Formation of Higher-dimensional Topological Black Holes

We study higher-dimensional gravitational collapse to topological black holes in two steps. First, we construct some (n + 2)-dimensional collapsing space–times, which include generalised Lemaître–Tolman–Bondi-like solutions, and we prove that these can be matched to static Λ-vacuum exterior space–times. We then investigate the global properties of the matched solutions which, besides black holes, may include the existence of naked singularities and wormholes. Second, we consider as interiors classes of 5-dimensional collapsing solutions built on Riemannian Bianchi IX spatial metrics matched to radiating exteriors given by the Bizoń–Chmaj–Schmidt metric. In some cases, the data at the boundary for the exterior can be chosen to be close to the data for the Schwarzschild solution.     

Reduction of the n-dimensional static vacuum Einstein equation and generalized Schwarzschild solutions

We consider the static vacuum Einstein spacetime when the spatial factor is conformal to a n-dimensional pseudo-Euclidean space. The most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described. We obtain the entire set of solutions of the reduced system, where the classical Schwarzschild solution arises as a particular solution. In addition, we show that the Riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature.     

Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one‐periodic and two‐periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.     

On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

In this paper, we derive a class of doubly periodic standing wave solutions for a coupled Higgs field equation by employing the Hirota bilinear method and theta function identities. Such solutions are expressed in terms of theta functions with variable separation form. Moreover, it is shown that these solutions can be converted into Jacobi elliptic function representations, and their long‐wave limit yields collision of dark solitons. In comparing with known solutions of the canonical evolution equation, three new aspects will be developed in this paper. First, the periods in the spatial and temporal directions, measured in terms of the theta function parameters τ and τ₁, are independent of each other, quite unlike most similar solutions found earlier in the literature. Second, the doubly periodic wave solutions possess two families of the local maxima, whose locations and types are then examined in detail. Third, we obtain new doubly periodic standing wave solutions for the Davey–Stewartson equation through its similarity transformation to the coupled Higgs field equation.     

Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge–Wheeler and Zerilli equations obtained recently by Anderson et al. (Ann. Henri Poincaré 21:61–813, 2020) are sufficient to obtain the conformal scattering.

     

Dynamics in the Schwarzschild Isosceles Three Body Problem

The Schwarzschild potential, defined as \(U(r)=-A/r-B/r^3\) , where \(r\) is the relative distance between two mass points and \(A,B>0\) , models astrophysical and stellar dynamics systems in a classical context. In this paper we present a qualitative study of a three mass point system with mutual Schwarzschild interaction where the motion is restricted to isosceles configurations at all times. We retrieve the relative equilibria and provide the energy–momentum diagram. We further employ appropriate regularization transformations to analyze the behavior of the flow near triple collision. We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for nonzero total angular momenta (e.g., when two of the mass points spin infinitely many times around the center of mass). This phenomenon is known in celestial mechanics as the black-hole effect and is understood as an analog in the classical context of behavior near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits that are neither homothetic nor homographic.     

The application of trigonal curve to the Mikhailov–Shabat–Sokolov flows

Resorting to the characteristic polynomial of Lax matrix for the Mikhailov–Shabat–Sokolov hierarchy associated with a \({3 \times 3}\) matrix spectral problem, we introduce a trigonal curve, from which we deduce the associated Baker–Akhiezer function, meromorphic functions and Dubrovin-type equations. The straightening out of the Mikhailov–Shabat–Sokolov flows is exactly given through the Abel map. On the basis of these results and the theory of trigonal curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic functions, and in particular, that of solutions for the entire Mikhailov–Shabat–Sokolov hierarchy.     

Analytic and arithmetic properties of the $$(\Gamma ,\chi )$$ ( Γ , χ ) -automorphic reproducing kernel function and associated Hermite–Gauss series

We consider the reproducing kernel function of the theta Bargmann–Fock Hilbert space associated with given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and the discreteness of its zeros are examined. The analytic sets of zeros of the theta Bargmann–Fock space inside a given fundamental cell is characterized and shown to be finite and of cardinal less or equal to its dimension. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite–Gauss coefficients. Some of them generalize some of the arithmetic identities given by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite–Gauss coefficients are nontrivial examples of the so-called lattice’s functions according the Serre terminology. The perfect use of the basic properties of the complex Hermite polynomials is crucial in this framework.

     

Quasi-periodic Solutions of the Kaup–Kupershmidt Hierarchy

Based on solving the Lenard recursion equations and the zero-curvature equation, we derive the Kaup–Kupershmidt hierarchy associated with a 3×3 matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the Kaup–Kupershmidt hierarchy, we introduce a trigonal curve $\mathcal {K}_{m-1}$ and present the corresponding Baker–Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the Kaup–Kupershmidt flows. With the aid of the properties of the Baker–Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. The Riemann–Jacobi inversion problem is achieved by comparing the asymptotic expansion of the Baker–Akhiezer function and its Riemann theta function representation, from which quasi-periodic solutions of the entire Kaup–Kupershmidt hierarchy are obtained in terms of the Riemann theta functions.     

On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.     

Embeddings for solutions of Einstein equations

We study isometric embeddings of some solutions of the Einstein equations with sufficiently high symmetries into a flat ambient space. We briefly describe a method for constructing surfaces with a given symmetry. We discuss all minimal embeddings of the Schwarzschild metric obtained using this method and show how the method can be used to construct all minimal embeddings for the Friedmann models. We classify all the embeddings in terms of realizations of symmetries of the corresponding solutions.     

On the existence of dynamics in Wheeler–Feynman electromagnetism

Wheeler–Feynman electrodynamics (WF) is an action-at-a-distance theory about world-lines of charges that in contrary to the textbook formulation of classical electrodynamics is free of ultraviolet singularities and is capable of explaining the irreversible nature of radiation. In WF, the world-lines of charges obey the so-called Fokker–Schwarzschild–Tetrode (FST) equations, a coupled set of nonlinear and neutral differential equations that involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation of this theory in terms of Maxwell–Lorentz electrodynamics without self-interaction that we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These conditional solutions solve the FST equations on any finite time interval with prescribed continuations outside of this interval. As a byproduct, we also prove existence and uniqueness of solutions to the Synge equations on the time half-line for a given history of charge world-lines.     

Stefan problems for reflected SPDEs driven by space–time white noise

We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space–time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space–time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary.     

The formation of black holes in spherically symmetric gravitational collapse

We consider the spherically symmetric, asymptotically flat Einstein–Vlasov system. We find explicit conditions on the initial data, with ADM mass M, such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where the area radius r along each geodesic is bounded by 2M, the timelike lines \({r=c\in [0,2M]}\) are incomplete, and for r > 2M the metric converges asymptotically to the Schwarzschild metric with mass M. The initial data that we construct guarantee the formation of a black hole in the evolution. We give examples of such initial data with the additional property that the solutions exist for all r ≥ 0 and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model characterized by conditions on the matter quantities.     

New approach to calculating the spectrum of a quantum space–time

We study the dynamics of a massive pointlike particle coupled to gravity in four space–time dimensions. It has the same degrees of freedom as an ordinary particle: its coordinates with respect to a chosen origin (observer) and the canonically conjugate momenta. The effect of gravity is that such a particle is a black hole: its momentum becomes spacelike at a distances to the origin less than the Schwarzschild radius. This happens because the phase space of the particle has a nontrivial structure: the momentum space has curvature, and this curvature depends on the position in the coordinate space. The momentum space curvature in turn leads to the coordinate operator in quantum theory having a nontrivial spectrum. This spectrum is independent of the particle mass and determines the accessible points of space–time.     

An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality

We prove a sharp Alexandrov–Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic n-space, n ≥ 3. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter–Schwarzschild solution. This sharpens previous results by Dahl–Gicquaud–Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space–times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension n ≥ 3. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chru?ciel and Simon on the validity of a Penrose-type inequality for exotic black holes.     

Exact Relativistic Treatment of Stationary Counter-Rotating Dust Disks: Axis,Disk, and Limiting Cases

We continue to study the construction of explicit solutions of the stationary axisymmetric Einstein equations, interpretable as counterrotating disks of dust. We discuss the previously constructed class of solutions for disks with constant angular velocity and constant relative density. The metric for these space–times is given in terms of theta functions on a Riemann surface of genus two. We discuss the metric functions at the axis of symmetry and on the disk. Interesting limiting cases are the Newtonian, static, and ultrarelativistic limits (in the latter limit, the central red shift diverges).     

Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions

We use the fixed point index theory of condensing mapping in cones discuss the existence of positive solutions for the following boundary value problem of fractional differential equations in a Banach space E

$$\begin{aligned} \left\{ \begin{array}{ll} -D^{\,\beta }_{0^{+}}u(t)=f(t,u(t)),\quad t\in J, \\ u(0)=u^{\prime }(0)=\theta ,\quad u(1)=\rho \int _{0}^{1}u(t)dt,\\ \end{array} \right. \end{aligned}$$

where both \(2<\beta \le 3\) and \(0<\rho <\beta \) are real numbers, \(J=[0,1]\), \(D^{\,\beta }_{0^{+}}\) is the Riemann–Liouville fractional derivative, \(f : J\times K \rightarrow K\) is continuous, K is a normal cone in Banach space E, \(\theta \) is the zero element of E. Under more general conditions of growth and noncompactness measure about nonlinearity f, we obtain the existence of positive solutions.