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Generalized Burgers equations and Euler-Painlev transcendents. I

Sachdev, PL and Nair, KRC and Tikekar, VG (1986) Generalized Burgers equations and Euler-Painlev transcendents. I. In: Journal of Mathematical Physics, 27 (6). pp. 1506-1522.

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Initial-value problems for the generalized Burgers equation (GBE) ut+u betaux+lambdaualpha =(delta/2)uxx are discussed for the single hump type of initial data both continuous and discontinuous. The numerical solution is carried to the self-similar ``intermediate asymptotic'' regime when the solution is given analytically by the self-similar form. The nonlinear (transformed) ordinary differential equations (ODE's) describing the self-similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE's are new, and it is postulated that they characterize GBE's in the same manner as the Painlev equations categorize the Kortweg-de Vries (KdV) type. A connection problem for some related ODE's satisfying proper asymptotic conditions at x=±[infinity], is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient lambda, are also discussed. The results are compared with those holding for the modified KdV equation with damping. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Item Type: Journal Article
Publication: Journal of Mathematical Physics
Publisher: American Institute of Physics
Additional Information: Copyright of this article belongs to American Institute of Physics.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 11 Aug 2009 10:23
Last Modified: 19 Sep 2010 05:39
URI: http://eprints.iisc.ac.in/id/eprint/21989

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