the modulus m → 1 or m → 0, these solutions degenerate into hyperbolic function solutions and trigonometric function solutions respectively. In long wave limit, solitonic solutions including bright soliton and dark soliton solutions are also obtained.

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“…2. For the doubly periodic wave solution u 2,n expressed by (24), in the long wave limit m → 1, the black soliton solution can be derived in the following…”

Section: Sd ξ N CD ξ N and Nd ξ N Expansionmentioning

confidence: 99%

“…The AL system has N-soliton solutions and a rich mathematical structure, and there are many works about it and some its modifications [17,[23][24][25].…”

Section: Exact Solutions For Al Systemmentioning

confidence: 99%

Doubly Periodic Wave Solutions and Soliton Solutions of Ablowitz–Ladik Lattice System

Huang

Liu

2007

Int J Theor Phys

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“…It is convenient to present the real-valued second-order difference equation (5) in a form of a two-dimensional area-preserving map T [1,3,17,18] T :…”

Section: Basic Notations and Some Auxiliary Statementsmentioning

confidence: 99%

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

Alfimov

Brazhnyi

Konotop

2004

Physica D: Nonlinear Phenomena

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We consider localized modes (discrete breathers) of the discrete nonlinear Schrödin-ger equation i dψn dt = ψ n+1 + ψ n−1 − 2ψ n + σ|ψ n | 2 ψ n , σ = ±1, n ∈ Z. We study the diversity of the steady-state solutions of the form ψ n (t) = e iωt v n and the intervals of the frequency, ω, of their existence. The base for the analysis is provided by the anticontinuous limit (ω negative and large enough) where all the solutions can be coded by the sequences of three symbols "-", "0" and "+". Using dynamical systems approach we show that this coding is valid for ω < ω * ≈ −3.4533 and the point ω * is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for ω > ω * and give the complete table of them for the solutions with codes consisting of less than four symbols.

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“…[1][2][3][4] These solutions have been termed intrinsically localized modes, reflecting the fact that no external defects are needed for their creation, or discrete breathers, in analogy with the well-known solutions to integrable nonlinear partial differential equations, which have similar properties. In the particular case of the one-dimensional ͑1D͒ discrete nonlinear Schrödinger ͑DNLS͒ equation, which will be considered in this paper, the creation of these modes is an example of the well-known discrete self-trapping ͑DST͒ phenomenon, [5][6][7] by which an initially localized excitation remains localized around the initially excited site for all times.…”

Section: Introductionmentioning

confidence: 99%

Breatherlike excitations in discrete lattices with noise and nonlinear damping

Christiansen¹,

Gaididei²,

Johansson

et al. 1997

Phys. Rev. B

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We discuss the stability of highly localized, ''breatherlike,'' excitations in discrete nonlinear lattices under the influence of thermal fluctuations. The particular model considered is the discrete nonlinear Schrödinger equation in the regime of high nonlinearity, where temperature effects are included as multiplicative white noise and nonlinear damping. Numerical analysis shows that the lifetime of the breather is always finite and, in a large parameter regime, inversely proportional to the noise variance for fixed damping and nonlinearity. We also find that the decay rate of the breather decreases with increasing nonlinearity and with increasing damping. Using a collective-coordinate approximation, we show how the qualitative features of the numerical results can be analytically understood. Finally, in the dimer case we show that the multiplicative noise can be transformed into additive noise, and an exact stationary solution to the Fokker-Planck equation is obtained. From this solution, the dimer system is found to exhibit a noise ͑temperature͒ induced phase transition. ͓S0163-1829͑97͒07409-2͔

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Spatial properties of integrable and nonintegrable discrete nonlinear Schrödinger equations

Cited by 38 publications

References 51 publications

Doubly Periodic Wave Solutions and Soliton Solutions of Ablowitz–Ladik Lattice System

Doubly Periodic Wave Solutions and Soliton Solutions of Ablowitz–Ladik Lattice System

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

Breatherlike excitations in discrete lattices with noise and nonlinear damping

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