Exact localized solutions of quintic discrete nonlinear Schrödinger equation

Maruno, Ken Ichi; Ohta, Yuichi; Joshi, Nalini

doi:10.1016/s0375-9601(03)00499-7

Cited by 16 publications

(19 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These simulations at κ = 1/2 reinforce our belief that our two dynamical criteria for understanding the stability of the solitary wave, namely the stability curve p(v) as well as studying orbits in the phase portrait are accurate indications of the stability of the solitary waves as obtained by numerical simulations of the FNLSE. Our results are likely to shed light on the behavior of a number of physical systems varying from optical fibers [1], Bragg gratings [2], BECs [3,4], nonlinear optics, and photonic crystals [7].…”

Section: Discussionmentioning

confidence: 95%

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

et al. 2012

View full text Add to dashboard Cite

We consider the nonlinear Schrödinger equation (NLSE) in 1 + 1 dimension with scalar-scalar self-interaction g 2 κ+1 (ψ ψ) κ+1 in the presence of the external forcing terms of the form re −i(kx+θ) − δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v k = 2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r → 0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq(t) < 0, where p(t) is the normalized canonical momentum p(t) = 1 M(t)∂L ∂q , anḋ q(t) is the solitary wave velocity. Here M(t) = dxψ (x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.

show abstract

Section: Discussionmentioning

confidence: 95%

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

et al. 2012

View full text Add to dashboard Cite

show abstract

“…Applying this method to Ablowitz-Ladik (AL) lattice system [21], we derived abundant Jacobian elliptic function doubly periodic wave solutions. When the modulus of elliptic functions m → 1 or 0, solitonic solutions including bright soliton and dark soliton solutions are generated.…”

Section: Introductionmentioning

confidence: 99%

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

Huang

Liu

2007

Int J Theor Phys

View full text Add to dashboard Cite

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.

show abstract

“…These terms appear in different physical contexts such as Bose gases with hard core interactions in the Tonks-Girardeau regime [28] and low dimensional Bose-Einstein condensates in which quintic nonlinearities in the NLS equation are used to model three-body interactions [29]. A self-focusing cubic-quintic NLS equation is also used in nonlinear optics as a model for photonic crystals [30].…”

Section: The Dcgl Equation Withmentioning

confidence: 99%

Dissipative Discrete System with Nearest-Neighbor Interaction for the Nonlinear Electrical Lattice

Abdoulkary¹,

Béda²,

Doka³

et al. 2012

JMP

View full text Add to dashboard Cite

A generalized dissipative discrete complex Ginzburg-Landau equation that governs the wave propagation in dissipative discrete nonlinear electrical transmission line with negative nonlinear resistance is derived. This equation presents arbitrarily nearest-neighbor nonlinearities. We analyze the properties of such model both in connection to their modulational stability, as well as in regard to the generation of intrinsic localized modes. We present a generalized discrete Lange-Newell criterion. Numerical simulations are performed and we show that discrete breathers are generated through modulational instability.

show abstract

Exact localized solutions of quintic discrete nonlinear Schrödinger equation

Cited by 16 publications

References 40 publications

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

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

Dissipative Discrete System with Nearest-Neighbor Interaction for the Nonlinear Electrical Lattice

Contact Info

Product

Resources

About