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

In this article, we use the improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method with computerized symbolic computation to construct more new exact solutions of a generalized time-dependent variable coefficients KdV- mKdV equation. As a result, new generalized Jacobi elliptic function-like solutions, soliton-like solutions and trigonometric function solutions are obtained by using this method. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics. Key words: Exact solutions, generalized time-dependent variable coefficients KdV- mKdV equation, improved general mapping deformation method, Jacobi elliptic functions.

An extended Jacobi elliptic function rational expansion method and its application to ([formula omitted])-dimensional dispersive long wave equation

Jacobi elliptic function solutions for two variant Boussinesq equations

In this paper, the Jacobi elliptic function expansion technique is used to obtain the exact solutions of the sixth order (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. Modulation instability is also discussed for this equation. The main purpose is to find novel exact solutions to this equation by means of a finite series expansion of degree n in terms of Jacobi elliptic functions. Single and combined Jacobi elliptic function solutions are obtained. The JEFE method is found to be highly effective for exact analytical solutions of nonlinear partial differential equations and its flexibility permits the development of several variations for specific problem types. The studied equation is reduced to nonlinear ordinary differential equation of integer order by using the traveling wave transformation. We observe that the solutions obtained are precise, and include periodic wave solutions, quasi-periodic wave solutions and solitary waves. Oscillatory phenomena in systems such as plasma physics and optics can be described by periodic wave solutions. Quasi periodic solutions occur in complex systems with multiple interacting frequencies, which are important in turbulence and nonlinear resonance. Solitary waves (solitons) are stable, localized waves that are critical to fluid dynamics, nonlinear optics, and plasma physics, and that model stable wave propagation in many applications. In addition, graphical representations of some solutions are presented to show the direct viewing analysis of the solutions. The results confirm that the proposed technique is a powerful tool for solving a large variety of NPDEs in mathematical physics, and may have applications to other nonlinear evolution equations.

In this paper, we investigate the variable coefficients Sasa-Satsuma model, which can describe the propagation of a light pulse in a cylindrical fiber. We study this model and obtain rich solutions using two separate methods. We obtain analytical Weierstrass elliptic function solutions using the Weierstrass elliptic function expansion method. Some Jacobi elliptic function solutions are obtained using the modified Jacobi elliptic function expansion method. When the Jacobi elliptic function degenerates, we obtain the corresponding trigonometric, hyperbolic function solutions and periodic solutions. We also try to take the coefficients of the equation as some functions and obtain some more complicated exact solutions, which have not appeared in previous studies. Finally, we simulate some waveform diagrams of the solutions using the computer software Mathematica and obtain periodic waves, bright and dark soliton, double solitons and some complex periodic waves. With these waveform diagrams, we can observe the dynamical behavior of the solutions more clearly.

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.

This work aims to look into the dynamic research of coupled NLS-type equations with three components. The optical solitons, including the periodic function, trigonometric function, exponential function, solitary wave, and elliptic function solutions are built using the Jacobi elliptic function (JEF) method. The investigations will aid in improving comprehension of the soliton dynamics system’s overall illustration. Using Mathematica software, we visually represent some solutions found in 3D, contour, and 2D graphs for tangible demonstration and visual presentation. These results are helpful in optical fiber, signal processing and data transmission.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

With the aid of symbolic computation, the sinh-Gordon equation expansion method is extended to seek Jacobi elliptic function solutions of (2+1)-dimensional long wave-short wave resonance interaction equation, which describe the long and short waves propagation at an angle to each other in a two-layer fluid. As a result, new Jacobi elliptic function solutions are obtained. When the modulus m of Jacobi elliptic functions approaches 1, we also deduce the singular oliton solutions; while when the modulus m→0, we get the trigonometric function solutions. - PACS: 02.30.Jr, 03.40.Kf

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.

Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation

Chirped optical soliton and Jacobian elliptic function solutions in a metamaterial waveguide

New exact traveling wave solutions for the two-dimensional KdV–Burgers and Boussinesq equations

A new general Jacobi elliptic function rational expansion method, which is more general and powerful than the tanh method, the sine-cosine method, the Jacobi elliptic function method and the extended Jacobi elliptic function method, and the extended Jacobi elliptic function rational expansion method, is proposed to construct abundant rational formal exact doubly periodic wave solutions of systems of nonlinear wave equations. When the modulus m $\rightarrow$ $1$ or $0$, these rational formal doubly periodic solutions degenerate as solitary wave solutions and trigonometric function solutions. And the method can be automatically carried out in computer.

Stability of kink, anti kink and dark soliton solution of nonlocal Kundu–Eckhaus equation