Lax pairs for higher-dimensional evolution PDEs and a 3+1 dimensional integrable generalization of the Burgers equation
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- by M. Rudnev, A. V. Yurov and V. A. Yurov PDF
- Proc. Amer. Math. Soc. 135 (2007), 731-741 Request permission
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.References
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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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 731-741
- MSC (2000): Primary 35Q53, 35Q58
- DOI: https://doi.org/10.1090/S0002-9939-06-08560-1
- MathSciNet review: 2262869