Bell-polynomial construction of Bäcklund transformations with auxiliary independent variable for some soliton equations with one Tau-function

Lü, Xing; Tian, Bo; Feng, Qi

doi:10.1016/j.nonrwa.2011.09.006

Cited by 40 publications

(20 citation statements)

References 41 publications

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“…We have found the integrable condition for (1), i.e., constraint (10). Based on bilinear form (13), soliton solutions (17), (19), and (20) have been derived through the Hirota bilinear method.…”

Section: Discussionmentioning

confidence: 99%

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

Lin

Tian

Wang

et al. 2012

Zeitschrift Für Naturforschung A

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Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlevé analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes. We also present the collisions among three solitons of atomic matter waves.

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Section: Discussionmentioning

confidence: 99%

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

Lin

Tian

Wang

et al. 2012

Zeitschrift Für Naturforschung A

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“…obeying a different partition rule obtained by restricting the Bell recipe to the even part partitions [32]:…”

Section: Bell Polynomialsmentioning

confidence: 99%

“…With c = 6, α = β, by virtue of Expressions (15), we can get the following form: With the introduction of an auxiliary independent variable r satisfying P 4x (Q) = μP xr (Q) [32], Equation (16) can be reduced to…”

Section: Binary Bell Polynomial Formsmentioning

confidence: 99%

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

Xiao

Tian

Zhen

et al. 2016

Waves in Random and Complex Media

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2016): Multisoliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid, Waves in Random and Complex Media, ABSTRACTIn this paper, we investigate a two-mode Korteweg-de Vries equation, which describes the one-dimensional propagation of shallow water waves with two modes in a weakly nonlinear and dispersive fluid system. With the binary Bell polynomial and an auxiliary variable, bilinear forms, multi-soliton solutions in the two-wave modes and Bell polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton propagation and collisions between the two solitons are presented. Based on the graphic analysis, it is shown that the increase in s can lead to the increase in the soliton velocities under the condition of α = β = 1, but the soliton amplitudes remain unchanged when s changes, where s means the difference between the phase velocities of twomode waves, α and β are the nonlinearity parameter and dispersion parameter respectively. Elastic collisions between the two solitons in both two modes are analyzed with the help of graphic analysis. ARTICLE HISTORY

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“…However, there is no universal method to get the bilinear forms and bilinear BT. In recent years, Bell polynomials approach is widely used to derive integrable properties for a large number of nonlinear equations [11][12][13][14][15][16]. Fan extended this method to the variable-coefficient and supersymmetric soliton equations [7,8].…”

Section: Introductionmentioning

confidence: 99%

Integrability of a (2+1)-dimensional generalized breaking soliton equation

2015

Applied Mathematics Letters

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Bell-polynomial construction of Bäcklund transformations with auxiliary independent variable for some soliton equations with one Tau-function

Cited by 40 publications

References 41 publications

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

Integrability of a (2+1)-dimensional generalized breaking soliton equation

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