Generalized Burgers equations and Euler–Painlevé transcendents. II

Abstract

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler–Painlevé equation yy"+ay'²+f(x)yy'+g(x) y²+by'+c=0 represents the generalized Burgers equations (GBE's) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE u_t+u^au_x+Ju/2t =(gd/2)u_xx (the nonplanar Burgers equation) is considered. It is found that its self-similar form is again governed by the Euler–Painlevé equation. The ranges of the parameter a 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–Painlevé equation with respect to GBE's.