On Some Exact Solutions for a Generalized form of Yang’s Euclidean R-Gauge Equations and their Relation with Painleve’ Property

Abstract

It is difficult to get solutions for non-linear partial differential equations. A strategy for overcoming the difficulty in getting solutions that works well in some situation may not work equally well in some other (apparently) similar situation. Such an example for a generalized version of Yang’s Euclidean R-gauge equations is presented here. In one situation one gets solutions in explicit quadrature form and in another situation (different only for the presence of an innocent looking parameter) one cannot invert and therefore cannot represent the solution even in quadrature form. It is observed from a comparative study that the equations admit Painleve’ properties in the previous situation when we get explicit solutions and in the later case the equations are deprived of Painleve’ properties when we do not get solutions even in quadrature form