Sachdev, PL and Nair, KRC (1987) Generalized Burgers equations and Euler-Painleve transcendents. II. In: Journal of Mathematical Physics, 28 (5). pp. 997-1004.
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It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler PainlevÃ© equation yy[script `]+ay[script ']2+ f(x)yy[script ']+g(x) y2+by[script ']+c=0 represents the generalized Burgers equations (GBE's) in the same manner as PainlevÃ© equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE ut+uaux+Ju/2t =(gd/2)uxx (the nonplanar Burgers equation) is considered. It is found that its self-similar form is again governed by the Euler PainlevÃ© equation. The ranges of the parameter alpha for which solutions of the connection problem to the self-similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE's. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well-known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler PainlevÃ© equation with respect to GBE's. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
|Item Type:||Journal Article|
|Additional Information:||Copyright of this article belongs to American Institute of Physics.|
|Department/Centre:||Division of Physical & Mathematical Sciences > Mathematics|
|Date Deposited:||22 Jul 2009 12:21|
|Last Modified:||19 Sep 2010 05:37|
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