New exact solutions of a nonlinear long–short wave interaction system

In this paper, new exact solutions of the time fractional KdV–Khokhlov–Zabolotskaya–Kuznetsov (KdV–KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann–Liouville derivative is used to convert the nonlinear time fractional KdV–KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV–KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV–KZK equation.

This paper deals with the soliton solutions for beam movement within a multimode optical fiber featuring a parabolic index shape. It is considered that a Two-Dimensional Nonlinear Schrödinger Equation (2D-NLSE) with an instantaneous Kerr nonlinearity of the kind can represent the beam dynamics. Nonlinear Multimode Optical Fibers (MMFs) of this kind are gaining popularity because they provide novel approaches to control the spectral, temporal, and spatial characteristics of ultrashort light pulses. We gain the optical soliton solutions for the nonlinear evolution beam dynamics using the Jacobi Elliptic Function Expansion (JEFE) method. The exact analytical solution of Nonlinear Partial Differential Equations (NLPDEs) can be achieved with wide application using the effective JEFE approach. These solutions are obtained in the form of dark, bright, combined dark–bright, complex combo, periodic, and plane wave solutions. Additional solutions for Jacobi elliptic functions, encompassing both single and dual function solutions, have been acquired. This approach is based on Jacobi elliptic functions, which will provide us the exact soliton solutions to nonlinear problems. Additionally, we will analyze the Modulation Instability (MI) for the underlying model. Moreover, we show the physical behavior of the beam propagation in a multimode optical fiber the three-dimensional, two-dimensional, and their corresponding contour plots are dispatched using the different values of parameters.

In this paper, we introduce the Noether approach for studying the Peyrard–Bishop (PB) DNA dynamic model equation. The Jacobi elliptic function expansion (JEFE) method is used to analyze this model. The obtained soliton solutions are hyperbolic, trigonometric, and rational functions. The dynamics of the obtained solutions conform to the properties of shock waves, periodic waves, solitary waves, double periodic waves, and rational solutions. A comparison between the results obtained by the JEFE method and the results existing in the literature shows that our results are more general, and novel, and have never been reported earlier. The wave profiles are investigated by Wolfram Mathematica simulations for the physical wave behavior.

The JEFE method and periodic solutions of two kinds of nonlinear wave equations

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by using our extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and another three families of new doubly periodic solutions (Jacobian elliptic function solutions) are found again by using a new transformation, which and our extended Jacobian elliptic function expansion method form a new method still called the extended Jacobian elliptic function expansion method. The new method can be more powerful to be applied to other nonlinear differential equations.

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

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.

Nonlinear pulse propagation for novel optical solitons modeled by Fokas system in monomode optical fibers

In this paper, the modified Kudryashov method is proposed to solve fractional differential equations, and Jumarie’s modified Riemann-Liouville derivative is used to convert nonlinear partial fractional differential equation to nonlinear ordinary differential equations. The modified Kudryashov method is applied to compute an approximation to the solutions of the space-time fractional modified Benjamin-Bona-Mahony equation and the space-time fractional potential Kadomtsev-Petviashvili equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions. This method is powerful, efficient, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

Highly dispersive singular optical solitons with Kerr law nonlinearity by Jacobi's elliptic ds function expansion

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system

Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey–Stewartson-type equation via a new method

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

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota–Satsuma KdV system