Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. M. J. Ablowitz, H. Segur. Solitons and the inverse scattering transform. SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981. x+425 pp. MR 0642018 (84a:35251)
  • 2. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari. Extended resolvent and inverse scattering with an application to KPI. Integrability, topological solitons and beyond. J. Math. Phys. 44 (2003), no. 8, 3309-3340. MR 2006753 (2004j:37137)
  • 3. J. M. Burgers. The nonlinear diffusion equation. D. Reidel, Massachusetts, 1974.
  • 4. C. Haselwandter, D. D. Vvedensky. Fluctuations in the lattice gas for Burgers' equation. J. Phys. A 35 (2002), no. 41, L579-L584. MR 1946960 (2003k:82078)
  • 5. U. Frisch, J. Bec. ``Burgulence". Turbulence: nouveaux aspects/New trends in turbulence (Les Houches, 2000), 341-383, EDP Sci., Les Ulis, 2001. MR 1999766 (2004g:76075)
  • 6. J. D. Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9 (1951) 225-236. MR 0042889 (13:178c)
  • 7. M. M. Crum. Associated Sturm-Liouville systems. Quart. J. Math. Oxford Ser. (2) 6 (1955) 121-127. MR 0072332 (17:266g)
  • 8. F. Gesztesy, H. Holden. The Cole-Hopf and Miura transformations revisited. Mathematical physics and stochastic analysis (Lisbon, 1998), 198-214, World Sci. Publishing, River Edge, NJ, 2000. MR 1893107 (2003a:37107)
  • 9. E. Hopf. The partial differential equation $ u\sb t+uu\sb x=u\sb {xx}$. Comm. Pure Appl. Math. 3 (1950) 201-230. MR 0047234 (13:846c)
  • 10. S. B. Leble, M. A. Salle, A. V. Yurov. Darboux transforms for Davey-Stewartson equations and solitons in multidimensions. Inverse problems 4 (1992) 207-218. MR 1158176 (93h:35191)
  • 11. V. B. Matveev, M. A. Salle. Darboux Transformation and Solitons. Springer Verlag, Berlin-Heidelberg 1991. MR 1146435 (93d:35136)
  • 12. A. C. Newell. Solitons in mathematics and physics. CBMS-NSF Regional Conference Series in Applied Mathematics, 48. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. xvi+244 pp. MR 0847245 (87h:35314)
  • 13. P. J. E. Peebles. Principles of Physical Cosmology. Princeton University Press, 1993. MR 1216520 (94d:83076)
  • 14. S. Yu. Sakovich. On the Thomas equation. J. Phys. A 21 (1988) L1123-L1126. MR 0980551 (90a:35209)
  • 15. A. V. Yurov. BLP dissipative structures in the plane. Phys. Letters A 262 (1999) no. 6, 445-452. MR 1729501 (2000i:37131)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35Q53, 35Q58

Retrieve articles in all journals with MSC (2000): 35Q53, 35Q58

Additional Information

M. Rudnev
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

A. V. Yurov
Affiliation: Department of Theoretical Physics, Kaliningrad State University, Aleksandra Nevskogo 14, Kaliningrad 236041, Russia

V. A. Yurov
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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.

American Mathematical Society