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In this paper, null geodesics for a symmergent black hole are investigated. The geodesic equation of this space time is solved analytically according to the Weier-strass elliptic function. Also, the effective potential is obtained and plotted. Finally, using the form of effective potential and obtained analytical solution of geodesic equations, some possible types of orbits related to null geodesics are demonstrated. [ABSTRACT FROM AUTHOR]

We study Hill’s differential equation with potential expressed by elliptic functions which arises in some problems of physics and mathematics. Analytical method can be applied to study the local properties of the potential in asymptotic regions of the parameter space. The locations of the saddle points of the potential are determined, the locations of turning points can be determined too when they are close to a saddle point. Combined with the quadratic differential associated with the differential equation, these local data give a qualitative explanation for the asymptotic eigensolutions obtained recently. A relevant topic is about the generalisation of Floquet theorem for ODE with doubly-periodic elliptic function coefficient which bears some new features compared to the case of ODE with real valued singly-periodic coefficient. Beyond the local asymptotic regions, global properties of the elliptic potential are studied using numerical method. [ABSTRACT FROM AUTHOR]

An effective technique is proposed for obtaining exact formulas for estimating the area of flow regions in two-dimensional fluid flow problems with free boundaries, that allow an exact solution in terms of elliptic functions. The effectiveness of the technique is demonstrated using a specific example of the problem of capillary waves on the surface of a liquid of finite depth. This example is characterized by mirror symmetry of the flow region, but the technique can be generalized to the case of other symmetry of the flow region. [ABSTRACT FROM AUTHOR]

The nonlinear massive plane wave solution of the classical scalar field in the Higgs potential is revisited to study the mass generation and particle creation. In particular, by assuming that the Higgs system is in the slightly excited state in early universe and it is described by the nonlinear solution, we study the mass generation mechanism for massive vector bosons and a heavy fermion in the quantum field theory around the nonlinear massive classical field. The nonlinear massive classical solution gives the transition from the vacuum to a pair of vector bosons and fermions. We present the new formulae of the probability density of the production process for particles in the standard model of elementary particle physics. The probability densities of the particle productions vanish when the nonlinear massive solution reduces to the constant solution (the classical vacuum expectation value); while the probability densities are expressed as the function of the free parameter in the classical solution in general case. We discuss the behavior of the probability densities for the three oscillating modes in the classical solution.

We obtain a power series representation of the Weierstrass sigma function arising in the theory of elliptic functions and derive a differential–recurrence equation as well as closed-form expressions for its coefficients. These results are obtained using the Weierstrass system of partial differential equations satisfied by the sigma function. [ABSTRACT FROM AUTHOR]

In this paper, we prove a logarithmic conjugation theorem on finitely connected tori. The theorem states that a harmonic function can be written as the real part of a function whose derivative is analytic and a finite sum of terms involving the logarithm of the modulus of a modified Weierstrass sigma function. We implement the method using arbitrary precision and use the result to find approximate solutions to the Laplace problem and the Steklov eigenvalue problem. Using a posteriori estimation, we show that the solution of the Laplace problem on a torus with a few holes has error less than 10-100 using a few hundred degrees of freedom and the Steklov eigenvalues have similar error. [ABSTRACT FROM AUTHOR]

In the present paper, by the complex method, the meromorphic solutions of the higher order ordinary differential equation w (5) + a w ″ + b w 2 − c w + d = 0 are investigated, where a , b , c , d are constant complex numbers, and b ≠ 0. Furthermore, by Theorem 1.1, we built elliptic and multiple-valued solutions for the higher order ordinary differential equations u (6) − u (5) + u ′ 2 − 2 u ′ u + u 2 + 2 u ′ − 2 u + 1 = 0 and u (6) − u (5) + a u ‴ − a u ″ + b u ′ 2 − 2 b u ′ u + b u 2 − c u ′ + c u + d = 0. At the end, we give some new meromorphic solutions for two higher-order KdV-like equations. [ABSTRACT FROM AUTHOR]

In the present paper, by the complex method, the meromorphic solutions of the higher order ordinary differential equation $ w^{(5)}+aw^{''}+bw^2-cw+d = 0 $ are investigated, where $ a, b, c, d $ are constant complex numbers, and $ b \neq0 $. Furthermore, by Theorem 1.1, we built elliptic and multiple-valued solutions for the higher order ordinary differential equations $ u^{(6)}-u^{(5)}+u'^2-2u'u+u^2+2u'-2u+1 = 0 $ and $ u^{(6)}-u^{(5)}+au^{'''}-au''+bu'^2-2bu'u+bu^2-cu'+cu+d = 0 $. At the end, we give some new meromorphic solutions for two higher-order KdV-like equations.

An attempt was made to reproduce, using the Schwartz function method, the well-known exact solution of Kinnersley to the problem of capillary waves on the surface of a liquid of finite depth. However, as a result, a new exact solution was obtained, which does not coincide with the solution of Kinnersley, although it is expressed in the same terms of Jacobi elliptic functions. The results of an independent numerical verification of the new solution are presented, confirming its reliability. The parametric analysis of the solution revealed, in particular, a nonmonotonic dependence of the wavelength and its amplitude on the Weber number. [ABSTRACT FROM AUTHOR]

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases. [ABSTRACT FROM AUTHOR]

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's 1 ψ 1 summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identity is proved. [ABSTRACT FROM AUTHOR]

The solutions of multi-breather and high-order rogue waves for the Hirota equation are systematically constructed on the elliptic function background, which were not previously studied. The spectral problem with the elliptic function potential is solved by the modified squared wave function approach and the traveling wave transformation. The fundamental solution of spectral problem is expressed in terms of theta functions. By the elementary Darboux transformation, the multi-breather solutions on different elliptic function backgrounds are constructed including the general breathers, Kuznetsov–Ma breathers and Akhmediev breathers, and their interaction patterns are exhibited by several examples. At branch points for breather solutions, higher-order rogue wave solutions on the elliptic function background are presented by the general Darboux transformation and some limit techniques. It is expected that these nonlinear dynamics of the breathers and rogue waves on the periodic background may help explain the isolated steep wave events in deep ocean or other nonlinear dispersive systems. [ABSTRACT FROM AUTHOR]

The nonlinear massive plane wave solution of the classical scalar field in the Higgs potential is revisited to study the mass generation and particle creation. In particular, by assuming that the Higgs system is in the slightly excited state in early universe and it is described by the nonlinear solution, we study the mass generation mechanism for massive vector bosons and a heavy fermion in the quantum field theory around the nonlinear massive classical field. The nonlinear massive classical solution gives the transition from the vacuum to a pair of vector bosons and fermions. We present the new formulae of the probability density of the production process for particles in the standard model of elementary particle physics. The probability densities of the particle productions vanish when the nonlinear massive solution reduces to the constant solution (the classical vacuum expectation value); while the probability densities are expressed as the function of the free parameter in the classical solution in general case. We discuss the behavior of the probability densities for the three oscillating modes in the classical solution. [ABSTRACT FROM AUTHOR]

In this article, we prove that the 〈 p , q 〉 condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions u r (x + ν t) and simply periodic solutions u s , 1 (x + ν t) of Kawahara equation are solitary wave solutions, while simply periodic solutions u s , 2 (x + ν t) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods. [ABSTRACT FROM AUTHOR]

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

The elliptically oscillating solutions in the Abelian–Higgs model are presented and the classical massive-dispersion-relation through the nonlinear dynamics is discussed. The generated massive-dispersion-relation including a field value of the scalar field is derived as the consequence of the equation of motions. We discuss the property of the new solutions and its Hamiltonian density. In addition, we calculate the electromagnetic property of the system, in particular, we derive the relation between the field value and the electric field and the electric current density. [ABSTRACT FROM AUTHOR]

In recent years, Rogers and Zudilin developed a method to write L -values attached to elliptic curves as periods. In order to apply this method to a broader collection of L -values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the L -values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at k > 1 explicitly as a period. As a side result, we give simple integral expressions for the generating functions of L (E , k) when even (or odd) k runs over positive integers. [ABSTRACT FROM AUTHOR]

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider's theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers. [ABSTRACT FROM AUTHOR]

Based on the values of the Weierstrass elliptic function ℘ (z | τ) at z = π τ / 2 , (π + π τ) / 2 , (π + π τ) / 4 , (π + 2 π τ) / 4 and the theory of modular forms on the arithmetic group Γ 0 (2) , we decompose θ 2 2 n (τ) as sum of Eisenstein series and cusp forms. Using the recurrence relation of ℘ (2 n) (z | τ) , we provide an algorithm to determine the exact form of these cusp forms. [ABSTRACT FROM AUTHOR]

This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are constructed explicitly. In particular, the solutions of four types are shown to be elliptic functions. Also, all transcendental meromorphic solutions that are locally injective or possess a Picard exceptional value are characterized for the remaining canonical type. [ABSTRACT FROM AUTHOR]

The article discusses and expands the known elements of the eight-vertex model, paying special attention to the parameterization of the matrix. The matrix values are interconnected with the knot through the braids and this model is valid on finite square lattices in two-dimensional space. A new solution of the parametrized eight-vertex model of free fermions with a complex version of elliptic functions, which is valid on a finite lattice, will be constructed. The range of applicability of the eight-vertex model with elements of the Jacobi elliptic function and the construction of a knot invariant on its basis is discussed by comparing the results obtained analytically for the model. The construction of the knot invariant using the Clebsch-Gordan coefficients and the main tool of statistical mechanics of the Yang-Baxter equation will be studied in detail. [ABSTRACT FROM AUTHOR]

Two cases of degeneration of elliptic functions are well known: degeneration into trigonometric functions and degeneration into hyperbolic functions. Approximations of elliptic functions in a neighborhood of a degeneration are usually examined by means of series in the modulus of the elliptic function. For applications of the theory of elliptic functions in the theory of dynamical systems, uniform approximations with respect to the modulus and the independent variable are important. This review contains methods for constructing uniform asymptotics. [ABSTRACT FROM AUTHOR]

The function $ \xi(z)$ is obtained from the logarithmic derivative function $\sigma(z)$. The elliptic function $ \wp(z) $ is also derived from the $ \xi(z) $ function. The function $ \wp(z) $ is a function of double periodic and meromorphic function on lattices region. The function $ \wp(z) $ is also double function. The function $ \varphi(z) $ meromorphic and univalent function was obtained by the serial expansion of the function $ \wp(z)$. The function $ \varphi(z) $ obtained here is shown to be a convex function.

In this article, we prove that the ⟨p,q⟩ condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions ur(x+νt) and simply periodic solutions us,1(x+νt) of Kawahara equation are solitary wave solutions, while simply periodic solutions us,2(x+νt) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.

We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function ℘ (z | τ) as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups Γ 0 (N) and Γ 1 (N) . [ABSTRACT FROM AUTHOR]

In this paper, a quasi-static quintication method for approximate periodic solution of strong nonlinear oscillators was presented. The basic idea of the method is to replace the original nonlinear oscillator with an equivalent cubic-quintic oscillator, whose stiffness constants are derived based on quasi-static equilibrium principles. The exact solution of the equivalent cubic-quintic oscillator was then applied to obtain the approximate solution to the original oscillator. The advantage of the present method is that its quintication constants (i.e. stiffness constants of the equivalent oscillator) are always in the form of elementary functions and are simpler than the corresponding constants derived by the Chebyshev quintication method. The present quintication method was verified using examples of strong nonlinear oscillators and was found to give very accurate results.

We investigate smooth one-parameter families of complex tori over the Riemann sphere. The main problem is to describe such families in terms of projections of their branch-points. Earlier we investigated the problem for the case where, for every torus of the family, there is only one point lying over infinity. Here we consider the general case. We show that the uniformizing functions satisfy a partial differential equation and derive a system of differential equations for their critical points, poles, and moduli of tori. Based on the system we suggest an approximate method allowing to find an elliptic function uniformizing a given genus one ramified covering of the Riemann sphere. [ABSTRACT FROM AUTHOR]

We investigate the problem of describing a one-parametric family of elliptic functions which uniformizes a given family of ramified coverings of the Riemann sphere with maximal possible ramification over infinity. We find a PDE for the family of functions and use it to deduce a system of ODEs for their critical points.

A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given.

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The following functional equation is solved: f (x 1 + z) ⋯ f (x 2 + z) f (x 1 + ⋯ + x s − 1 − z) = ϕ 1 (x) ψ 1 (z) + ⋯ + ϕ m (x) ψ m (z) , where x =(x1,...,xs−1), for the unknowns f , ψ j : ℂ → ℂ and ϕ j : ℂ s − 1 → ℂ for s ≥ 3 and m ≤ 4s − 5. [ABSTRACT FROM AUTHOR]

It is found the spectrum of a singular integral operator on the compound contour with elliptic function in the kernel. Explicit solution of the corresponding singular integral equation is constructed by applying its reduction to the boundary value problem on the Riemann surface. [ABSTRACT FROM AUTHOR]