Generalized Burgers equations and Euler–Painlevé transcendents. III

Sachdev, PL and Nair, KRC and Tikekar, VG (1988) Generalized Burgers equations and Euler–Painlevé transcendents. III. In: Journal of Mathematical Physics, 29 (11). pp. 2397-2404.

PDF Generalized_Burgers.pdf - Published Version Restricted to Registered users only Download (506kB) | Request a copy

Abstract

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986); P. L. Sachdev and K. R. C. Nair, ibid. 28, 977 (1987)] that the Euler–Painlevé equations  y(d2y/dη2)+a(dy/dη)2 +f(η)y(dy/dη)+g(η)y2+b(dy/dη) +c=0 represent generalized Burgers equations (GBE’s) in the same way as Painlevé equations represent the Korteweg–de Vries type of equations. The earlier studies were carried out in the context of GBE’s with damping and those with spherical and cylindrical symmetry. In the present paper, GBE’s with variable coefficients of viscosity and those with inhomogeneous terms are considered for their possible connection to Euler–Painlevé equations. It is found that the Euler–Painlevé equation, which represents the GBE ut+uβux=(δ/2)g(t)uxx, g(t)=(1+t)n, β>0, has solutions, which either decay or oscillate at η=±∞, only when −1<n<1. The solutions are shocklike when n=1. On the other hand, they oscillate over the whole real line when n=−1. Furthermore, the solutions monotonically decay both at η=+∞ and η=−∞, that is, they have a single hump form if β≥βn=(1−n)/(1+n). For β<βn, the solutions have an oscillatory behavior either at η=+∞ or at η=−∞, or at η=+∞ and η=−∞. For β=βn, there exists a single parameter family of exact single hump solutions, similar to those found for the nonplanar Burgers equations in Paper II. Thus the parametric value β=βn seems to bifurcate the families of solutions, which remain bounded at η=±∞. Other GBE’s considered here are also found to be reducible to Euler–Painlevé equations.

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:	21 Sep 2010 09:53
Last Modified:	21 Sep 2010 09:53
URI:	http://eprints.iisc.ac.in/id/eprint/32202

Actions (login required)

View Item