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 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–Painlevé equation. The ranges of the parameter α 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.

On the generalized Burgers equation

“…Here ε is a positive constant, f (t, z), g(t, z) and φ(x) are given functions. Equation (1.1) appears in different applications, see [2,[6][7][8][9][10][11][12][13]16,17]. The goal of the present paper is to obtain an a priori estimate of |u x | independent of ε and T and on the basis of this estimate to prove the existence and the uniqueness of the classical solution of problem (1.1)-(1.3) and to investigate the behavior of this solution when ε → 0 and when T → +∞.…”

“…Example 1. The simplest case of (1.1) is the following equation (see [6][7][8][9][10][11][12][13]16])…”

Painlev� analysis and similarity solutions of Burgers' equation with variable coefficients

“…It has to be remarked that all the reduced similarity equations (namely equations (4.7), (4.12), (4.18), (4.21) and (4.30)) to be solved in order to get the requested invariant solutions are easily transformed, by using as new dependent variable the old one raised to the power -1, to particular forms of the Euler-Painlev6 equations identified in [10,11,12] as characterizing a wide class of generalized Burgers' equations.…”

“…Various generalized Burgers' equations have been analyzed in [10,11,12] where, through the use of a self-similar approach, the link has been shown with a class of second order ordinary differential equations defining the so-called Euler-Painlev6 transcendents that play the same role as the Painlev6 equations for the Korteweg-de Vries type of equations.…”

Large-time behaviour of solutions of the inviscid non-planar Burgers equation

“…A reference may be made to the work of Srinivasa Rao et al [20] for a detailed discussion on the existence and non-existence of self-similar solutions of (1.4). They proved that positive self-similar solutions of (1.4) vanishing at x = ±∞ exist only for the parametric regime 1/( j + 1) ≤ α < 1/j (see also [21]). It is also interesting to refer to the work of Sachdev et al [22], wherein the constructed N -wave solution of (1.4) embeds both inviscid solution and the old age solution of (1.4).…”