A Linearization Approach for Rational Nonlinear Models in Mathematical Physics

Abstract

In this paper, a novel method for linearization of rational second order nonlinear models is discussed. In particular, we discuss an application of the δ expansion method (created to deal with problems in Quantum Field Theory) which will enable both the linearization and perturbation expansion of such equations. Such a method allows for one to quickly obtain the order zero perturbation theory in terms of certain special functions which are governed by linear equations. Higher order perturbation theories can then be obtained in terms of such special functions. One benefit to such a method is that it may be applied even to models without small physical parameters, as the perturbation is given in terms of the degree of nonlinearity, rather than any physical parameter. As an application, we discuss a method of linearizing the six Painlevé equations by an application of the method. In addition to highlighting the benefits of the method, we discuss certain shortcomings of the method.