Fuchs Equations Research Articles

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Studying the Upper Bounds of the Numbers of Zeros of Abelian Integrals by the Law of Polynomials

For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree [Formula: see text], the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.

An upper bound for the number of small-amplitude limit cycles in non-smooth Liénard system

The non-smooth Liénard system, a well-known nonlinear model, appears in a natural way in physics, chemistry, biology, and so on, in which periodic phenomena play a relevant role. In this paper, we investigate the small-amplitude limit cycles generated by the Hopf bifurcation of the non-smooth Liénard system. By utilizing the Picard–Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles for a generic non-smooth Liénard system. Finally, an example is given to illustrate the efficiency of the theoretical results.

Apéry extensions

AbstractThe Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau–Ginzburg (LG) models — and thus, in particular, as periods. We also construct an Apéry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard–Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG‐models mirror to several Fano threefolds. By describing the “elementary” Apéry numbers in terms of regulators of higher cycles (i.e., algebraic ‐theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG‐models are modular families of surfaces, and the distinction between multiples of and (or ) translates ultimately into one between algebraic and of the family.

On Multivariate Picard–Fuchs Systems and Equations

In this paper, we studied the Picard–Fuchs systems and equations which appear in the theory of Gauss–Manin systems and connections associated with deformations of isolated singularities. Among other things, we describe some interesting properties of such systems and relationships between them. Then we show how to calculate the fundamental solutions to the Gauss–Manin system for Aμ-singularities and to the corresponding generalized Legendre equations in terms of the multidimensional Horn’s hypergeometric functions. In conclusion, some important questions concerning basic properties of the local and global Picard–Fuchs systems of Pfaffian type, involving integrability conditions and commuting relations, are discussed in some detail.

Algorithms for minimal Picard–Fuchs operators of Feynman integrals

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard--Fuchs operator for the generic massive two-point $n-1$-loop sunset integral in two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi--Yau of dimensions $n-2$. We have checked this explicitly till six loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from $K3$ surface with Picard number 11. We determine as well Picard--Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.

Cyclicity of a Class of Hamiltonian Systems under Perturbations of Piecewise Smooth Polynomials

In the study of the weakened Hilbert’s 16th problem, there usually appears period annulus surrounding more than one singular point, implying that there are often more than three generate functions in the Abelian integrals (or the first order Melnikov functions). It is also known that there are usually more than three generate functions in the study of the bifurcation of limit cycles in piecewise smooth Hamiltonian systems. This paper aims to explore a method so that we can estimate the upper bound of limit cycles bifurcating from the period annulus if the unperturbed system has some good properties such as Picard–Fuchs equations. This method is based on [Gavrilov & Iliev, 2009] and is useful for smooth and piecewise smooth polynomial vector fields of a general degree. Finally, we present an example to illustrate an application of the theoretical results.

Limit Cycles from Perturbing a Piecewise Smooth System with a Center and a Homoclinic Loop

We study bifurcations of limit cycles arising after perturbations of a special piecewise smooth system, which has a center and a homoclinic loop. By using the Picard–Fuchs equation, we give an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop. Furthermore, by applying the method of first-order Melnikov function we obtain a lower bound of the maximum number of limit cycles bifurcated from the center.

On a theory of mass transfer during the evaporation of a spherical droplet

An analytic model of evaporation of a spherical droplet into the surrounding gas without consideration of the Stefan flow and vapor buoyancy is presented. This model is based on the use of the diffusion equation and contains a kinetic description of the evaporation process. In addition, this model is based on the assumption that droplet temperature does not change over time. In the framework of the presented model, an equation for the rate of droplet evaporation that is an alternative to the Fuchs equation is obtained. Analysis of the model showed that, depending on droplet size, evaporation can take place in three regimes.

Bending of light and inhomogeneous Picard–Fuchs equation

The bending of light rays by gravitational sources is one of the first evidence of general relativity. When the gravitational source is a stationary massive object such as a black hole, the bending angle has an integral representation, from which various series expansions up to a finite order in terms of the parameters of orbit and the background spacetime has been derived. However, it has not been clear that it has any analytic expansion. In this paper, we show that such an analytic expansion can be obtained for the case of a Schwarzschild black hole by solving an inhomogeneous Picard–Fuchs equation, which has been applied to compute effective superpotentials on D-branes in the Calabi–Yau manifolds. From the analytic expression of the bending angle, the full order expansions in both weak and strong deflection limits are obtained. We show that the result can be obtained by the direct integration approach as well. We also discuss how the charge of the gravitational source affects the bending angle and show that a similar analytic expression can be obtained for the extremal Reissner–Nordström spacetime.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Picard–Fuchs equations for Shimura curves over Q

We show that Picard-Fuchs equations of periods of certain families of abelian surfaces with quaternionic multiplication have fractional powers of algebraic modular forms as solutions. We give sever ...

Picard–Fuchs Equation Applied to Quadratic Isochronous Systems with Two Switching Lines

The present paper is devoted to study the problem of limit cycle bifurcations for nonsmooth integrable differential systems with two perpendicular switching lines. By using the Picard–Fuchs equation, we obtain the upper bounds of the number of limit cycles bifurcating from the period annuli of the quadratic isochronous systems [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], when they are perturbed inside a class of all discontinuous polynomial differential systems of degree [Formula: see text]. This method can be applied to study the limit cycle bifurcations of other integrable differential systems.

Generalized Picard—Fuchs Operators From Whitham Hierarchy in $${\mathcal N} = 2$$ Supersymmetric Gauge Theory with Massless Hypermultiplets

Using the Whitham hierarchy, we obtain the Picard—Fuchs equations in$${\mathcal N} = 2$$supersymmetric Yang—Mills theory for a classical gauge group with Nfmassless hypermultiplets. In the general case for Nf ≠ 0, there are at least r−2 Picard—Fuchs equations that can be computed exactly from the commutation relations of the meromorphic differentials defined up to a linear combination of holomorphic differentials on the Seiberg—Witten hyperelliptic curve. Using Euler operator techniques, we study the Picard—Fuchs equations, including instanton corrections. Moreover, using symbolic computer calculations, we can obtain a complete set of Picard—Fuchs equations.

Limit Cycle Bifurcations from a Quadratic Center with Two Switching Lines

In this paper, we study limit cycle bifurcations for a differential system with two switching lines by using Picard–Fuchs equation. We obtain a detailed expression of the corresponding first order Melnikov function which can be used to get the upper bound of the number of limit cycles. It is worth noting that we greatly simplify the computation. Our results also show that the number of switching lines has essential impact on the number of limit cycles bifurcating from a period annulus.

On the number of limit cycles for a class of piecewise smooth Hamiltonian systems with discontinuous perturbations

By analyzing the corresponding Picard–Fuchs equations, we obtain an upper bound of the number of limit cycles for a class of piecewise smooth Hamiltonian systems when they are perturbed inside discontinuous polynomials of degree n. Finally, we present an example to illustrate an application of the theoretical results.

Geometric aspect of three-dimensional Chern–Simons-like gravity

In this paper, we detailed study the geometric aspect of Chern–Simons-like gravity in three dimensions. The vacua are provided by three dimensional conformally flat manifold, which admit a special configuration, a two dimensional system (M2,h,Φ) consisting of metric h and scalar field Φ, by dimensional reduction. For this system we define the quasi-local mass. An interesting observation is that this system contains certain two dimensional dilaton gravity at the classical level. Via AdS/CFT, we check the Weyl anomaly and diffeomorphism anomaly for the boundary theory. We study the linearization of Chern–Simons-like gravity around the background with constant scalar curvature via linear Cotton tensor. If the background manifold is of positive constant scalar curvature, we show that there is no solutions for linearized vacuum equation. To find the solutions of the linear gravity with respect to the background with negative constant scalar curvature, we need to solve some Schrödinger-type equations. Related to supersymmetric quantum mechanics, one can find some exact solutions and some dualities between different components or modes of solutions. And it is also can be related to Seiberg–Witten theory via Picard–Fuchs equation in terms of WKB approximation. We discuss the ADM-type charge under the context of Chern–Simons-like gravity. Finally, we extend the Chern–Simons-like gravity to the supermanifolds by embedding the structure group of three-manifold into the body of othosymplectic supergroup.

Construction of the Fuchs equation with four given finite critical points and a given reducible monodromy group in the resonance case

One inverse problem of the analytic theory of linear differential equations is considered. Namely, the completely integrable Fuchs equation with four given finite critical points and a given reducible monodromy group of rank 2 on the complex projective line is constructed. Reducibility of the monodromy group of rank 2 means that 2×2-monodromy matrices (the generators of the monodromy group) can be simultaneously reduced by a linear nonsingular transformation to an upper triangular form. In so doing we study the case when the eigenvalue ξj of the diagonal matrix of the monodromy formal exponent at a corresponding Fuchs critical point is equal to an integer different from zero (resonance takes place).

Isochronous cosmological solutions of the Friedmann–Robertson–Walker model

In this paper, the periodic solutions of the equation of Friedmann–Robertson–Walker cosmology with a cosmological constant are investigated. Using variable transformation, the original second-order ordinary differential equation is converted to a planar dynamical system with cosmic time t. Numerical simulations indicate that period function T(h) of this dynamical system is monotonically increasing. However, a new planar dynamical system could be deduced by using conformal time variable [Formula: see text]. We prove that the new planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Explicitly, we find that there exist two families of periodic solutions with equal period for the new planar dynamical system which is derived from the Friedmann–Robertson–Walker model.

On the Number of Zeros of Abelian Integral for a Class of Cubic Hamilton Systems with the Phase Portrait “Butterfly”

The present paper is devoted to study the number of zeros of Abelian integral for the near-Hamilton system $$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x} = 2y(bx^2+2cy^2)+\varepsilon f(x,y),\\ \dot{y} = 2x(1-2ax^2-by^2)+\varepsilon g(x,y), \end{array}\right. } \end{aligned}$$ where \(a,b,c\in \mathbb {R}\), \(b 0\), \(b^2<4ac\), \(0<|\varepsilon |\ll 1\), f(x, y) and g(x, y) are polynomials in (x, y) of degree n. The generators of the corresponding Abelian integral satisfy three different Picard–Fuchs equations. We obtain an upper bound of the number of isolated zeros of the Abelian integral.