On a direct bilinearization method: Kaup's higher-order water wave equation as a modified nonlocal Boussinesq equation

Lambert, Franklin; Loris, Ignace; Springael, Johan; Willer, R

doi:10.1088/0305-4470/27/15/028

Cited by 99 publications

(57 citation statements)

References 14 publications

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“…(11), and comparing the like powers of the coefficients of , we can derive multisoliton solutions of Eq. (4) as follows.…”

Section: Gauge Transformation and Multisoliton Solutionsmentioning

confidence: 99%

“…In the study of nonlinear partial differential equations of mathematical physics, the Bäcklund transformation (BT) provides a means to construct new solutions from known ones [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. BTs can connect solutions of different equations or the same equation [2,3].…”

Section: Introductionmentioning

confidence: 99%

“…Generally speaking, there exist some difficulties in applying the Hirota bilinear method to construct bilinear BTs for nonlinear equations. In some cases, employing the Bell-polynomial technique and introducing the auxiliary independent variable(s) are helpful [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model

Lü

Khalique

2015

Applied Mathematics Letters

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Please cite this article as: X. Lü, W.-X. Ma, C.M. Khalique, A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg-de Vries-like model, Appl. Math. Lett. (2015), http://dx. AbstractWe will directly construct a bilinear Bäcklund transformation (BT) of a (2+1)-dimensional Korteweg-de Vries-like model. The construction is based on a so-called quadrilinear representation. The resulting bilinear BT is in accordance with the auxiliary-independentvariable-involved one derived with the Bell-polynomial scheme. Moreover, by applying the gauge transformation and the Hirota perturbation technique, multisoliton solutions are iteratively computed.

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“…(11), and comparing the like powers of the coefficients of , we can derive multisoliton solutions of Eq. (4) as follows.…”

Section: Gauge Transformation and Multisoliton Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model

Lü

Khalique

2015

Applied Mathematics Letters

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“…This system has been known as the classical Boussinesq system [13] or the Kaup system [18], [12], [19], [16].…”

Section: Iv2 Non-local Boussinesq Equation Versus the Kaup Systemmentioning

confidence: 99%

Miura transformation between two non-linear equations in 2+1 dimensions

Cerveró

Estévez

1998

Journal of Mathematical Physics

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A Dispersive Wave Equation in 2 + 1 dimensions (2LDW) widely discussed by different authors is shown to be nothing but the modified version of the Generalized Dispersive Wave Equation (GLDW). Using Singularity Analysis and techniques based upon the Painlevé Property leading to the Double Singular Manifold Expansion we shall find the Miura Transformation which converts the 2LDW Equation into the GLDW Equation. Through this Miura Transformation we shall also present the Lax pair of the 2LDW Equation as well as some interesting reductions to several already known integrable sytems in 1+ 1 dimensions. As the 2LDW Equation arises from a Miura Transformation we propose that it should be treated conventionally as a Modified Equation. In this case, we propose its designation as The MGLDW Equation PACS Numbers 02.30 and 03.40K

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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 bilinearization method: Kaup's higher-order water wave equation as a modified nonlocal Boussinesq equation

Cited by 99 publications

References 14 publications

A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model

A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model

Miura transformation between two non-linear equations in 2+1 dimensions

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

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