Terms Of Weierstrass Research Articles

Analytical representation and efficient computation of the effective conductivity of two‐phase composite materials

AbstractMany engineered materials display ordered or disordered microstructures. Such materials exhibit transport properties which are unmatched by their single‐phase homogeneous counterparts. These properties are obtained by the mixture of two or more phases typically characterized by a large contrast in their properties. For the development of these materials, it is critical to develop a robust computational framework in order to provide a fundamental understanding of how microstructure affects performance. This hinges on predicting their macroscopic properties, given the constitutive laws and spatial distribution of their constituents. To this end, this work presents a computational framework based on formulating periodic conduction transport problems in terms of boundary integral equations whose kernel is expressed in terms of Weierstrass zeta‐function. The components of the effective conductivity tensor are then sought in the form of power series expansions of a conductivity contrast parameter. To accelerate their convergence, these expansions are transformed into Padé approximants. Presently restricted to the case of two‐dimensional, two‐phase microstructures, this framework is shown to yield accurate results over the entire range of the contrast parameter. Representation of the kernel as a lattice sum allows the use the fast multipole method, thereby making computations significantly more efficient.

Optical soliton perturbation with Kudryashov’s law of refractive index by modified sub-ODE approach

This paper implements the sub-ODE method and a wide spectrum of solitons are recovered for Kudryashov’s law of refractive index. The self-phase modulation comprises of four nonlinear components of refractive index. The perturbation terms are all of Hamiltonian type and are considered with maximum intensity. The solutions are written in terms of Weierstrass’ elliptic functions and Jacobi’s elliptic function. With the modulus of ellipticity approaching zero or unity, soliton solutions emerge.

Solitons and conservation laws in magneto–optic waveguides with generalized Kudryashov’s equation

The modified sub–ODE approach secures optical soliton solutions in magneto–optic waveguides with generalized Kudryashov’s equation. The solutions are initially drafted in terms of Jacobi’s elliptic functions. The limiting process, when the modulus of ellipticity approaches zero or unity, the soliton solutions emerge. A few solutions in terms of Weierstrass’ elliptic functions are also revealed. Finally, the conservation laws are computed for the model using the multiplier approach.

Solitons and conservation laws in magneto-optic waveguides with polynomial law nonlinearity

Two prolific integration algorithms, extended Kudryashov's method and generalized Jacobi's elliptic function expansion, led to optical soliton solutions to magneto-optic waveguides maintaining polynomial law of nonlinear refractive index. Bright, singular and combo soliton solutions have thus emerged. A couple of solutions in terms of Weierstrass’ elliptic functions are also presented. Finally, the corresponding conservation laws are enumerated.

English

English

Existence of Meromorphic Solutions of First-Order Difference Equations

It is shown that if f(z+1)n=R(z,f),\\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}$$\\begin{aligned} f(z+1)^n=R(z,f), \\end{aligned}$$\\end{document}where R(z, f) is rational in f with meromorphic coefficients and deg _f(R(z,f))=n, has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if f(z+1)^n=R(z,f), where the assumption deg _f(R(z,f))=n has been discarded, has rational coefficients and a transcendental meromorphic solution f of hyper-order <1, then either f satisfies a difference linear or Riccati equation with rational coefficients, or the equation above can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions that are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz’ generalization of Malmquist’s theorem.

Parameter dependence and stability of guided TE-waves in a lossless nonlinear dielectric slab structure

The nonlinear Schrödinger equation is the basis of the traditional stability analysis of nonstationary guided waves in a nonlinear three-layer slab structure. The stationary (independent of the propagation distance) solutions of the nonlinear Schrödinger equation are used as ”initial data” in this analysis. In the present paper, we propose a method to investigate the dependence of these solutions on the experimental parameters and discuss their stability with respect to the parameters. The method is based on the phase diagram condition (PDC) and compact representation (in terms of Weierstrass’ elliptic function and its derivative) of the dispersion relation (DR). The problem’s parameters are constrained to certain regions in parameter space by the PDC. Dispersion curves inside (or at boundaries) of these regions correspond to possible physical solutions of Maxwell’s equations as ”start” solutions for a traditional stability analysis. Numerical evaluations of the PDC, DR, and power flow including their parameter dependence are presented.

Order parameter profiles in a system with Neumann – Neumann boundary conditions

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

Representation of solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of Weierstrass - and -functions and nonintegrability of the Hess case by quadratures

A method for the representation of Delaunay's solutions and some other particular solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of the Weierstrass - and -functions is put forward. The Hess case in the problem of the motion of a heavy rigid body is shown to be nonintegrable by quadratures. Bibliography: 24 titles.

Theory of TE-polarized waves in a lossless cubic-quintic nonlinear planar waveguide

TE-polarized electromagnetic waves, guided by a three-layer slab structure consisting of a central film with quartic permittivity placed between two half spaces with Kerr permittivity, are studied. Traveling-wave solutions of Maxwell's equations are expressed in terms of Weierstrass's elliptic function $\ensuremath{\wp}$. A general dispersion relation is derived and evaluated by using a phase diagram analysis. Emphasis is placed on the conditions of existence and solvability of the dispersion relation. Numerical results are presented.

Representation of solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of Weierstrass $\zeta$- and $\wp$-functions and nonintegrability of the Hess case by quadratures

Предложен способ представления решений Делоне и некоторых других частных решений задачи о движении тяжелого твердого тела в случае Ковалевской в $\zeta$- и $\wp$-функциях Вейерштрасса. Для случая Гесса задачи о движении тяжелого твердого тела доказана невозможность решения в квадратурах. Библиография: 24 названия.

Weierstrass traveling wave solutions for dissipative Benjamin, Bona, and Mahony (BBM) equation

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified Benjamin, Bona, and Mahony (BBM) equation by viscosity. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant makes the equation integrable in terms of Weierstrass ℘ functions. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of ℘ functions, from which all the other known solutions can be obtained via simplifying assumptions. Using ODE (ordinary differential equations) analysis we show that the traveling wave speed is a bifurcation parameter that makes transition between different classes of waves.

On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system

A 2+1-dimensional version of a non-isothermal gas dynamic system with origins in the work of Ovsiannikov and Dyson on spinning gas clouds is shown to admit a Hamiltonian reduction which is completely integrable when the adiabatic index γ = 2. This nonlinear dynamical subsystem is obtained via an elliptic vortex ansatz which is intimately related to the construction of a Lax pair in the integrable case. The general solution of the gas dynamic system is derived in terms of Weierstrass (elliptic) functions. The latter derivation makes use of a connection with a stationary nonlinear Schrödinger equation and a Steen–Ermakov–Pinney equation, the superposition principle of which is based on the classical Lamé equation.

Periodic Solutions to KdV–Burgers–Kuramoto Equation

In this paper, a new special ansatz solution, where ellipticequation satisfied by elliptic functions is taken as anintermediate transformation, is applied to solve theKdV-Burgers-Kuramoto equation, and many more new periodicsolutions are obtained, including solutions expressed in terms ofJacobi elliptic functions, solution expressed in terms ofWeierstrass elliptic function.

Squared wavefunctions approach to periodic solutions of vector nonlinear Schr dinger equation

We develop a method of squared wavefunctions for the vector nonlinear Schrodinger equation. The squared wavefunctions of the octet representation of SU(3) group give periodic solutions in terms of Weierstrass' elliptic functions. Specific limits of the obtained solution are the plane wave, the soliton and cnoidal waves, which were previously obtained using the ansatz of stationary motion.

Lump scattering on the torus

Head-on collisions between two solitons in the pure \(\) model on a flat torus are investigated via numerical simulations. The charge-two lumps, written out in terms of Weierstrass' elliptic \(\)-function, are found to scatter at 90\(\). The phenomenon of singularity formation is also seen.

Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation.

A subset of the exact analytical traveling-wave solutions of the equation i${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}}$+${\mathrm{\ensuremath{\Psi}}}_{\mathit{zz}}$=${\mathit{a}}_{1}$\ensuremath{\Psi}|\ensuremath{\Psi}${\mathrm{|}}^{2}$+${\mathit{a}}_{2}$\ensuremath{\Psi}|\ensuremath{\Psi}${\mathrm{|}}^{4}$ is presented in compact form. The solutions are expressed in terms of Weierstrass's elliptic function \ensuremath{\wp} and include periodic and solitary-wave-like solutions. The algebraic conditions for both cases are given. Examples are presented for simple cases. \textcopyright{} 1996 The American Physical Society.

On the theory of TE-polarized waves guided by a nonlinear three-layer structure

TE-polarized electromagnetic waves guided by a three-layer structure consisting of a film surrounded by semi-infinite media (all media are characterized by a Kerrlike dielectric function) are investigated by expressing the solution of the field equations in terms of Weierstrass' elliptic functions. Evaluation leads to a universal dispersion relation and its solutions and a universal expression for the power flow. Numerical results are presented for the effective wave number as a function of the intensity of the electric field at the lower surface of the nonlinear film, for various profiles of the field intensity, and for the power flow as a function of the effective wave number.

On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab

For the transverse electric polarization case (TE) we present a treatment of the optical reflectivity and transmissivity of a slab whose dielectric coefficient is a real valued function of the light intensity. If this function is numerically integrable with respect to the light intensity, our treatment can serve as an algorithm for a numerical solution of the nonlinear wave equation. If the dielectric function is proportional to the intensity, an analytical solution of the cubic wave equation is given for the electric field strength and for the phase of the field in terms of Weierstrass' elliptic functions and first elliptic theta functions, respectively. Evaluating this solution by means of a computer algebra system yields the reflectivity, transmissivity and phase dependency on the incident field intensity and on parameters characteristic for the problem. Certain combinations of the parameters lead to bistable and multivalued behavior. The solution found is used to determine the relative extrema of the reflectivity and the critical values of the thickness and of the incident intensity. The results are a generalization of linear optics results. Application of the analysis to the cubic-quintic wave equation yields the general analytic solution which is used to detemine the reflectivity of a semi-infinite nonlinear medium.

Vacuum structure of the electroweak theory in high magnetic fields

In the electroweak theory one can reach the unbroken phase SU(2) × U Y (1) by pumping enough magnetic energy into the system. The whole energy is then carried by the fields associated with U Y (1), whereas the fields corresponding to SU(2) are in a vacuum state. We show that the vacuum is non-trivial in the sense that it consists of a condesate of zero-field twists which arise in a smooth way from a condensate of vortex lines existing in the broken phase. An explicit vacuum solution is constructed in terms of Weierstrass' elliptic function. The results mentioned above are strictly valid only in the case when the Z mass equals the Higgs mass.