Jacobian Elliptic Research Articles

Soliton and Other Function Solutions of The Potential KdV Equation with Jacobi Elliptic Function Method

The current study is concerned analytical solutions of the nonlinear potential KdV equation. Here, we implemented the Jacobi elliptic function method for soliton, hyperbolic and periodic solutions. Moreover, we illustrate our results with some graphs.

Solving a Class of Nonlinear Evolution Equations Using Jacobi Elliptic Functions

Nonlinear evolution equations play a crucial role in many applied scientific disciplines. Finding the exact solutions to such equations can contribute to a wide variety of applied disciplines. To be able to calculate exact periodic solutions of nonlinear evolution equations, this paper discusses the expansion of the Jacobi elliptic function and finds the proper expansion of the function. Additionally, by analyzing the applicable conditions of the function according to its specific properties, we find the exact periodic solutions to nonlinear evolution equations with constant coefficients and variable coefficients that satisfy the applicable conditions of the Jacobi elliptic function.

Variety of optical soliton solutions via sub-ODE approach to embedded soliton generating model in quadratic nonlinear media

This paper studies the soliton solutions for Embedded soliton (ES) generating model with [Formula: see text] nonlinear susceptibility. The bright, rational, Jacobi elliptic, periodic, dark, Weierstrass, hyperbolic solitary wave solutions will be found with the aid of sub-ODE technique under certain conditions. The main objective behind the sub-ODE method is to find the wave solutions of a complex model with the help of simple and solvable ODEs called sub-ODEs. The resulting wave solutions are presented graphically for suitable values of different parameters.

On Travelling Wave Solutions of Dullin-Gottwald-Holm Dynamical Equation and Strain Wave Equation

In this study, extended trial equation method (ETEM) is implemented to obtain exact solutions of the Dullin-Gottwald-Holm Dynamical equation (DGHDE) and the strain wave equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic, periodic wave solutions and hyperbolic function solutions of these equations via ETEM. Then, we present results that we obtained by using this method.

Some advanced chirped pulses for generalized mixed nonlinear Schrödinger dynamical equation

In an optical fibre, we investigate the propagation properties of nonlinear periodic waves (PW) for a generalized mixed nonlinear Schrödinger (GMNLS) equation. The Jacobi elliptic (JE) solutions will be used to find the nonlinear chirp. The chirp varies with two intensity-dependent chirping terms are included in the linear section of the pulse chirp. The presence of the newly discovered periodic waves will be discussed in terms of fibre parameter conditions. The long-wave limit generates a wide range of solitary pulse forms, including topological, non topological, dark, kink, hyperbolic and periodic solitary waves (SW). Our findings show that for periodic and solitary waves, a nonlinear chirp obtains. Finally, under finite perturbations, the stability of these nonlinearly chirped solutions will be quantitatively investigated.

The Weierstrass and Jacobi elliptic solutions along with multiwave, homoclinic breather, kink-periodic-cross rational and other solitary wave solutions to Fornberg Whitham equation

We investigate multiwave, homoclinic breather, M-shaped, Kink cross-rational, periodic cross kink solutions for the nonlinear generalized Fornberg–Whitham (gFW)-equation by using appropriate polynomial function schemes. We also obtain some interactions for governing model including; interaction of M-shaped soliton with single exp-function, interactional solution with double exp-functions. In addition, we obtain dark, periodic soliton, Weierstrass and Jacobi elliptic, bell and kink type solutions for gFW-equation via newly developed sub-ODE method.

The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

This article considers the stochastic fractional Radhakrishnan-Kundu-Lakshmanan equation (SFRKLE), which is a higher order nonlinear Schrödinger equation with cubic nonlinear terms in Kerr law. To find novel elliptic, trigonometric, rational, and stochastic fractional solutions, the Jacobi elliptic function technique is applied. Due to the Radhakrishnan-Kundu-Lakshmanan equation’s importance in modeling the propagation of solitons along an optical fiber, the derived solutions are vital for characterizing a number of key physical processes. Additionally, to show the impact of multiplicative noise on these solutions, we employ MATLAB tools to present some of the collected solutions in 2D and 3D graphs. Finally, we demonstrate that multiplicative noise stabilizes the analytical solutions of SFRKLE at zero.

Investigation of Nonlinear Forced Vibrations of the “Rotor-Movable Foundation” System on Rolling Bearings by the Jacobi Elliptic Functions Method

The paper considers a rotor system with a nonlinear characteristic. Its equations of motion are a kind of Duffing class equations with multiple degrees of freedom. The paper shows the advantage of using the method of elliptic functions for solving problems of this type. This method enables us to take into account not only vibrations of the rotor installed in elastic nonlinear supports, but also vibrations of the foundation. A comparative analysis of application of the method of elliptic functions proposed by the authors is carried out by comparing the derived equations of motion of the system, as well as by comparing the obtained amplitude-frequency characteristics with the results obtained by the numerical Runge–Kutta–Fehlberg’s 4-order method and the approximate analytical Van der Pol method. The regions of resonant frequencies for superharmonic oscillations and bifurcation regimes are determined. It is concluded that the method proposed by the authors is a more accurate and general case than the previously used approximate methods.

Weierstrass and Jacobi elliptic, bell and kink type, lumps, Ma and Kuznetsov breathers with rogue wave solutions to the dissipative nonlinear Schrödinger equation

This paper exposes Weierstrass elliptic, Jacobi elliptic, Bell type, kink type, bright soliton, periodic and some other soliton solutions for (1 + 1)-dimensional dissipative nonlinear Schrödinger equation (d-NSLE) by applying newly developed sub-ODE method. The d-NLSE is applied to model the dissipative self-modulating monochromatic waves with dispersion. The lump solutions, lump with one kink, lump periodic, interaction with kink, manifold periodic, Ma breather, Kuznetsov-Ma breather and rogue wave solutions for governing model are derived via symbolic computation with the appropriate transformation function technique.

Constructing new solitary wave solutions to the strain wave model in micro-structured solids

In the present article, the improved modified extended tanh function method is implemented to investigate the strain wave model which governs the wave propagation in micro-structured solids. Solitary and other wave solutions are provided for this model such as singular and bright solitary type solutions. In addition Kink, periodic, Jacobi elliptic, hyperbolic, exponential and Weierstrass elliptic type solutions are constructed. To show the physical characteristics of the raised solutions, graphical illustration of some solutions is presented.

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

Generalized Jacobi Elliptic Solutions for the KdV Equation with Dual Power Law Non-Linearity and for the Power Law KdV-Burger Equation with the Source

Generalized Jacobi Elliptic Solutions for the KdV Equation with Dual Power Law Non-Linearity and for the Power Law KdV-Burger Equation with the Source

Computational extracting solutions for the perturbed Gerdjikov-Ivanov equation by using improved modified extended analytical approach

In this work, the improved modified extended tanh function method will be applied to study the perturbed Gerdjikov-Ivanov equation (PGI) which describe the dynamics of the soliton in optical fibers. New exact solutions for the PGI equation will be introduced including Weierstrass elliptic, Jacobi elliptic, dark and bright solitons, singular periodic and rational type solutions. Moreover, for the physical illustration of the obtained solutions, 3D and 2D graphs are presented.

Pressure wave characteristics in a bubble-liquid mixture via Kudryashov–Sinelshchikov equation

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system.

Weierstrass and Jacobi elliptic solutions with some new dromions to Maccari system

This paper studies Attilio Maccari (AM) system for some new solitary wave solutions. We used sub-ODE scheme to find kink shape, bell type, Weierstrass and Jacobi elliptic, hyperbolic, periodic and other solitary wave solutions under some constraint conditions. We will also present our results graphically in distinct dimensions.

New exact solitary wave solutions for the 3D-FWBBM model in arising shallow water waves by two analytical methods

In this article, we investigate the exact solitary wave solutions to the 3D fractional Wazwaz-Benjamin- Bona-Mahony (3D-FWBBM) equation in emerging shallow-water waves. Different kinds of solutions such as hyperbolic, trigonometric, Jacobi elliptic, and rational function including some special known solitary waves like shock, singular, combo shock-solitary wave, and multiple soliton solutions are achieved by the utilization of the sound computational integration tools namely the new Φ6-model expansion method and modified direct algebraic method (MDAM). In addition, we also secure mixed combined solitons and singular periodic solutions and the constraint conditions also emerge which provide the guarantee to the reported solutions. Some results are figured out graphically in 3D, 2D, and their corresponding contour profiles by selecting appropriate parametric values to anticipate the wave dynamics of the solutions. The obtained outcomes are more general and fresh to show that these applied methods are concise, direct, elementary, and can be imposed in more complex phenomena with the assistance of symbolic computations.

New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions

The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, by using a direct method based on the Jacobi elliptic functions, the exact solutions of the space-time fractional symmetric regularized long wave (SRLW) equation have been obtained. The elliptic function solutions of a nonlinear ordinary differential (auxiliary) equation $\left({dF}/{d \xi}\right) ^{2} = PF^{4} (\xi)+QF^{2} (\xi) + R$ have also been examined. Besides, the solutions have been found in general form including rational, trigonometric and hyperbolic functions. Moreover, the complex valued solutions, periodic solutions, and soliton solutions, have also been gained. Some solutions have been illustrated by the graphics.

Construction of Exotical Soliton-Like for a Fractional Nonlinear Electrical Circuit Equation Using Differential-Difference Jacobi Elliptic Functions Sub-Equation Method

Construction of Exotical Soliton-Like for a Fractional Nonlinear Electrical Circuit Equation Using Differential-Difference Jacobi Elliptic Functions Sub-Equation Method

Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of Some Nonlinear Schrödinger’s Equations

In this study, we found the traveling wave solutions of these equations by applying (3+1)-dimensional nonlinear Schrödinger’s equation and coupled nonlinear Schrödinger’s equation to Generalized Jacobi elliptic function method. We have expressed these solutions both as Jacobi elliptical solutions and trigonometric and hyperbolic solutions. We present two and three dimensional graphics of some solutions we have found. We also state some studies on these equations.