Stability and quasi-equidistant propagation of NLS soliton trains

Gerdjikov, Vladimir S.; Evstatiev, Evstati; Kaup, D. J.; Diankov, Georgi; Uzunov, Ivan M.

doi:10.1016/s0375-9601(98)00152-2

Cited by 37 publications

(94 citation statements)

References 18 publications

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“…[3,5]. These papers propose a method to determine the asymptotic dynamical regime of the CTC for a given set of initial parameters ν k (0), µ k (0), ξ k (0) and δ k (0).…”

Section: Analysis Of the Perturbed Ctc And Comparison With Numerical mentioning

confidence: 99%

“…On the contrary the IC (32) lead to singular solutions for the CTC (see [2,5,23]). The singularities of the exact solutions for the CTC coincide with the positions of the collisions.…”

Section: Analysis Of the Perturbed Ctc And Comparison With Numerical mentioning

confidence: 99%

“…started with the pioneering paper [1], by now has been extensively studied (see [2,3,4,5,6] and references therein). Several other nonlinear evolution equations (NLEE) were also studied, among them the modified NLS equation [8,9,10,11,12], some higher NLS equations [6], the Ablowitz-Ladik system [7] and others.…”

Section: Introductionmentioning

confidence: 99%

“…The adiabatic approximation uses as a small parameter ε 0 ≪ 1 the soliton overlap which falls off exponentially with the distance between the solitons. Then the soliton parameters must satisfy [1]: One can expect that the approximation holds only for such times t for which the set of 4N parameters of the soliton train satisfy (5). Equation (1) finds a number of applications to nonlinear optics and for R[u] ≡ 0 is integrable via the inverse scattering transform method [13,14].…”

Section: Introductionmentioning

confidence: 99%

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Modeling Adiabatic N-Soliton Interactions and Perturbations

2005

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We analyze perturbed version of the complex Toda chain (CTC) in an attempt to describe the adiabatic N -soliton train interactions of the perturbed nonlinear Schrödinger equation (NLS). We study perturbations with weak quadratic and periodic external potentials by both analytical and numerical means. The perturbed CTC adequately models the N -soliton train dynamics for both types of potentials. As an application of the developed theory we consider the dynamics of a train of matter -wave solitons confined in a parabolic trap and optical lattice, as well as tilted periodic potentials.

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“…[3,5]. These papers propose a method to determine the asymptotic dynamical regime of the CTC for a given set of initial parameters ν k (0), µ k (0), ξ k (0) and δ k (0).…”

Section: Analysis Of the Perturbed Ctc And Comparison With Numerical mentioning

confidence: 99%

“…On the contrary the IC (32) lead to singular solutions for the CTC (see [2,5,23]). The singularities of the exact solutions for the CTC coincide with the positions of the collisions.…”

Section: Analysis Of the Perturbed Ctc And Comparison With Numerical mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modeling Adiabatic N-Soliton Interactions and Perturbations

2005

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“…Extensive use of the fact that the CTC is a completely integrable model permits to classify soliton parameter regions with different asymptotic regimes of the Nsoliton train [5,6,7,8]. It was also shown in [8] that the CTC can be associated with any equation from the NSE hierarchy.…”

Section: Introductionmentioning

confidence: 99%

Adiabatic interaction ofNultrashort solitons: Universality of the complex Toda chain model

Gerdjikov

Yang

2001

Phys. Rev. E

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Using the Karpman-Solov'ev method we derive the equations for the 2-soliton adiabatic interaction for solitons of the modified nonlinear Schrödin-ger equation (MNSE). Then we generalize these equations to the case of N interacting solitons with almost equal velocities and widths. On the basis of this result we prove that the N MNSE-soliton train interaction (N > 2) can be modeled by the completely integrable complex Toda chain (CTC). This is an argument in favor of universality of the complex Toda chain which was previously shown to model the soliton train interaction for nonlinear Schrödinger solitons. The integrability of the CTC is used to describe all possible dynamical regimes of the N -soliton trains which include asymptotically free propagation of all N solitons, N -soliton bound states, various mixed regimes, etc. It allows also to describe analytically the manifolds in the 4N -dimensional space of initial soliton parameters which are responsible for each of the regimes mentioned above. We compare the results of the CTC model with the numerical solutions of the MNSE for 2 and 3-soliton interactions and find a very good agreement.

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On Modeling Adiabatic N-Soliton Interactions

Gerdjikov

Nonlinear Waves: Classical and Quantum Aspects

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Stability and quasi-equidistant propagation of NLS soliton trains

Cited by 37 publications

References 18 publications

Modeling Adiabatic N-Soliton Interactions and Perturbations

Modeling Adiabatic N-Soliton Interactions and Perturbations

Adiabatic interaction ofNultrashort solitons: Universality of the complex Toda chain model

On Modeling Adiabatic N-Soliton Interactions

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