Spherical nebulons and B�cklund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation

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This paper is to investigate the (2 + 1)-dimensional variable-coefficient KonopelchenkoDubrovsky equations, which can be applied to the phenomena in stratified shear flow, internal and shallow-water waves, plasmas, and other fields. The bilinear-form equations are transformed from the original equations, and soliton solutions are derived via symbolic computation. Soliton solutions and collisions are illustrated. The bilinear-form Bäcklund transformation and another soliton solution are obtained. Wronskian solutions are constructed via the Bäcklund transformation and solution.

Section: Wronskian Solutionmentioning

confidence: 77%

Section: Discussionmentioning

confidence: 99%

Soliton Solutions, Bäcklund Transformation and Wronskian Solutions for the (2+1)-Dimensional Variable-Coefficient Konopelchenko–Dubrovsky Equations in Fluid Mechanics

Gao

2012

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“…The CNLSEs [i.e., (1)] and CBEs [i.e., (7)] are, respectively, a model for certain birefringent optical fibers with Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. In this paper, with dependent-variable transformations (8) and (27), we have converted (1) to the Manakov system [i.e., (9)], and (7) to a linear equation [i.e., (30)] together with Expression (29).…”

Section: Discussionmentioning

confidence: 99%

“…In Section 2, based on the symbolic computation [6,7,9,10], we will demonstrate that (1) can be reduced into a Manakov system [22 -27] and obtain some new analytic solutions. Especially, the one-dark-soliton-like solution will be given, against [18]'s claim.…”

Section: Introductionmentioning

confidence: 99%

Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

Liu

Tian

Qin

et al. 2011

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Under investigation in this paper are the coupled nonlinear Schrödinger equations (CNLSEs) and coupled Burgers-type equations (CBEs), which are, respectively, a model for certain birefringent optical fibers Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. Special attention should be paid to the existing claim that the solitons for the CNLSEs do not exist. Through certain dependent-variable transformations, the CNLSEs are reduced to a Manakov system and the CBEs are linearized. In that way, some new solutions of the CNLSEs and CBEs are obtained via symbolic computation. Especially the one-dark-soliton-like solutions for the CNLSEs have been found, against the existing claim.

“…It was demonstrated that the only parameter A of the system plays two roles: A controls the width of the quasi-stationary solution as shown in Figure 1 and acts as the coefficient of the dispersion term in (1). Before closing, we mention that the stability of the numerical solution may shed light on finding some analytical solutions by some symbolic computations [15] or approximate analytical solutions by variational analysis approach [16].…”

mentioning

confidence: 96%

Dynamics of Paired-Solitons in Quantum Electron Plasmas

Hong

Jung

2011

We show the existence of new localized nonlinear structures for the electrostatic potential and the electron density in the form of bright and W-shaped solitons in quantum electron plasmas, respectively, which is modelled by the coupled nonlinear Schrödinger-Poisson equations. The robustness and the conservation of the energy of the solitons are demonstrated by numerical simulations. The sensitivity of the coupling constant on the stability of the paired solitons in the quantum electron plasmas are investigated. Quantum mechanical effects in dense plasmas become increasingly important when the de Broglie length of the charge carriers is comparable to the dimension of the plasma system. Quantum hydrodynamic (QHD) models for charged particle systems and collective interactions have been proposed for describing the behaviours of dense quantum plasmas [1 -5]. Ubiquitous presences of quantum plasmas in micromechanical systems and ultra-small electronic devices, in laser and microplasmas [6], in dense astrophysical environments [7], as well as in next generation intense laser-solid density plasma interaction experiments and in quantum X-ray free-electron lasers, have brought active research interests (see [8] for review).Shukla and Eliasson [9] have recently investigated the formation and dynamics of dark and gray envelope solitons and two-dimensional vortices in quantum electron plasmas with fixed ion background, based on a nonlinear Schrödinger-Poisson (SP) system of equations [1 -5, 8], in the form ofwhere the wave function Ψ is normalized by the equilibrium electron density √ n 0 , the electrostatic potential φ by T F /e, the time t byh/T F , and the space coordinate r by the Debye length λ D , where0 /m e is the Fermi temperature (neglecting relevant dimensionless constant), m e is the electron mass, e is the magnitude of the electron charge, ε 0 is the electric permittivity,h is the Plank constant divided by 2π [8 -10]. The quantum coupling constant A = Γ Q /2, where0 , is the most important parameter for the system, which can be both smaller and larger than unity depending on the physical situation under consideration [8]. More recently, we showed that the localized nonlinear structures in the form of domainwall (DW) solitons can exist in (1) -(2), admitting a set of conserved quantities [11]. Quasi-stationary solutions in the form of DW solitons, which show anticorrelation between the electron density and the electrostatic potential, were tested numerically to check their robustness [11].In this work, we show the existence of new localized nonlinear structures for the electrostatic potential and the electron density in the form of bright and Wshaped solitons in quantum electron plasmas, in the SP system, respectively. We demonstrate their robustness by numerical simulations, identify the boundary condition and the quantum coupling constant range for such stable paired solitons.