Abstract
The first part of the paper proves that a subset of the general set of Ermakov-Pinney equationscan be obtained by differentiation of a first-order non-linear differential equation. The second partof the paper proves that, similarly, the equation for the amplitude function for the parametrix ofthe scalar wave equation can be obtained by covariant differentiation of a first-order non-linearequation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi) = 0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2 = 0. The auxiliary 1-forms (psi,rho) are evaluated explicitlyin Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained.Thus, the novel method developed in this paper can be used with profit in physical applications.
Highlights
The modern theoretical description of gravitational interactions [1] has completely superseded Newtonian gravity, the investigation of ordinary differential equations provides an invaluable tool in the analysis of chaotic dynamical systems [2] and in studying the interplay between linear and non-linear differential equations [3]
This is of special interest because, as was proved by Cohen and Kegeles [4], the evaluation of electromagnetic fields in curved space-times can be reduced to solving a complex linear scalar wave equation
Since α should be the counterpart of u (x), and the divergenceless condition, Eq 24, the counterpart of π = 0 in Section 3, we are led to consider the first-order non-linear equation
Summary
The modern theoretical description of gravitational interactions [1] has completely superseded Newtonian gravity, the investigation of ordinary differential equations provides an invaluable tool in the analysis of chaotic dynamical systems [2] and in studying the interplay between linear and non-linear differential equations [3]. Ordinary differential equations may prove useful in developing methods which are part of the framework necessary to solve more difficult cases of partial differential equations. We are going to provide a concrete example of application: the scalar wave equation in curved space-time. This is of special interest because, as was proved by Cohen and Kegeles [4], the evaluation of electromagnetic fields in curved space-times can be reduced to solving a complex linear scalar wave equation.
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