On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations

Lambert, Franklin; Springael, Johan

doi:10.1016/s0960-0779(01)00096-0

Cited by 82 publications

(41 citation statements)

References 10 publications

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“…Let g = g(x, y, z, t) be a C ∞ function, and the multi-dimensional Bell polynomials are written as [40][41][42][43][44] , , ,…”

Section: Bell Polynomials and Bilinear Equationmentioning

confidence: 99%

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On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

Zuo¹,

Gao

et al. 2015

Zeitschrift Für Naturforschung A

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The Boiti-Leon-Manna-Pempinelli (BLMP) equation is seen as a model for the incompressible fluid. In this article, a (3+1)-dimensional BLMP equation is investigated. With the aid of the Bell polynomials, bilinear form of such an equation is obtained. By virtue of the bilinear form, two kinds of soliton solutions with different nonlinear dispersion relations and another kind of analytic solutions are derived. Lax pairs and Bäcklund transformations are also constructed. Soliton propagation and interaction are analysed: (i) solitions with different nonlinear dispersion relations have different velocities and backgrounds; (ii) for another kind of analytic solutions with different nonlinear dispersion relations, the periodic property is displayed.

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“…Let g = g(x, y, z, t) be a C ∞ function, and the multi-dimensional Bell polynomials are written as [40][41][42][43][44] , , ,…”

Section: Bell Polynomials and Bilinear Equationmentioning

confidence: 99%

“…The P-polynomials can be characterized by an equallyrecognisable even-part-partitional structures [41][42][43][44] …”

Section: Bell Polynomials and Bilinear Equationmentioning

confidence: 99%

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

Zuo¹,

Gao

et al. 2015

Zeitschrift Für Naturforschung A

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“…the inverse scattering transformation, Hirota method, Bell-polynomial approach and Wronskian technology. [9][10][11][12][13][14] When people add some inhomogeneous terms reflecting the inhomogeneities of the media and non-uniformities of the boundaries to Eq. (1), [15][16][17][18][19][20] such as the damping or gaining terms with the constant or time/space-dependent coefficients, 16 or certain density profiles, 17-20 the inhomogeneous NLS-type equations are used to describe the wave propagation, claimed to be more effective in the realistic situations than Eq.…”

Section: Introductionmentioning

confidence: 99%

Integrability and soliton dynamics for an inhomogeneous nonlinear Schrödinger equation in an inhomogeneous plasma

Wang

Tian

2015

Mod. Phys. Lett. B

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Under investigation in this paper is an inhomogeneous nonlinear Schrödinger equation, which describes the propagation of a large-wavelength small-amplitude electron plasma wave in a parabolic-distributed and constant-interactional-damping inhomogeneous plasma. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear form, Bäcklund transformation and N -soliton solutions are obtained. Influence of the linear density coefficient α and damping coefficient β on the soliton envelopes is also discussed, i.e. α can affect the soliton position, while β is related to the soliton intensity, velocity and phase shift. Periodically attractive and repulsive interactions are shown. Asymptotic analysis shows that the interactions between/among the solitons are elastic.

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“…In the early 1930s, the classical Bell polynomials were introduced by Bell which are specified by a generating function and exhibiting some important properties [7]. Recently, Lambert and coworkers have proposed a relatively convenient procedure based on Bell polynomials which enables us to obtain bilinear forms, bilinear BTs, Lax pairs, and Darboux covariant Lax pairs for NEEs [8][9][10][11]. It is shown that Bell polynomials play an important role in the characterization of bilinearizable equations and a deep relation between the integrability of an NEE and the Bell polynomials.…”

Section: Introductionmentioning

confidence: 99%

Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

Cheng

Chen

2014

Abstract and Applied Analysis

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The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1)-dimensional variable-coefficient Caudrey-DoddGibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.

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On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations

Cited by 82 publications

References 10 publications

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

Integrability and soliton dynamics for an inhomogeneous nonlinear Schrödinger equation in an inhomogeneous plasma

Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

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