dorsal/arxiv
View SchemaPainlev\'e analysis for nonlinear partial differential equations
| Authors | M. Musette |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9804003 |
| URL | https://arxiv.org/abs/solv-int/9804003 |
Abstract
The Painlev\'e 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 Painlev\'e 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 B\"acklund 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 % Conte 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.
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"abstract": "The Painlev\\\u0027e analysis introduced by Weiss, Tabor and Carnevale (WTC) in\n1983 for nonlinear partial differential equations (PDE\u0027s) is an extension of\nthe method initiated by Painlev\\\u0027e and Gambier at the beginning of this century\nfor the classification of algebraic nonlinear differential equations (ODE\u0027s)\nwithout movable critical points. In these lectures we explain the WTC method in\nits invariant version introduced by Conte in 1989 and its application to\nsolitonic equations in order to find algorithmically their associated\nB\\\"acklund transformation. A lot of remarkable properties are shared by these\nso-called ``integrable\u0027\u0027 equations but they are generically no more valid for\nequations modelising physical phenomema. Belonging to this second class, some\nequations called ``partially integrable\u0027\u0027 sometimes keep remnants of\nintegrability. In that case, the singularity analysis may also be useful for\nbuilding closed form analytic solutions, which necessarily % Conte agree with\nthe singularity structure of the equations. We display the privileged role\nplayed by the Riccati equation and systems of Riccati equations which are\nlinearisable, as well as the importance of the Weierstrass elliptic function,\nfor building solitary waves or more elaborate solutions.",
"arxiv_id": "solv-int/9804003",
"authors": [
"M. Musette"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Painlev\\\u0027e analysis for nonlinear partial differential equations",
"url": "https://arxiv.org/abs/solv-int/9804003"
},
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