Strong Singularity Research Articles

Singularity analysis of igneous zircon U-Pb age and Hf isotopic record in the Zhongdian arc, northwest Yunnan, China: Implications for Indosinian magmatic flare-up and the formation of porphyry copper deposits

Situated in the southern part of Yidun arc in the southeastern Tibetan Plateau, the Zhongdian arc records magmatic activity that ranges from approximately 270 to 190 Ma, which is likely attributable to subduction of the Garze-Litang slab and involved steady low-volume magmatic lulls punctuated by short-lived high-volume flare-ups. Many porphyry copper deposits have been found in these magmatic rocks, but the temporal-spatial relationships among magmatism, porphyry copper deposits, and subduction processes around the Zhongdian arc have long been debated. Frequency distributions calculated from igneous zircon U-Pb ages are commonly utilized to interpret the duration of magmatic events. Recently, a new concept of fractal density and a local singularity analysis (LSA) method have been applied to analyze the geometric properties of zircon U-Pb age peak anomalies, which are linked to deeply rooted avalanches associated with short spurts of convection during the formation of supercontinents and continental crust growth. In this paper, the igneous zircon U-Pb dating data collected from the Zhongdian arc are analyzed by the LSA method from the perspective of fractal density to interpret the causational relationship between age peak and subduction process of the Garze-Litang oceanic crust. The results show that the age density around the age peak of 215 Ma can be well fitted and described by power law functions, and the strong singularity (k > 0.8) indicates the occurrence of extreme geological events. Meanwhile, systematic changes in the hafnium isotopic composition of igneous zircons with time reflect the upwelling of asthenospheric mantle materials, characterized by a rapid increase in εHf(t) values that reach the highest around the 215 Ma peak. Moreover, the log–log plot that shows the cumulative number-age distribution reveals another transitional age of 235 Ma. By comparison of the magmatic intensity between the western and eastern belts of the Zhongdian arc, it is indicated that the eastward magmatic migration might have begun at ca. 235 Ma. Combined with the previous work, we propose that the Indosinian magmatic flare-up in the Zhongdian arc might have been caused by superimposition of slab subduction, steepening of slab dip, and slab breakoff. This long-term magmatic hydrothermal evolution system and extreme singularity events also provide indispensable conditions for the formation of large-scale and high-grade porphyry copper deposits in the region. Our findings provide new insights into the subduction process of the Garze-Litang oceanic crust and Indosinian magmatic framework around the Zhongdian arc. It is believed that the LSA method has potential applications in depicting the variation of magmatic activity and predicting mineral resources.

Zipoy-Voorhees Gravitational Object as a Source of High-Energy Relativistic Particles

The Zipoy-Voorhees solution is known as the γ-metric and/or q-metric being static and axisymmetric vacuum solution of Einstein field equations which becomes strong curvature naked singularity. The metric is characterized by two parameters, namely, the mass M and the dimensionless deformation parameter γ. It is shown that the velocity of test particle orbiting around the central γ-object can reach the speed of light, consequently, the total energy of the particle will be very high for a specific value the deformation parameter of the spacetime. It is also shown that causality problem arises in the interior region of the physical singularity for the specific value of the deformation parameter when test particles can move with superluminal velocity being greater than the speed of light that might be an additional tool for explaining the existence of tachyons for γ>1/2 which are invisible for an observer.

Spherically Symmetric Exact Vacuum Solutions in Einstein-Aether Theory

We study spherically symmetric spacetimes in Einstein-aether theory in three different coordinate systems, the isotropic, Painlevè-Gullstrand, and Schwarzschild coordinates, in which the aether is always comoving, and present both time-dependent and time-independent exact vacuum solutions. In particular, in the isotropic coordinates we find a class of exact static solutions characterized by a single parameter c14 in closed forms, which satisfies all the current observational constraints of the theory, and reduces to the Schwarzschild vacuum black hole solution in the decoupling limit (c14=0). However, as long as c14≠0, a marginally trapped throat with a finite non-zero radius always exists, and on one side of it the spacetime is asymptotically flat, while on the other side the spacetime becomes singular within a finite proper distance from the throat, although the geometric area is infinitely large at the singularity. Moreover, the singularity is a strong and spacetime curvature singularity, at which both of the Ricci and Kretschmann scalars become infinitely large.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

Asymptotic soliton-like solutions to the Benjamin–Bona–Mahony equation with variable coefficients and a strong singularity

The paper deals with constructing an asymptotic one-phase soliton-like solution to the Benjamin--Bona--Mahony equation with variable coefficients and a strong singularity making use of the non-linear WKB technique. The influence of the small-parameter value on the structure and the qualitative properties of the asymptotic solution, as well as the accuracy with which the solution satisfies the considerable equation, have been analyzed. It was demonstrated that due to the strong singularity, it is possible to write explicitly not only the main term of the asymptotics but at least its first-order term.

Electron holes in a κ distribution background with singularities

The pseudo-potential method is applied to derive diverse propagating electron–hole structures in a nonthermal or κ particle distribution function background. The associated distribution function Ansatz reproduces the Schamel distribution of [H. Schamel, Phys. Plasmas 22, 042301 (2015)] in the Maxwellian (κ→∞) limit, providing a significant generalization of it for plasmas where superthermal electrons are ubiquitous, such as space plasmas. The pseudo-potential and the nonlinear dispersion relation are evaluated. The role of the spectral index κ on the nonlinear dispersion relation is investigated, in what concerns the wave amplitude, for instance. The energy-like first integral from Poisson's equation is applied to analyze the properties of diverse classes of solutions: with the absence of trapped electrons, with a non-analytic distribution of trapped electrons, or with a surplus of trapped electrons. Special attention is, therefore, paid to the non-orthodox case where the electrons distribution function exhibits strong singularities, being discontinuous or non-analytic.

The fractional p-Laplacian evolution equation in $${\mathbb {R}}^N$$ in the sublinear case

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter $$1<p<2$$ and fractional exponent $$s\in (0,1)$$ . Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $$L^q$$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case $$p>2$$ has been dealt with in a recent paper. We study here the “fast” regime $$1<p<2$$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass value M and any p in the subrange $$p_c=2N/(N+s)<p<2$$ , and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all $$L^q$$ spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for $$p_c<p<p_1$$ , where $$p_1$$ is a new critical number that we introduce, $$p_1\in (p_c,2)$$ . We extend this type of singular solutions to the “very fast range” $$1<p<p_c$$ . They represent examples of weak solutions having finite-time extinction in that lower p range. We briefly examine the situation in the limit case $$p=p_c$$ . Finally, we show that very singular solutions are related to fractional elliptic problems of the nonlinear eigenvalue type, of interest in their own right.

Existence of Positive Solutions for the Nonhomogeneous Schrödinger-Poisson System with Strong Singularity

Existence of Positive Solutions for the Nonhomogeneous Schrödinger-Poisson System with Strong Singularity

Singular elliptic equations with quadratic gradient term

We establish the existence of a non-trivial weak solution to the following singular quasilinear equation with Hardy potential and singular quadratic gradient term:{−Δu−μu|x|2=|∇u|2u+f(x,u)inΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN is a smooth bounded domain, μ>0,0∈Ω,N≥3. We show that there exists a solution u∈H01(Ω) to the above problem. The notable characteristic of this problem is that it includes quadratic gradient nonlinearity and strong singularity.

A boundary shape function iterative method for solving nonlinear singular boundary value problems

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

Lebesgue spectrum of countable multiplicity for conservative flows on the torus

We study the spectral measures of conservative mixing flows on the 2 2 -torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: (1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, (2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows.

An existence result for singular nonlocal fractional Kirchhoff–Schrödinger–Poisson system

In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff–Schrödinger–Poisson system involving a weak singularity, i.e. when . Further, we obtain the existence of a solution with a strong singularity, i.e. when . We employ variational techniques to prove the existence and multiplicity results. Moreover, an estimate is obtained by using the Moser iteration method.

Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity

In this paper, we consider the following fractional Kirchhoff problem with strong singularity: {(1+b∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy)(−Δ)su+V(x)u=f(x)u−γ,x∈R3,u>0,x∈R3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (1+ b\\int _{\\mathbb{R}^{3}}\\int _{\\mathbb{R}^{3}} \\frac{ \\vert u(x)-u(y) \\vert ^{2}}{ \\vert x-y \\vert ^{3+2s}}\\,\\mathrm{d}x \\,\\mathrm{d}y )(-\\Delta )^{s} u+V(x)u = f(x)u^{-\\gamma }, & x \\in \\mathbb{R}^{3}, \\\\ u>0,& x\\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where (-Delta )^{s} is the fractional Laplacian with 0< s<1, b>0 is a constant, and gamma >1. Since gamma >1, the energy functional is not well defined on the work space, which is quite different with the situation of 0<gamma <1 and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution u_{b} by using variational methods and the Nehari manifold method. We also give a convergence property of u_{b} as brightarrow 0, where b is regarded as a positive parameter.

Multiple results to $$\phi $$-Laplacian singular Liénard equation and applications

In this paper, we investigate the existence of a positive periodic solution for the following $$\phi $$ -Laplacian generalized Lienard equation with a singularity: $$\begin{aligned} (\phi (u'))'+f(t,u)u'+ \frac{b(t)}{u^\rho }=h(t)u^m, \end{aligned}$$ where $$\rho $$ is a positive constant and m is a constant. Our proof is based on the Manasevich–Mawhin continuation theorem, and the results are applicable to weak and strong singularities of attractive type (or repulsive type, attractive-repulsive type). Besides, these results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. As applications, we study the existence of positive periodic solutions for a Rayleigh–Plesset equation. Finally, three examples are given to show applications of these theorems.

Positive Periodic Solution for a Second-Order Damped Singular Equation via Fixed Point Theorem in Cones

The aim of this paper is to show that fixed point theorem in cones can be applied to singular equations. Using the positivity of Green’s function and the external force e(t), we prove the existence of a positive periodic solution for a damped singular equation with sub-linearity, semi-linearity and super-linearity conditions, and these results are applicable to weak and strong singularities. As applications, we consider the existence of a positive periodic solution for nonlinear elasticity model and Ermakov–Pinney equation

Fractional Schrödinger Equation with Singular Potentials of Higher Order

In this paper the space-fractional Schrödinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles accumulating effect is observed in the singular cases. From the mathematical point of view a “splitting of the strong singularity” phenomena is also observed.

Some thermodynamical peculiarities at the Lifshitz topological transitions in trigonally warped AB-stacked bilayer graphene and graphite near K points

ABSTRACT Similarity has been proven between the Lifshitz topological transitions (LTT) in AB-stacked trigonally warped bilayer graphene (TWBG) and graphite near K points. The density of states (DOS) has been shown to have the van Hove singularities (VHS) of type at LTT in AB-stacked (TWBG) and graphite near K points. The topology evolutions of the iso-energetic lines at LTT have been established, and transitions are realised via four stages. The LTT transition energy in TWBG is ϵc ≈ 1 meV, while in graphite ϵc ≈ 10 meV. Thermodynamical characteristics are investigated of AB-stacked TWBG and graphite near K points at LTT. Thermodynamical parameters possess of the strongest singularities at LTT near the K points: the electron specific heat Ce and compressibility δ diverge as ; and thermal coefficient of pressure δ diverges as (here z = μ–ϵc , and μ is the chemical potential). The similarity between the band structure of graphite near the K point and that of the bilayer graphene logically suggests probing of Lifshitz transitions in the advance study of both systems. The developed methodology can be used for exploration of LTT in other bilayer and multilayer structures, like hBN, silicene, germanene, etc.

Adaptive pseudo-time methods for the Poisson–Boltzmann equation with Eulerian solvent excluded surface

This work further improves the pseudo-transient approach for the Poisson Boltzmann equation (PBE) in the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is known to involve many difficulties, such as exponential nonlinear term, strong singularity by the source terms, and complex dielectric interface. Recently, a pseudo-time ghost-fluid method (GFM) has been developed in [S. Ahmed Ullah and S. Zhao, Applied Mathematics and Computation, 380, 125267, (2020)], by analytically handling both nonlinearity and singular sources. The GFM interface treatment not only captures the discontinuity in the regularized potential and its flux across the molecular surface, but also guarantees the stability and efficiency of the time integration. However, the molecular surface definition based on the MSMS package is known to induce instability in some cases, and a nontrivial Lagrangian-to-Eulerian conversion is indispensable for the GFM finite difference discretization. In this paper, an Eulerian Solvent Excluded Surface (ESES) is implemented to replace the MSMS for defining the dielectric interface. The electrostatic analysis shows that the ESES free energy is more accurate than that of the MSMS, while being free of instability issues. Moreover, this work explores, for the first time in the PBE literature, adaptive time integration techniques for the pseudo-transient simulations. A major finding is that the time increment $\Delta t$ should become smaller as the time increases, in order to maintain the temporal accuracy. This is opposite to the common practice for the steady state convergence, and is believed to be due to the PBE nonlinearity and its time splitting treatment. Effective adaptive schemes have been constructed so that the pseudo-time GFM methods become more efficient than the constant $\Delta t$ ones.

Control and inverse problems for the heat equation with strong singularities

We consider a linear system composed of N + 1 one dimensional heat equations connected by point-mass-like interface conditions. Assume an L 2 Dirichlet boundary control at one end, and Dirichlet boundary condition on the other end. Given any L 2 -type initial temperature distribution, we show that the system is null controllable in arbitrarily small time. The proof uses known results for exact controllability for the associated wave equation. An argument using the Fourier Method reduces the control problem for both the heat equation and the wave equation to certain moment problems. Controllability is then proved by relating minimality properties of the family of exponential functions associated to the wave with the family associated to the heat equation. Based on the controllability result we solve the dynamical inverse problem, i.e. recover unknown parameters of the system from the Dirichlet-to-Neumann map given at a boundary point.

Factorizations and Hardy–Rellich inequalities on stratified groups

In this paper, we obtain Hardy, Hardy–Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn [9]. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues of Gesztesy and Littlejohn’s 2-parameter version of the Rellich inequality on stratified groups and on the Heisenberg group, and a new two-parameter estimate on \mathbb R^n which can be regarded as a counterpart to the Gesztesy and Littlejohn estimate. Moreover, we give some applications of Hardy and Rellich type inequalities to the lower semiboundedness and to form boundedness for interactions with countably many strong singularities on stratified groups.