Jacobi Elliptic Function Expansion Method Research Articles

New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions

In this study, new families of analytical exact solutions of the space fractional Korteweg-de Vries (KdV) equation are presented. Here, the fractional derivative is considered in conformable sense. By utilizing the Jacobi elliptic function expansion method, the solutions are obtained in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions are obtained and some solutions of this equation are demonstrated.

Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation

Abstract In this work we study the Korteweg-de Vries-Benjamin-Bona-Mahony (KdV-BBM) equation, which describes the two-way propagation of waves. Using Lie symmetry method together with Jacobi elliptic function expansion and Kudryashov methods we construct its travelling wave solutions. Also, we derive conservation laws of the KdV-BBM equation using the variational derivative approach. In this method, we begin by computing second-order multipliers for the KdV-BBM equation followed by a derivation of the respective conservation laws for each multiplier.

Optical solitons, nonlinear self-adjointness and conservation laws for Kundu–Eckhaus equation

In this article, Kundu–Eckhaus equation (KE) is studied from the perspective of modified tanh-coth method (MTC), extended Jacobi elliptic function expansion method (EJEF), Lie symmetry analysis, nonlinear self-adjointness and conservation laws (Cls). New soliton solutions like combined dark-bright, dark, periodic wave and singular soliton solutions are obtained. The equation is found to be a nonlinear self-adjoint, we construct the Cls using the new conservation theorem presented by Ibragimov. Physical interpretation for some of the obtained solutions are illustrated in Figures.

Exact solutions for the Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equations

In this article, exact solutions of Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equations have been obtained by using the Jacobian elliptic function expansion method. We have used the Hermite transform for transforming the Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equation into a deterministic partial differential equation. Also, we have applied the inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.

The investigation of soliton solutions of the coupled sine-Gordon equation in nonlinear optics

The coupled sine-Gordon equation, which can be used to describe the propagation of an optical pulse in fibre waveguide, has been investigated analytically. The hyperbolic and trigonometric function solutions are constructed by the extended expansion method, and then the Jacobi elliptic, hyperbolic and trigonometric function solutions are derived with the help of the extended Jacobi elliptic function expansion method.

Soliton solutions for Davydov solitons in α-helix proteins

The propagation equation for describing Davydov solitons in α-helix proteins has been investigated analytically. There are seven integration tools to extract analytical soliton solutions. They are the Ricatti equation expansion approach, ansatz scheme, improved extended tanh-equation method, the extend exp(−Ψ(τ))-expansion method, the extended Jacobi elliptic function expansion method, the extended trial equation method and the extended G'/G− expansion method.

Explicit solitons in the parabolic law nonlinear negative-index materials

This paper deals with the existence of explicit ultra-short solitons in the negative-index materials with third-order dispersion and higher-order nonlinearities. Four integration algorithms, that are the improved modified extended tanh-function method, the extended trial equation method, the extended Jacobi elliptic function expansion method and the \(\exp \left( -\Phi \left( \eta \right) \right) \)-expansion method, are employed to extract analytical traveling wave solutions. The presented results show that exact solitons can exist in this physical model.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Mathematical Physics

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to some nonlinear evolution equations which play an important role inmathematical physics.

New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method

In this paper, the Jacobi elliptic function expansion method is proposed for the first time to construct the exact solutions of the time conformable fractional two-dimensional Boussinesq equation and the combined KdV-mKdV equation. New exact solutions are found. This method is based on Jacobi elliptic functions. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.

New solitary solutions in FPU-β atom chain

Solitons in one-dimensional FPU-[Formula: see text] lattice chains were studied by the Jacobi elliptic function expansion method under the condition of quasi-continuum approximation. It is found that there exhibits a novel nonlinear elementary excitation, i.e. exact solitary solutions. On the basis of the above, the influence of the nonlinear intensity on the solitons was analyzed. It is found that the modulation amplitude of soliton is related closely to the nonlinear parameter also.

General Jacobi elliptic function expansion method applied to the generalized (3 + 1)-dimensional nonlinear Schrödinger equation

We generalize the Jacobi elliptic function expansion method used to solve the (3 + 1)-dimensional nonlinear Schrodinger equation for the case of an arbitrary inverse of an elliptic integral. Among the obtained solutions are functions based on the Weierstrass elliptic function and the inverses of Carlson’s elliptic integrals.

New Analytic Solutions for the (N + 1)-Dimensional Generalized Boussinesq Equation

In this paper, the generalized Jacobi elliptic functions expansion method with computerized symbolic computation are employed to investigate explicitly analytic solutions of the (N + 1)-dimensional generalized Boussinesq equation. The exact solutions to the equation are constructed analytically under certain circumstances, some of these solutions are degenerated to soliton-like solutions and trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic function solutions tends to 0 and 1, which shows that the applied method is more powerful and will be used in further works to establish more entirely new exact solutions for other kinds of higher-dimensional nonlinear partial differential equations in mathematical physics.

An improved Jacobian elliptic function expansion method to find new traveling wave solutions for a class of nonlinear wave equations

An improved Jacobian elliptic function expansion method to find new traveling wave solutions for a class of nonlinear wave equations

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution**Supported by Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant No. 41421005, National Natural Science Foundation of China under Grant Nos. 41376030, 41376029, 41476019, NSFC–Shandong Joint Fund for Marine Science Research Centers Grant (U1406401), Special Funds for Theoretical Physics of

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt–Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.

A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

The first elliptic function equation is used in this article to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing the nonlinear low-pass electrical lines are obtained with the aid of computer algebraic system Maple. Based on Kirchhoff’s law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.

Extended Jacobian Elliptic Function Expansion Method and its Applications for Solving some Nonlinear Evolution Equations in Mathematical Physics

Extended Jacobian elliptic function expansion method is employed to find the exact traveling wave solutions involving parameters for nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that extended Jacobian elliptic function expansion method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Biology

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to Dynamical system in a new Double-Chain Model of DNA and a diffusive predator-prey system which play an important role in biology.

Exact Solutions for Some Fractional Differential Equations

The extended Jacobi elliptic function expansion method is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential equations.

Exact Traveling Wave Solutions for Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells and Nano-Ionic Currents of MTs

In this work, the extended Jacobian elliptic function expansion method is used as the first time to evaluate the exact traveling wave solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to nano-solitons of ionic waves propagation along microtubules in living cells and nano-ionic currents of MTs which play an important role in biology.