On the absence of logarithmic singularities in the solutions of Lamé-type equations

The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations. These equations will be called Lamé-type equations. The question arises under what conditions the general solution of Lamé-type equations contains no logarithms. For this purpose, in the present paper, the solutions of Lamé-type equations are investigated and the conditions are found that make it possible to judge the presence or absence of logarithmic singularities in the solutions of the equations under study. An example of an equation with an irregular singularity having a solution with an logarithmic singularity is given, since the equation, defining it, has a multiple root.

On the Asymptotics and Distribution of Values of the Jacobi Theta Functions and the Estimate of the Type of the Weierstrass Sigma Functions

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.

Quantum oscillations between weakly coupled Bose–Einstein condensates: Evolution in a non-degenerate double well

In this paper, we discuss coherent atomic oscillations between two weakly coupled Bose–Einstein condensates that are energetically different. The weak link is notionally provided by a laser barrier in a (possibly asymmetric) multi-well trap or by Raman coupling between condensates in different hyperfine levels. The resultant boson Josephson junction dynamics is described by a double-well nonlinear Gross–Pitaevskii equation. On the basis of a new set of Jacobian elliptic function solutions, we describe the period of the oscillations as well as associated quantities and predict novel observable consequences of the interplay of the energy difference and initial phase difference between the two condensate populations.

Dispersive soliton solutions for the Salerno equation for the nonlinear discrete electrical lattice in the forbidden bandgaps

In this paper, the modified Jacobi elliptic function method is applied for Salerno equation which describes the nonlinear discrete electrical lattice in the forbidden bandgaps. Dark and bright solitons are obtained. Also, periodic solutions and periodic Jacobi elliptic function solutions are reported. Moreover, for the physical illustration of the obtained solutions, three-dimensional and two-dimensional graphs are presented.

Approximation of Elliptic Functions by Means of Trigonometric Functions with Applications

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

Applications of Jacobi’s Elliptic Functions in Forced Vibration in Nonlinear Rotor Dynamics

In this paper studied a rotary system with a nonlinearity, the equations of motion of which is one of the types of the Duffing’s equations with multidegree-of-freedom, and also consider the advantages of using the elliptic function method to solve problems of this type. The system studied in this paper is also distinguished by the fact that in addition to the rotor vibrations in elastic supports with a nonlinearity, the vibrations of the foundation are also taken into account. The article presents a comparative analysis of the numerical Runge-Kutta-Fehlberg's 4-order method with an error estimate of 5-order, the approximate analytical Van der Pol’s method and the elliptic function method proposed by the authors by comparing the obtained equations of motion of the system, as well as by comparing the constructed frequency response characteristic. From the results obtained, it follows that the method proposed by the authors can serve as a more accurate, general case of previously used approximate methods.