A wave equation in 2+1: Painleve analysis and solutions

In this paper the nonlinear equation mty=(myxx+mxmy)x is thoroughly analysed. The Painleve test is performed yielding a positive result. The Backlund transformations are found and the Darboux-Moutard-Matveev formalism arises in the context of this analysis. The singular manifold method, based upon the Painleve analysis, is proved to be a useful tool for generating solutions. Some interesting explicit expressions for one and two solitons are obtained and analysed in such a way.

[1]  P. R. Gordoa,et al.  Modified Singular Manifold Expansion: Application to the Boussinesq and Mikhailov-Shabat Systems , 1993 .

[2]  Vladimir E. Zakharov,et al.  What Is Integrability , 1991 .

[3]  On the characterization of a new soliton sector in the classical Boussinesq system , 1994 .

[4]  V. Matveev,et al.  Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters , 1979 .

[5]  P. G. Estévez.  Non-classical symmetries and the singular manifold method: the Burgers and the Burgers-Huxley equations , 1994 .

[6]  P. Clarkson.  The Painleve´ property, a modified Boussinesq equation and a modified Kadomtsev-Petviashvili equation , 1986 .

[7]  N. V. Ustinov,et al.  Third order spectral problems: reductions and Darboux transformations , 1994 .

[8]  H. Schaefer,et al.  Cyclopentadienylidene in interstellar space , 1990 .

[9]  L. Solombrino,et al.  On the Painlevé property of nonlinear field equations in 2+1 dimensions: The Davey–Stewartson system , 1988 .

[10]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[11]  N. A. Kudryashov.  Singular manifold equations and exact solutions for some nonlinear partial differential equations , 1993 .

[12]  B. Konopelchenko,et al.  On the structure and properties of the singularity manifold equations of the KP hierarchy , 1991 .

[13]  M. Tabor,et al.  Painlevé expansions for nonintegrable evolution equations , 1989 .

[14]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[15]  Elizabeth L. Mansfield,et al.  On a Shallow Water Wave Equation , 1994, solv-int/9401003.

[16]  M. Tabor,et al.  A unified approach to Painleve´ expansions , 1987 .

[17]  R. Hirota,et al.  N-Soliton Solutions of Model Equations for Shallow Water Waves , 1976 .

[18]  C. Athorne.  On the characterization of Moutard transformations , 1993 .

[19]  P. R. Gordoa,et al.  Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples , 1995 .

[20]  M. Lakshmanan,et al.  Singularity-structure analysis and Hirota's bilinearisation of the Davey-Stewartson equation , 1987 .

[21]  M. Boiti,et al.  On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions , 1986 .