Elliptic Functions Cn Research Articles

Rogue waves on the periodic background for a higher-order nonlinear Schrödinger–Maxwell–Bloch system

In this paper, we construct the rogue wave solutions on the background of the Jacobian elliptic functions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system. The Jacobian elliptic function traveling wave solutions as the seed solutions are considered. Through the approach of the nonlinearization of the Lax pair and Darboux transformation method, the rogue waves and the line rogue waves on the Jacobian elliptic functions dn and cn background are obtained, respectively.

Rogue waves for the (2+1)-dimensional Myrzakulov–Lakshmanan-IV equation on a periodic background

In this paper, the rogue wave solutions of the (2+1)-dimensional Myrzakulov–Lakshmanan (ML)-IV equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation (DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained. The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Rogue waves on the periodic wave background in the Kadomtsev–Petviashvili I equation

Rogue waves arising on the background of two families of periodic standing waves in the Kadomtsev–Petviashvili I (KPI) equation are investigated. By the nonlinearization of spectral problem and Darboux transformation approach, the rogue wave solutions of the KPI equation on the Jacobian elliptic functions dn and cn background are derived. Darboux transformation is also used to introduce trigonometric function periodic background for the higher-order rogue wave in the KPI equation. On the plane wave background, some rogue waves, breathers and hybrid waves in the KPI equation are obtained as a byproduct in this paper. Moreover, the results represented in this paper enrich the dynamics of (2+1) dimensional nonlinear wave equations.

Solitons, breathers and periodic rogue waves for the variable-coefficient seventh-order nonlinear Schrödinger equation

Through Darboux transformation (DT) method, Several nonlinear wave solutions of seventh-order variable-coefficient nonlinear Schrödinger (vcNLS) equation are obtained, including solitons, breathers and rogue periodic waves. When the coefficients are linear, parabolic and periodic functions, the parabolic, cubic and quasi-periodic solitons and breathers can be constructed. Then we investigate their effects on the solutions, the variation of the coefficients affects the shape of the solutions. On this basis, the interactions between two solitons are studied and the interactions between two types of breathers are constructed. Next, through the approaches of the nonlinearization of spectral problem and DT method, rogue waves on the background of the Jacobi elliptic functions dn and cn for a seventh-order vcNLS equation are constructed. When the coefficients are selected as linear, exponential and periodic functions, the nonlinear dynamics of two kinds of rogue periodic waves are analysed.

Rogue wave solutions on different periodic backgrounds for the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation

In this paper, we will present exact rogue wave solutions on different periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation, which can govern nonlinear spin dynamics of the (2+1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit. We will derive different spectral parameters and corresponding periodic eigenfunctions of Lax pair according to the nonlinearization of spectral problem. Via Darboux transformation method, first- and second-order rogue wave solutions will be derived under two different periodic seed solutions, which can be expressed by the Jacobian elliptic functions dn and cn. Meanwhile, dynamic features of these various solutions with different parameters on two kinds periodic backgrounds taking different elliptic modulus will be analytically and graphically analyzed.

Rogue waves of the AB system on the periodic background

In this paper, we study rogue wave solutions of the [Formula: see text] system under the background of Jacobian elliptic functions. First, we introduce the Jacobian elliptic functions dn and cn through traveling wave transformation. Next, we derive the expression of spectral parameters [Formula: see text] and rogue wave with periodic background by nonlinearization of the Lax pair and the Darboux transformation (DT) method. At last, we present the dynamic evolution process of the [Formula: see text] system in some figures.

Solution to a Damped Duffing Equation Using He’s Frequency Approach

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt–Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.

Rogue waves of the sixth-order nonlinear Schrödinger equation on a periodic background

In this paper, we construct the rogue wave solutions of the sixth-order nonlinear Schrödinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Breathers on elliptic function background for a generalized nonlinear Schrödinger equation with higher-order terms

A generalized nonlinear Schrödinger equation with higher-order terms, which is derived as a model for the nonlinear spin excitations in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin, is investigated. Firstly, in view of Riccati equations associated with the Lax pair, the Darboux transformation of a generalized nonlinear Schrödinger equation is presented. Secondly, the complicated Jacobi elliptic functions as seed solutions are considered so that much more fascinating solutions and dynamical properties can be obtained. Based on the above discussion, breathers in the presence of two kinds of Jacobian elliptic functions dn and cn are constructed. Finally, the dynamical properties of such solutions are analyzed by drawing the three-dimensional figures. The structures of these solutions are influenced by the higher-order operator. More importantly, the method provided in this paper can also be adopted to construct breathers on the elliptic functions background of other higher-order nonlinear integrable equations.

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.

Lax pair, Darboux transformation and rogue-periodic waves of a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics

For a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics, we derive a Lax pair, a Darboux transformation and two families of the periodic-wave solutions via the Jacobian elliptic functions dn and cn. We construct the linearly-independent and non-periodic solutions of that Lax pair, and substitute those solutions into the Darboux transformation to get the rogue-periodic-wave solutions. When the third-order dispersion or group velocity dispersion (GVD) or inter-modal dispersion (IMD) increases, the maximum amplitude of the rogue-periodic wave remains unchanged. From the rogue-dn-periodic-wave solutions, when the GVD decreases, the minimum amplitude of the rogue-dn-periodic wave decreases. When the third-order dispersion decreases, the minimum amplitude of the rogue-dn-periodic wave rises. Decrease of the IMD causes the period of the rogue-dn-periodic wave to decrease. From the rogue-cn-periodic-wave solutions, when the GVD increases, the minimum amplitude of the rogue-cn-periodic wave decreases. Increase of the third-order dispersion or IMD leads to the decrease of the period.

Dynamics of mechanical metamaterials: A framework to connect phonons, nonlinear periodic waves and solitons

Flexible mechanical metamaterials have been recently shown to support a rich nonlinear dynamic response. In particular, it has been demonstrated that the behavior of rotating-square architected systems in the continuum limit can be described by nonlinear Klein–Gordon equations. Here, we report on a general class of solutions of these nonlinear Klein–Gordon equations, namely cnoidal waves based on the Jacobi elliptic functions sn, cn and dn. By analyzing theoretically and numerically their validity and stability in the design- and wave-parameter space, we show that these cnoidal wave solutions extend from linear waves (or phonons) to solitons, while covering also a wide family of nonlinear periodic waves. The presented results thus reunite under the same framework different concepts of linear and non-linear waves and offer a fertile ground for extending the range of possible control strategies for nonlinear elastic waves and vibrations.

On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method

In this paper, a strongly nonlinear oscillator equation with higher-order nonlinear (cubic) restoring force, namely damped Duffing equation with higher-order nonlinearities, has been solved and analyzed analytically and numerically using some effectiveness and high-accuracy methods. Firstly, the physical oscillator system consisting of a spring–mass moves under cubic resorting force with constant spring is considered for getting damped Duffing equation. After that a semi-analytical solution is obtained in terms of the Jacobian elliptic function cn. Also, the numerical approximate solution using Runge–Kutta-4th (RK4) procedure is carried out and compared to the semi-analytical solution. It is found that the agreement between both solutions is very good especially if the coefficient of the cubic nonlinear term becomes larger than the coefficient of the linear one. However, we made some improvements to the semi-analytical solution using the numerical moving boundary method to get higher accuracy solution. Furthermore, the error analysis is estimated for both the semi-analytical solution, improved semi-analytical solution, and RK4 solution. The method that is used for getting the semi-analytical solution is elementary, i.e., allows to obtain only integrable case without using any Lie group theory. In another meaning, our approach gives exact/analytical solution for the integrable case such as undamped Duffing equation. Moreover, the t-intercepts and amplitude points are estimated precisely. Furthermore, the plasma oscillations are discussed in the framework of damped Duffing equation. Firstly, we reduced the fluid basic equations of pair-ion plasmas with Maxwellian electrons using reductive perturbation technique to a modified korteweg de-Vries Burgers (mKdVB) equation, and by using traveling wave transformation the mKdVB equation is converted to the damped Duffing equation. After that we compared between all mentioned solutions using plasma parameters.

Nonlinear periodic coupled Alfvén and ion-acoustic wave structures in plasmas

The nonlinear propagation of coupled Alfvén and ion-acoustic waves is investigated in homogeneous, magnetized and collisionless electron-ion plasma. The ion parallel motion is included in the model so that ion-acoustic wave couples with Alfvén wave in plasmas. In linear limit, the dispersion relation of coupled Alfvén and ion-acoustic waves is derived and its dispersion effects are investigated analytically as well as numerically using laboratory plasma parameters, given in the literature. Using reductive perturbation method, the Korteweg–de Vries (KdV) equation is derived for nonlinear coupled Alfvén and ion-acoustic waves in plasmas by applying appropriate periodic boundary conditions. The cnoidal wave solution of coupled Alfvén and ion-acoustic waves in terms of Jacobian elliptic function (cn) is obtained and its soliton solution is also investigated. The numerical plots of cnoidal wave and soliton structures with variations of plasma β (defined as the ratio of kinetic pressure to magnetic pressure) and obliqueness of the wave propagation with respect to external magnetic field are presented for illustration. It is found that low amplitude compressive structures of nonlinear coupled Alfvén and ion-acoustic waves exist in a magnetized plasma, which moves with sub-Alfvénic wave speed.

Solution for Boundedness, Time Period and Graphs of Rate Measures in Matlab of Undamped and Unforced Duffing Equation

The duffing equation is is a non -linear second order differential equation. In this paper my aim is to solve for boundedness and time period of the duffing equation (undamped ( and unforced / undriven ( )) by Jacobi elliptic functions cn and nc, nd, cd and dc and sn. Also I expressed the identities, the properties and graphs using MATLAB program of three Jacobi elliptic functions. I observed the three special cases solve for boundedness and time period. They are Case A : solve Cubic , Case B : Solve Initial Value Problem for Cubic duffing equation with Special Cases for boundedness and time period which cannot be solved by cosine function. So, it can be solved in the form of in terms of Jacobi elliptic functions cn and nc, dc and cd, nd having positive frequency and modulus in the interval [0, 1]. Case C : Solve Initial Value Problem for Linear . Also for practical and research purposes I introduced the graphs of velocity and acceleration of duffing equation (undamped ( and unforced / undriven ( )) using MATLAB.

THE POINT CHARGE OSCILLATOR: QUALITATIVE AND ANALYTICAL INVESTIGATIONS

We study the mathematical model of the point charge oscillator which has been derived by A. Beléndez et al. [2]. First we determine the global phase portrait of this model in the Poincaré disk. It consists of a family of closed orbits surrounding the unique finite equilibrium point and of a continuum of homoclinic orbits to the unique equilibrium point at infinity. Next we derive analytic expressions for the relationship between period (frequency) and amplitude. Further, we prove that the period increases monotone with the amplitude and derive an expression for its growth rate as the amplitude tends to infinity. Finally, we determine a relation between period and amplitude by means of the complete elliptic integral of the first kind K(k) and of the Jacobi elliptic function cn.

Rogue periodic waves of the modified KdV equation

Rogue periodic waves stand for rogue waves on a periodic background. Two families of travelling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions for the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the new solution constructed from the dn-periodic wave is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the new solution constructed from the cn-periodic wave is a rogue wave on the cn-periodic background, which generalizes the classical rogue wave (the so-called Peregrine’s breather) of the nonlinear Schrödinger equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains constant for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the spectral problem associated with the dn and cn periodic waves of the mKdV equation.

New solitons and periodic wave solutions for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

We consider a nonlinear Schrödinger type equation in (2+1) dimensions. The proposed equation describes the nonlinear spin dynamics of (2+1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit. We first construct three families of exact periodic solutions expressed in terms of Jacobi’s elliptic functions cn, sn and dn. We second consider the limit where the elliptic modulus approaches 1 to obtain bright and dark soliton solutions. Furthermore, we find a new type of soliton-like solution, illustrating the potentially rich set of wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Parametric conditions for the existence and uniqueness of exact solutions are presented. The derived structures of the obtained solutions offer a rich platform to study the nonlinear spin dynamics in magnetic materials.

Soluciones exactas a la ecuación del oscilador de helmholtz para circuitos eléctricos con no linealidad cuadrática

en este trabajo se obtiene una solución exacta a la ecuación de Helmholtz con condiciones iniciales y soluciones periódicas acotadas. Esta solución se expresa en términos de la función elíptica de Jacobi cn. Se utiliza esta solución exacta como una semilla para generar una buena aproximación a la solución trigonométrica analítica de la ecuación de Helmholtz para valores de módulo pequeño. Se resuelve numéricamente esta última ecuación y se compara esta solución numérica con la solución analítica obtenida a partir de la solución de la ecuación de Helmholtz. Se dan también algunos ejemplos ilustrativos

Response to “Comment on ‘Superposition of elliptic functions as solutions for a large number of nonlinear equations’” [J. Math. Phys. 56, 084101 (2015)

In a recent paper, Zhang and Li [J. Math. Phys. 56, 084101 (2015)] have doubted our claim that whenever a nonlinear equation has solutions in terms of the Jacobi elliptic functions cn(x, m) and dn(x, m), then the same nonlinear equation will necessarily also have solutions in terms of dn(x,m)±mcn(x,m). We point out the flaw in their argument and show why our assertion is indeed valid.