The Jacobi Elliptic Function Method for Solving Zakharov Equation

The Zakharov equation to describe the laser plasma interaction process has very important sense, this paper gives the solitary wave solutions for Zakharov equation by using Jacobi elliptic function method.

Addition Formulas of Leaf Functions and Hyperbolic Leaf Functions

Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov's method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.

Propagation of traveling wave solutions for nonlinear evolution equation through the implementation of the extended modi ed direct algebraic method

In this work,di erent kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D)nonlinear evolution equations(NEEs)through the implementation of the modi ed extended direct algebraic method.Bright-singular and dark-singular combo solitons,Jacobi's elliptic functions,Weierstrass elliptic functions,constant wave solutions and so on are attained beside their existing conditions.Physical interpretation of the solutions to the 3-D modi ed KdV-Zakharov-Kuznetsov equation are also given.

Relating Some Nonlinear Systems to a Cold Plasma Magnetoacoustic System

Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schrödinger equation.

Periodic Wave Solutions and Solitary Wave Solutions of the (2+1)-Dimensional Korteweg-de-Vries Equatio

In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different coefficients p, q and r in the elliptic equation. Then these solutions are coupled into an auxiliary equation and substituted into the (2+1)-dimensional KDV equation. As a result, a large number of complex Jacobi elliptic function solutions are obtained, and many of them have not been found in other documents. As , some complex solitary solutions are also obtained correspondingly. These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.

New physical structures and patterns to the optical solutions of the nonlinear Schrödinger equation with a higher dimension

It is commonly recognized that,despite current analytical approaches,many physical aspects of nonlinear models remain unknown.It is critical to build more efficient integration methods to design and construct numerous other unknown solutions and physical attributes for the nonlinear models,as well as for the benefit of the largest audience feasible.To achieve this goal,we propose a new extended unified auxiliary equation technique,a brand-new analytical method for solving nonlinear partial differential equations.The proposed method is applied to the nonlinear Schrödinger equation with a higher dimension in the anomalous dispersion.Many interesting solutions have been obtained.Moreover,to shed more light on the features of the obtained solutions,the figures for some obtained solutions are graphed.The propagation characteristics of the generated solutions are shown.The results show that the proper physical quantities and nonlinear wave qualities are connected to the parameter values.It is worth noting that the new method is very effective and efficient,and it may be applied in the realisation of novel solutions.

New Exact Solutions of Time Fractional Gardner Equation by Using New Version of F-Expansion Method

In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained.

New doubly-periodic solutions for the new integrable non local modified KdV equation

In this paper,new doubly-periodic solutions will be found for a new integrable nonlocal modified Korteweg-de Vries equation(mKdV).We adopt a technique using a travelling wave transformation which results in different solutions represented in Jacobi elliptic functions.

Soliton and other solutions to the (1+2)-dimensional chiral nonlinear Schrodinger equation

The(1+2)-dimensional chiral nonlinear Schr?dinger equation(2D-CNLSE)as a nonlinear evolution equation is considered and studied in a detailed manner.To this end,a complex transform is firstly adopted to arrive at the real and imaginary parts of the model,and then,the modified Jacobi elliptic expansion method is formally utilized to derive soliton and other solutions of the 2D-CNLSE.The exact solutions presented in this paper can be classified as topological and nontopological solitons as well as Jacobi elliptic function solutions.

On approach for obtaining approximate solution to highly nonlinear oscillatory system with singularity

This paper deals with further investigations of recently introduced so-called low-frequency pendulum mechanism,which represents an extended form of classical pendulum.Exact equation of motion,which is in Eksergian’s form,is a singular and highly nonlinear second order differential equation.It is transformed by suitable choice of a new “coordinates”into classical form of nonlinear conservative oscillator containing only inertial and restoring force terms.Also,due to the singularity of coefficient of governing equation that shows hyperbolic growth,Laurent series expansion was used.Using these,we derived a nonsingular nonlinear differential equation,for which there exists an exact solution in the form of a Jacobi elliptic function.By using this exact solution,and after returning to the original coordinate,both explicit expression for approximate natural period and solution of motion of mechanism were obtained.Comparison between approximate solution and solution is obtained by numerical integration of exact equation shows noticeable agreement.Analysis of impact of mechanism parameters on period is given.

Soliton Trains Induced by Adaptive Shaping with Periodic Traps in Four-Level Ultracold Atom Systems

It is well known that an optical trap can be imprinted by a light field in an ultracold-atom system embedded in an optical cavity,and driven by three different coherent fields.Of the three fields coexisting in the optical cavity there is an intense control field that induces a giant Kerr nonlinearity via electromagnetically-induced transparency,and another field that creates a periodic optical grating of strength proportional to the square of the associated Rabi frequency.In this work elliptic-soliton solutions to the nonlinear equation governing the propagation of the probe field are considered,with emphasis on the possible generation of optical soliton trains forming a discrete spectrum with well defined quantum numbers.The problem is treated assuming two distinct types of periodic optical gratings and taking into account the negative and positive signs of detunings(detuning above or below resonance).Results predict that the competition between the self-phase and cross-phase modulation nonlinearities gives rise to a rich family of temporal soliton train modes characterized by distinct quantum numbers.