Jacobian Elliptic Sine Research Articles

A note on the Jacobian elliptic sine function

Inequalities involving Jacobian elliptic sine function sn are established. Those results are obtained using log-convexity of the function 1/sn(t) (t∈(0, K]) (see Proposition 2.1), where K is Legendre's complete elliptic integral of the first kind.

RESHAPING MODULATION PROFILE TO DECREASE AND INCREASE THE THRESHOLD FOR CHAOTIC BEHAVIOR IN A PUMP-MODULATION Nd:YVO4 LASER

This study investigates the changes of the threshold of chaotic behavior in a pump-modulation Nd:YVO4 laser created by varying the shape of the modulation profile. The threshold for chaotic behavior can be increased and decreased when the modulation profiles are reshaped by triangular wave and Jacobi elliptic sine functions, respectively. Experimental results show that sinusoidal-wave, square-wave, and triangular-wave modulations have different chaotic thresholds, agreeing with numerical results. In addition to the dynamic parameters of modulation frequency and modulation depth, the shape of modulation is an important parameter for controlling the nonlinear dynamics in pump-modulation lasers.

ANALYTICAL STUDY ON NONLINEAR DIFFERENTIAL–DIFFERENCE EQUATIONS VIA A NEW METHOD

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

Improvement to the averaging method using the Jacobian elliptic function

The averaging method was improved by using the Jacobian elliptic sine (sn), cosine (cn) and delta (dn) functions as generating solutions in order to obtain a highly accurate periodic solution for a strongly nonlinear dynamical system. The proposed method can be applied to a relatively general nonlinear system based on the single degree-of-freedom Duffing equation. Two methods of stability analysis were applied and evaluated for the approximate solutions obtained by the proposed method. The approximate solutions obtained by the proposed method for practical examples were compared to the solutions obtained by the shooting method. The results confirmed that the proposed method provides a more accurate solution and more accurate stability analysis results than those obtained by the conventional averaging method that uses trigonometric functions as the generating solution.

Exact solution of a quadratic nonlinear oscillator

The exact solution of a quadratic nonlinear oscillator (QNO) is presented by using the Jacobian elliptic sine function. The constants in the solution are expressed in terms of the initial conditions. The formulas for the period of oscillation T and amplitude B (for x ⩽ 0 ) are also given. This exact solution can serve as a benchmark solution for various approximate solutions. It also can be utilized for educational purpose.

Nonlinear waves and periodic solution in finite deformation elastic rod

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.

Applications of Jacobi Elliptic Function Expansion Method for Nonlinear Differential-Difference Equations

The Jacobi elliptic function expansion method is extended to derive theexplicit periodic wave solutions for nonlineardifferential-difference equations. Three well-known examples arechosen to illustrate the application of the Jacobi ellipticfunction expansion method. As a result, three types of periodicwave solutions including Jacobi elliptic sine function, Jacobielliptic cosine function and the third elliptic function solutionsare obtained. It is shown that the shock wave solutions andsolitary wave solutions can be obtained at their limit condition.

Combinability of Travelling Wave Solutions to Nonlinear Evolution Equation

First, applying the Jacobi elliptic sine function expansion, basic travelling wave solutions of some nonlinear evolution equations are obtained. Second, according to the formal invariance of nonlinear evolution equations, the sister travelling wave solutions, which have the same wave speed as the basic ones, are derived. Finally, we show that a suitable linear combination of these two solutions results in a combined travelling wave solution, whose wave speed is different from those of the basic and sister travelling wave solutions. The combination of travelling wave solutions to nonlinear evolution equations must satisfy certain conditions. - PACS: 03.65.Ge.

Study on High Performance Averaging Method for System Based on Duffing Equation with Strong Nonlinearity. In Cases that the Jacobian Elliptic Sine and Delta Functions are Used.

The improved averaging method suggested in our previous report is developed in order to obtain a highly accurate periodic solution for a strongly nonlinear system. In this reports, the Jacobian elliptic sine and delta functions are applied as generating solutions for harmonically excited Duffing-type equations with softening spring and snap through spring, respectively. Two types of stability criterion for approximate solutions obtained by the present method are also discussed. The present method is applied to the systems with some kind of nonlinear spring. By comparing with the very accurate numerical solutions obtained by applying the shooting method, it is confirmed that the present method gives more accurate solution and more accurate results for stability of the solution than those obtained by the traditional averaging method using the trigonometric function as a generating solution.

Effects of one- and three-dimensional inhomogeneities on the wave spectrum of multilayers with finite interface thicknesses

To describe a partially randomized multilayer structure with arbitrary thicknesses of the interfaces between layers, we introduce a model in which the dependence of a material parameter along the axis of such a superlattice is described by a Jacobian elliptic sine function with a random spatial modulation of its period. Both one- and three-dimensional inhomogeneities of the period are considered. We develop the correlation function for this model, and investigate the dispersion law and damping of averaged waves in this superlattice. The dependencies of the widths of the gaps in the spectrum and the damping at the boundaries of all odd Brillouin zones, on the thicknesses of the interfaces, and on the dimensionality, intensity, and correlation wave number of the inhomogeneities are found. It is shown that experimental investigations of the widths of the gaps and damping for several Brillouin zones could permit, in principle determining all parameters of the superlattice as well as the parameters of the inhomogeneities from these spectral characteristics.

Wave spectrum of multilayers with finite thicknesses of interfaces

To describe a multilayer structure with arbitrary thicknesses of the interfaces between layers, we introduce a model in which the dependence of a material parameter along the axis of such a superlattice is described by a Jacobian elliptic sine function. Depending on the value of the modulus \ensuremath{\kappa} of the elliptic function, the model describes the limiting cases of multilayers with sharp interfaces $(\ensuremath{\kappa}=1,d/l=0,$ where d is the thickness of the interface, l is the period of the superlattice) and of sinusoidal superlattices $(\ensuremath{\kappa}=0,d/l=1/4),$ as well as all intermediate situations. We investigate the wave spectrum in such a superlattice. The dependences of the widths of the gaps in the spectrum at the boundaries of all odd Brillouin zones on the ratio $d/l$ are found. It is shown that the thicknesses of the interfaces can be determined if the experimental value of the relation between the widths of the first gap $\ensuremath{\Delta}{\ensuremath{\nu}}_{1}$ and any other gap $\ensuremath{\Delta}{\ensuremath{\nu}}_{n}$ is known.

A type of B�cklund-like invariance transformation for a class of second-order ordinary differential equations

Invariance conditions which link a class of ordinary differential equations with a type of Backlund-like transformation are developed and solved. These nonpoint transformations are used to obtain ‘super-position principles’ and as an illustration of their consequences a derivation of the duplication law for the Jacobian elliptic sine function is sketched. The group properties of these transformations are identified in all but one case and in one case it leads to a ‘double group’.