Behavior of a vacuum and naked singularity under a smooth gauge function in Lyra geometry

Lyra geometry is a conformal geometry that originated from Weyl geometry. In this article, we derive the exterior field equation under a spherically symmetric gauge function x0(r) and metric in Lyra geometry. When we impose a specific form of the gauge function x0(r), the radial differential equation of the metric component g00 will possess an irregular singular point (ISP) at r = 0. Moreover, we can apply the method of dominant balance to get the asymptotic behavior of the new space–time solution. The significance of this work is that we can use a series of smooth gauge functions x0(r) to modulate the degree of divergence of the singularity at r = 0, which will become a naked singularity under certain conditions. Furthermore, we investigate the physical meaning of this novel behavior of space–time in Lyra geometry and find out that no spaceship with finite integrated acceleration can arrive at this singularity at r = 0. The physical meaning of the gauge function and integrability is also discussed.

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in ϵ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

Bessel Equation in the Semiunbounded Intervalx∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point

This study expresses the solution of the Bessel equation in the neighbourhood ofx=∞as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval[x0,∞]. In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution atx→∞(not to mention a so-called formal solution), but a solution in the interval[x0,∞]with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.