Symbolic-computation study of the perturbed nonlinear Schrödinger model in inhomogeneous optical fibers

Tian, Bo; Gao, Yi–Tian

doi:10.1016/j.physleta.2005.05.041

Cited by 180 publications

(73 citation statements)

References 45 publications

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“…Their methods are consistent with the zero curvature condition in modern soliton theory. Using the hyperelliptic sigma function and defining natural sigma functions of more general algebraic curves, the authors in [21][22][23][24][25][26] have been constructing deeper theories of Abelian functions and soliton equations [35][36][37][38][39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Hyperelliptic function solutions with finite genus �� of coupled nonlinear differential equations*

Tian

Lü

Yang

et al. 2021

JNMP

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In this paper, using the properties of hyperelliptic σ-and ℘-functions, ℘ µν := ∂ µ ∂ ν log σ, we propose an algorithm to obtain particular solutions of the coupled nonlinear differential equations, such as a general (2+1)-dimensional breaking soliton equation and static Veselov-Novikov(SVN) equation, the solutions of which can be expressed in terms of the hyperelliptic Kleinian functions for a given curve y 2 = f (x) of (2g+1)-and (2g+2)-degree with genus G . In particular, owing to the idea of CK direct method, the algorithm can generate a series of new forms of hyperelliptic function solutions with the same genus G .

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Section: Introductionmentioning

confidence: 99%

Hyperelliptic function solutions with finite genus �� of coupled nonlinear differential equations*

Tian

Lü

Yang

et al. 2021

JNMP

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“…Seeking for the soliton solutions of the nonlinear evolution equations (NLEEs) is of importance since such equations can describe the diverse physical aspects [1][2][3][4][5][6][7][8][9][10][11][12]. Darboux transformations (DTs) based on the Lax pair are a method to get the soliton solutions of some NLEEs from the seeds .…”

Section: Introductionmentioning

confidence: 99%

“…Especially, the N -fold DT, which can be interpreted as the superposition of a single DT, has been applied to certain NLEEs for deriving the multi-soliton solutions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Advantage of the N -fold DT is that the problem solving of a NLEE is finally reduced to solve a linear system, which enables us to generate the multi-soliton solutions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] with symbolic computation [1][2][3][4][5][6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Vadermonde-Type Odd-Soliton Solutions for the Whitham–Broer–Kaup Model in the Shallow Water Small-Amplitude Regime

Wang¹,

Gao²,

Gai³

et al. 2021

JNMP

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Under investigation in this paper, with symbolic computation, is the Whitham-Broer-Kaup (WBK) system for the dispersive long waves in the shallow water small-amplitude regime. N -fold Darboux transformation (DT) for a spectral problem associated with the WBK system is constructed. Oddsoliton solutions in terms of the Vandermonde-like determinant for the WBK system are presented via the N -fold DT and evolution of the three-soliton solutions is graphically studied. Our results could be used to illustrate the bidirectional propagation of the waves in the shallow water smallamplitude regime.

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“…As an important model in optical communication systems, the nonlinear Schrödinger model with variable coefficients has directed the attention of many researchers. Tian and Gao [11] applied a direct method to get exact, analytic bright-solitonic solutions for the perturbed nonlinear Schrödinger model. In [12 -13] the variable coefficient higher order nonlinear Schrödinger equation has been studied from the viewpoint of bilinear form, Backlund transformation, brightons and symbolic computation.…”

Section: Introductionmentioning

confidence: 99%

A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type

Zhang

Liu

2009

Zeitschrift Für Naturforschung A

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In this paper, the generalized variable coefficient nonlinear Schrödinger (NLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients are directly reduced to simple and solvable ordinary differential equations by means of a direct transformation method. Taking advantage of the known solutions of the obtained ordinary differential equations, families of exact nontravelling wave solutions for the two equations have been constructed. The characteristic feature of the direct transformation method is, that without much extra effort, we circumvent the integration by directly reducing the variable coefficient nonlinear evolution equations to the known ordinary differential equations. Another advantage of the method is that it is independent of the integrability of the given nonlinear equation. The method used here can be applied to reduce other variable coefficient nonlinear evolution equations to ordinary differential equations.

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Symbolic-computation study of the perturbed nonlinear Schrödinger model in inhomogeneous optical fibers

Cited by 180 publications

References 45 publications

Hyperelliptic function solutions with finite genus �� of coupled nonlinear differential equations*

Hyperelliptic function solutions with finite genus �� of coupled nonlinear differential equations*

Vadermonde-Type Odd-Soliton Solutions for the Whitham–Broer–Kaup Model in the Shallow Water Small-Amplitude Regime

A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type

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Symbolic-computation study of the perturbed nonlinear Schrödinger model in inhomogeneous optical fibers

Cited by 180 publications

References 45 publications

Hyperelliptic function solutions with finite genus ������ of coupled nonlinear differential equations*

Hyperelliptic function solutions with finite genus ������ of coupled nonlinear differential equations*

Vadermonde-Type Odd-Soliton Solutions for the Whitham–Broer–Kaup Model in the Shallow Water Small-Amplitude Regime

A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type

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Resources

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Hyperelliptic function solutions with finite genus �� of coupled nonlinear differential equations*

Hyperelliptic function solutions with finite genus �� of coupled nonlinear differential equations*