The integration of Burgers and Korteweg-de Vries equations with nonuniformities

Brugarino, T.; Pantano, Pietro

doi:10.1016/0375-9601(80)90005-5

Cited by 31 publications

(9 citation statements)

References 13 publications

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“…which means that the new expansion variable Z is related to χ by an homographic transformation (as suggested in [99], formula (17)) such that in the neighbourhood of χ = 0, one has Z ∼ χ. such that u T ∼ 2i Log χ as χ → 0, Pickering obtains :…”

Section: Weiss Methods Plus Homographymentioning

confidence: 99%

Painlevé Analysis for Nonlinear Partial Differential Equations

Musette¹

1999

The Painlevé Property

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La propriété de Painlevé, un siècle après The Painlevé property, one century laterAbstract. The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated Bäcklund transformation. A lot of remarkable properties are shared by these so-called "integrable" equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called "partially integrable" sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and systems of Riccati equations which are linearisable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.The Painlevé property, one century later, ed. R. Conte, CRM series in mathematical physics (Springer-Verlag, Berlin, 1998) 31 March 1998 solv-int/9804003

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Section: Weiss Methods Plus Homographymentioning

confidence: 99%

Painlevé Analysis for Nonlinear Partial Differential Equations

Musette¹

1999

The Painlevé Property

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“…These exact analytic solutions may help physicists and engineers to examine the sensitivity of the model by adjusting some physical parameters, and give good enough support to numerical simulation. In the past few years, significant progression emerged in the development of powerful methods, such as inverse scattering transformation, truncated expansion, Jacobi elliptic function expansion [4][5][6][7], etc., which can be employed to obtain the solution of completely integrable evolution equation. Among these, truncated expansion and Jacobi elliptic function expansion are the most concise, effective and successful techniques for the solution of these equations.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions of KdV equation with relaxation effect of inhomogeneous medium

Xiang

2010

Applied Mathematics and Computation

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“…Their solution spaces are infnite-dimensional and contain diverse solution structures. In the past few years, wide variety of the powerful and direct methods to find all kinds of analysis solutions of nonlinear evolution equations had been developed [1][2][3][4][5][6][7][8][9][10][11][12][13]. The basic purpose of them is to construct new solitary wave solutions and periodic solutions.…”

Section: Introductionmentioning

confidence: 99%

Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

Xiang¹

2011

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Abstract(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.

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The integration of Burgers and Korteweg-de Vries equations with nonuniformities

Cited by 31 publications

References 13 publications

Painlevé Analysis for Nonlinear Partial Differential Equations

Painlevé Analysis for Nonlinear Partial Differential Equations

Analytical solutions of KdV equation with relaxation effect of inhomogeneous medium

Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

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