The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups

Anahita Eslami Rad; Enrique G. Reyes

doi:10.3934/jgm.2013.5.345

Article Contents

2013, Volume 5, Issue 3: 345-364. Doi: 10.3934/jgm.2013.5.345

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The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups

Anahita Eslami Rad^1, and
Enrique G. Reyes^2,

1.
Département de Mathématique, CP 218, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles,
2.
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago

Received: February 28, 2013

Revised: July 31, 2013

Published: August 2013

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Abstract

We present explicit formal solutions to the systems of equations in two independent variables $t_m$, $x$, $m =1,2,\dots$, of the Kadomtsev-Petviashvili hierarchy. The main tools used are a Birkhoff-like factorization of formal Lie groups due to M. Mulase, and the classical theory of A.G. Reyman and M.A. Semenov-Tian-Shansky on the integration of Hamiltonian systems on coadjoint orbits using $r$-matrices. Our paper also contains full proofs of Mulase's results.

Keywords:

Kadomtsev-Petviashvili hierarchy,
formal Lie groups,
Mulase factorization theorem,
$r$-matrices,
Reyman-Semenov Tian Shansky integration theorem.

Mathematics Subject Classification: Primary: 37K10, 37K30; Secondary: 17B80, 22E67.

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