Effect of discreteness on the continuum limit of the Heisenberg spin chain

Lakshmanan, M.; Porsezian, K.; Daniel, M.

doi:10.1016/0375-9601(88)90520-8

Cited by 254 publications

(119 citation statements)

References 11 publications

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“…The singular manifold method allows us to derive the Lax pair for this system. We have also presented an iterative procedure to build up lump solutions to the system of nonlinear equations (4). Most importantly, the proposed generalization involves third-and fourth-order dispersion terms that could pave the way to more elaborate models in the context of molecular energy transfer and nonlinear optics.…”

Section: Discussionmentioning

confidence: 99%

“…As is well known [19], the Painlevé property for (4) means that the fields u, w, and m can be expanded through a generalized Laurent expansion of the form…”

Section: Singular Manifold Methodsmentioning

confidence: 99%

“…This equation contains many integrable particular cases such as the standard NLSE (α = γ = 0) [7], the Hirota equation (γ = 0) [8], and the Lakshmanan-Porsezian-Daniel equation (α = 0) [4]. Soliton solutions and rogue wave for this equation can be found in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…[3] and [4]. More recently, Ankiewicz et al proposed a further generalization of the NLSE, adding a third-order dispersion term [5].…”

Section: Introductionmentioning

confidence: 99%

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Lump solitons in a higher-order nonlinear equation in2+1dimensions

et al. 2016

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We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrödinger equation to 2 + 1 dimensions. This integrable system of equations is a promising starting point to elaborate more accurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the system is derived after applying the singular manifold method. We also present an iterative procedure to construct the solutions from a seed solution. Solutions with one-, two-, and three-lump solitons are thoroughly discussed.

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Section: Discussionmentioning

confidence: 99%

“…As is well known [19], the Painlevé property for (4) means that the fields u, w, and m can be expanded through a generalized Laurent expansion of the form…”

Section: Singular Manifold Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[3] and [4]. More recently, Ankiewicz et al proposed a further generalization of the NLSE, adding a third-order dispersion term [5].…”

Section: Introductionmentioning

confidence: 99%

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Lump solitons in a higher-order nonlinear equation in2+1dimensions

et al. 2016

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“…(1) can be reduced to the Hirota equation for the propagation of a subpicosecond or femtosecond pulse [10]; (3) for α = δ = 0 and γ = 0, Eq. (1) can be reduced to a fourthorder dispersive NLS equation for the one-dimensional anisotropic Heisenberg ferromagnetic spin chain with the octuple-dipole interaction [27,28]; and (4) with α = γ = 0 and δ = 0, Eq. (1) can be reduced to a fifth-order NLS equation for the Heisenberg ferromagnetic spin system [22,28].…”

Section: Introductionmentioning

confidence: 99%

Breathers and rogue waves of the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

et al. 2015

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One-dimensional anisotropic Heisenberg ferromagnetic spin chain can be described by the fifthorder nonlinear Schrödinger equation, which is investigated in this paper. Through the Darboux transformation, we obtain the Akhmediev breathers (ABs), Kuznetsov-Ma (KM) solitons and rogue-wave solutions. Effects of the coefficients of the fourth-order dispersion, γ , and of the fifth-order dispersion, δ, on the properties of ABs, KM solitons and rogue waves are discussed: (1) With γ increasing, the AB exhibits stronger localization in time; (2) The propagation directions of an AB and a KM soliton change with the presence of δ; and (3) Enhancement of γ makes the existence time of the rogue waves shorter, while enhancement of δ increases the existence time of the rogue waves.

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Nonlinear Wave Molecules for the Lakshmanan–Porsezian–Daniel Equation in Nonlinear Optics and Biology

et al. 2022

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The nonlinear wave molecules of the Lakshmanan–Porsezian–Daniel (LPD) equation describing the propagation of ultrashort optical pulses through optical fibers and Davydov soliton in α‐helical proteins are investigated. Based on the analysis of characteristic lines, the breather molecules consisting of two, three, or even four different breather atoms are derived, and the synthesis modes of molecules by adjusting the values of the phase parameters are generated. The state conversion of the breather molecules is studied and a variety of converted wave molecules are produced. In particular, it is reported that the full conversion of the breather molecules does not exist in the LPD equation. The formation mechanisms of different types of nonlinear wave molecules through the nonlinear superposition are further explored and the influence of higher‐order effects on the breather molecules is discussed. Finally, the intricate interactions between the molecules and nonlinear waves are considered. It is found that the distances between atoms in the molecules as well as the shapes of the converted waves change after the interactions, and the essence of such shape‐changed interactions is revealed.

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Effect of discreteness on the continuum limit of the Heisenberg spin chain

Cited by 254 publications

References 11 publications

Lump solitons in a higher-order nonlinear equation in2+1dimensions

Lump solitons in a higher-order nonlinear equation in2+1dimensions

Breathers and rogue waves of the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

Nonlinear Wave Molecules for the Lakshmanan–Porsezian–Daniel Equation in Nonlinear Optics and Biology

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