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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lax pairs for higher-dimensional evolution PDEs and a 3+1 dimensional integrable generalization of the Burgers equation


Authors: M. Rudnev, A. V. Yurov and V. A. Yurov
Journal: Proc. Amer. Math. Soc. 135 (2007), 731-741
MSC (2000): Primary 35Q53, 35Q58
Published electronically: August 31, 2006
MathSciNet review: 2262869
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Abstract: We construct Lax pairs for general $ d+1$ dimensional evolution equations in the form $ u_t=F[u]$, where $ F[u]$ depends on the field $ u$ and its space derivatives. As an example we study a $ 3+1$ dimensional integrable generalization of the Burgers equation. We develop a procedure to generate some exact solutions of this equation, based on a class of discrete symmetries of the Darboux transformation type. In the one-dimensional limit, these symmetries reduce to the Cole-Hopf substitution for the Burgers equation. It is discussed how the technique can be used to construct exact solutions for higher-dimensional evolution PDEs in a broader context.


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Additional Information

M. Rudnev
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: m.rudnev@bris.ac.uk

A. V. Yurov
Affiliation: Department of Theoretical Physics, Kaliningrad State University, Aleksandra Nevskogo 14, Kaliningrad 236041, Russia
Email: artyom_yurov@mail.ru

V. A. Yurov
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: valerian@math.missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08560-1
PII: S 0002-9939(06)08560-1
Keywords: Integrable evolution equations, Lax pairs, Darboux transformation, Burgers equation
Received by editor(s): November 29, 2004
Received by editor(s) in revised form: May 16, 2005, and September 23, 2005
Published electronically: August 31, 2006
Communicated by: Mark J. Ablowitz
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.