Repulsive Singularity Research Articles

Accurate mapped trigonometric discrete variable representations for Coulomb singularities in molecules: Applications with H2+ and H2 in cylindrical coordinates

Mapped discrete variable representations using the sine and cosine polynomials are purposed to accurately solve the stationary electronic Shrödinger equation of molecules. In this method, the grid points densely cluster around the positions of the nuclei but sparse enough in other regions. The electronic states of H2+ and H2 molecules are calculated by a preconditioned inexact spectral transform method in cylindrical coordinates for illustrating. A spectral convergence behaviour of eigenvalues of the electronic state of the H2+ molecule but slower convergence behaviour for the H2 molecule are observed, due to the repulsive singularity between electrons in the H2 molecule.

Positive periodic solutions for planar differential systems with repulsive singularities on the axes

In this paper we provide an existence result of periodic solutions to planar systems having a repulsive singularity on both the axes. We require a nonresonance assumption, related to the spectrum of the periodic problem associated to the Steen's planar system{−y′=−cx3+μx,x′=−dy3+νy,x>0,y>0, which is isochronous (see [15]). As a particular case we obtain the existence of periodic solutions to a periodically perturbed Steen's planar system.

Periodic solutions of radially symmetric systems with a singularity

In this paper, we study the existence of infinitely many periodic solutions to planar radially symmetric systems with certain strong repulsive singularities near the origin and with some semilinear growth near infinity. The proof of the main result relies on topological degree theory. Recent results in the literature are generalized and complemented.

Studies on a damped differential equation with repulsive singularity

By an application of Leray–Schauder alternative principle, existence results of positive periodic solution are presented for a damped differential equation with repulsive singularity. Copyright © 2012 John Wiley & Sons, Ltd.

An elliptic singular system with nonlocal boundary conditions

We study the existence of solutions for the nonlinear second order elliptic system Δu+g(u)=f(x), where g∈C(RN∖S,RN) with S⊂RN bounded. Using topological degree methods, we prove an existence result under a geometric condition on g. Moreover, we analyze the particular case of an isolated repulsive singularity: under a Nirenberg type condition, we prove the existence of a sequence of solutions of appropriate approximated problems that converges to a generalized solution.

Nonlinear dynamics of differential equations with attractive–repulsive singularities and small time‐dependent coefficients

Motivated by some physical models with small parameters, in this paper, we proved the existence of periodic solutions (almost periodic solutions) for two classes of differential equations with attractive–repulsive singularities and small time‐dependent coefficients by the averaging method and the implicit function theorem. Copyright © 2012 John Wiley & Sons, Ltd.

K-independent vibrational bases for systems with large amplitude motion

For J > 0 calculations it would be advantageous to have a vibrational basis independent of rotational quantum numbers, but which can be applied to molecules or systems with large amplitude motion. Several authors have explored the possibility of using as bend functions (m = 0) Legendre polynomials. Their most obvious disadvantage is the existence of infinite matrix elements. Their behaviour near the θ = 0 and π singularities will also be inappropriate for some wavefunctions. In this paper, we test and analyse several rotational-index-independent vibrational bases and compare them to the standard basis of associated Legendre polynomials, , where m depends on K, the quantum number for the molecule-fixed z component of the angular momentum. We find that for three-atom systems with wavefunctions having both significant amplitude at linearity and important Θ m=odd components a Legendre basis is poor, despite the repulsive singularity at linear geometries. Similar problems occur for systems with more than three atoms.

Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling

Conservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles. We address the evolution and formation of these aggregating steady states when the interaction potential has both attractive and repulsive singularities. Currently, no existence theory for such potentials is available. We develop and compare two complementary solution methods, a continuous pseudoinverse method and a discrete stochastic lattice approach, and formally show a connection between the two. Interesting aggregation patterns involving multiple peaks for a simple doubly singular attractive-repulsive potential are determined. For a swarming Morse potential, characteristic slow-fast dynamics in the scaled inverse energy is observed in the evolution to steady state in both the continuous and discrete approaches. The discrete approach is found to be remarkably robust to modifications in movement rules, related to the potential function. The comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two- and three-dimensional nonlocal interaction conservation equations.

Periodic solutions of singular second order differential equations: Upper and lower functions

In this paper, we continue the study of the periodic problem for the second-order equation u ′ ′ + f ( u ) u ′ + g ( u ) = h ( t , u ) , where h is a Carathéodory function and f , g are continuous functions on ( 0 , + ∞ ) which may have singularities at zero. Both attractive and repulsive singularities are considered. The method relies on a novel technique of construction of lower and upper functions. As an application, we obtain new sufficient conditions for the existence of periodic solutions to the Rayleigh–Plesset equation.

A Existence Theory for Positive Solutions to Superlinear Repulsive Singular Differential Equations with Impulse Effects

In this paper, we study positive solutions to the repulsive singular perturbation Hill equations with impulse effects. It is proved that such a perturbation problem has at least one positive impulsive periodic solution by a nonlinear alternative of Leray--Schauder.

On direct product based discrete variable representations for angular coordinates and the treatment of singular terms in the kinetic energy operator

The use of curvilinear coordinates results in kinetic energy operators with singular terms. These singularities can be in the dynamically relevant region. The present work specifically considers 1/sin 2 θ singularities and wavefunctions showing significant amplitude close to θ = 0. It investigates the use of direct product-type discrete variable representations (DVRs) to describe such situations in quantum dynamics calculations and discusses the regularization of such repulsive singularities. A new scheme, the cot-DVR, is developed which can be used to construct well converging direct product DVRs for Hamiltonians showing a 1/sin 2 θ singularity. The cot-DVR is based on the diagonalization of the cot θ-matrix and adds two additional basis function, sin( θ) and sin(2 θ), to the standard basis of Legendre polynomials. It is particularly well suited for multi-configurational time-dependent Hartree calculations which employ one-dimensional (bottom layer) single-particle functions.

On periodic solutions of second-order differential equations with attractive–repulsive singularities

Sufficient conditions for the existence of a solution to the problem u ″ ( t ) = g ( t ) u μ ( t ) − h ( t ) u λ ( t ) + f ( t ) for a.e. t ∈ [ 0 , ω ] , u ( 0 ) = u ( ω ) , u ′ ( 0 ) = u ′ ( ω ) are established.

Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation $$\ddot{x}+f(x)\dot{x}+g(x)=c(t),$$ where f, g:ℝ+=(0,+∞)→ℝ are continuous functions, g has a repulsive singularity at the origin and c(t) is a continuous T-periodic function. The novelty of the present article is that for the first time we show that a weakly repulsive singularity enables the achievement of a new existence criterion of positive T-periodic solutions through a basic application of the coincidence degree theory. Recent results in the literature are generalized and significantly improved.

Quasi-periodic solutions of forced isochronous oscillators at resonance

We deal with the existence of quasi-periodic solutions of forced isochronous oscillators with a repulsive singularity, the nonlinearity is a bounded perturbation. Using a variant of Moser's twist theorem of invariant curves, due to Ortega [R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999) 381–413], we show that there are many quasi-periodic solutions and the boundedness of all solutions.

Periodic solutions for damped differential equations with a weak repulsive singularity

This paper deals with the existence of positive T -periodic solutions for the following damped differential equation x ̈ + p ( t ) x ̇ + q ( t ) x = f ( t , x ) + c ( t ) , where p , q , c ∈ L 1 ( R ) are T -periodic functions and f ∈ Car ( R × R + , R ) is T -periodic in the first variable. We will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of Schauder’s fixed point Theorem.

Multiple Positive Solutions to Superlinear Repulsive Singular Impulsive Differential Equations

The existence of multiple positive solutions to superlinear repulsive singular impulsive differential equations is discussed in this paper. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed-point theorem on compression and expansion of cone.

Periodic Solutions of Equations of Emarkov-Pinney Type

Abstract After establishing a relation between the Hill’s equations and the Emarkov-Pinney equations, we are able to use degree theory to remove a technical assumption in [14], and give a complete set of non-resonance conditions for differential equations with repulsive singularities.

Existence of periodic solutions for second-order differential equations with singularities and the strong force condition

We prove the existence of periodic solutions for the equation (1) u ″ + f ( u ) u ′ + g ( t , u ) = e ( t ) , where the nonlinearity g has a repulsive singularity at the origin. In previous papers dealing with this kind of problem it is usually assumed a nonintegrability condition on g near the origin. We provide a weaker condition that substitutes the nonintegrability of g. If f ≡ 0 the existence of subharmonic solutions is proved utilizing a variational method and when f ≠ 0 we prove the existence of a periodic solution using topological degree theory.

Periodic singular problem with quasilinear differential operator

We study the singular periodic boundary value problem of the form \[ \left(\phi (u^{\prime })\right)^{\prime }+h(u)u^{\prime }=g(u)+e(t),\quad u(0)=u(T),\quad u^{\prime }(0)=u^{\prime }(T), \] where $\phi \:\mathbb{R}\rightarrow \mathbb{R}$ is an increasing and odd homeomorphism such that $\phi (\mathbb{R})=\mathbb{R},$ $h\in C[0,\infty ),$ $e\in L_1J$ and $g\in C(0,\infty )$ can have a space singularity at $x=0,$ i.e. $\limsup _{x\rightarrow 0+}|g(x)|=\infty $ may hold. We prove new existence results both for the case of an attractive singularity, when $\liminf _{x\rightarrow 0+}g(x)=-\infty ,$ and for the case of a strong repulsive singularity, when $\lim _{x\rightarrow 0+}\int _x^1g(\xi )\hspace{0.56905pt}\text{d}\xi =\infty .$ In the latter case we assume that $\phi (y)=\phi _p(y)=|y|^{p-2}y,$ $p>1,$ is the well-known $p$-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.

Twist periodic solutions of repulsive singular equations

Motivated by the Lazer–Solimini equation and the Brillouin equation, which present a repulsive singularity at the origin, we will develop in this paper some criterion for the (positive) periodic solution to be of twist type. As an application of the criterion, we will give also some quantitative estimates to the region of parameters so that the equations have twist periodic solutions. These, together with the Moser Twist Theorem, imply that the equations have rich dynamics in the neighborhood of the twist periodic solutions.