Jacobi Elliptic Function Method Research Articles

Dark solitons with anti-cubic and generalized anti-cubic nonlinearities in an optical fiber

In this research work, anti-cubic and generalized anti-cubic nonlinearity are taken into consideration. The Jacobi elliptic function method and Jacobi elliptic function expansion method is to nonlinear optical fiber for obtaining dark soliton structure. Dark solitons emerged from this method. Furthermore, we establish the physical significance and dynamical behaviors of the dark soliton solutions by using the physical parameter namely dispersion coefficient.

New optical solitons based on the perturbed Chen-Lee-Liu model through Jacobi elliptic function method

New optical solitons based on the perturbed Chen-Lee-Liu model through Jacobi elliptic function method

Doubly periodic wave structure of the modified Schrödinger equation with fractional temporal evolution

Abundant Jacobi elliptic type solutions with distinct physical structures of complex nonlinear conformable time-fractional modified Schrödinger equation are obtained by using the generalized Jacobi elliptic function (GJEF) method. The Jacobi function expansions may lead to new doubly periodic wave solutions, soliton solutions, and triangular periodic solutions. Nowadays the conformable operator is being used for a better description of the dynamical systems. Motivated by the potential applications of the governed equation in nonlinear optics, biological sciences, and fluid dynamics, these solutions may be significant in the study of wave propagation in the desired field. Symbolic computations are made with the aid of Maple.

Soliton solutions and periodic solutions for two models arises in mathematical physics

<abstract><p>We aimed in this paper to acquire the periodic wave solutions and soliton solutions and other solutions such as kink-wave solutions for the cubic nonlinear Schrödinger equation with repulsive delta potential ($ \delta $-NLSE) and complex coupled Higgs field equation via two mathematical methods Jacobi elliptic function method and generalized Kudryashov method. Some of these solutions are degenerated to solitary wave solutions and periodic wave solutions in the limit case. We also gave the meaning of these solutions physically and the numerical simulation by some figures.</p></abstract>

Dispersive soliton solutions for the Salerno equation for the nonlinear discrete electrical lattice in the forbidden bandgaps

In this paper, the modified Jacobi elliptic function method is applied for Salerno equation which describes the nonlinear discrete electrical lattice in the forbidden bandgaps. Dark and bright solitons are obtained. Also, periodic solutions and periodic Jacobi elliptic function solutions are reported. Moreover, for the physical illustration of the obtained solutions, three-dimensional and two-dimensional graphs are presented.

Analytical novel solutions to the fractional optical dynamics in a medium with polynomial law nonlinearity and higher order dispersion with a new local fractional derivative

Novel solutions for the nonlinear dynamics of Schrödinger equation for polynomial law medium with third-order dispersion (TOD), fourth-order dispersion (FOD), and self-steepening are investigated based in a novel local fractional derivative of order α and the Jacobi elliptic function method which are combined into a novel fractional sub-equation method. The Jacobi elliptic function method will provide different types of analytical solutions and not only soliton type solutions for the propagation of ultra-short optical signals through a polynomial law medium. Solutions are illustrated in 3-D graphs, contour plots and 2-D plots under the obtained constraint conditions.

Assessment of nonlinear vibrations of thin plates undergoing large deflection and moderate rotation using Jacobi elliptic functions

In this article, nonlinear free/forced vibrations of a plate undergoing large deflection and moderate rotation have been investigated based on Jacobi elliptic functions. The formulation has been established using a generalized von-Karman’s strain in which nonlinear terms of deflection and rotations are simultaneously included. The governing equations have been derived with the use of extended Hamilton’s principle assuming that the plate is under an external force of elliptic type. It is proved that conventional approximate solutions of nonlinear vibrations of plates based on single-harmonic assumption are inadequate to consider higher harmonics. Whereas, Jacobi elliptic function method considers the effects of higher-order harmonic leading to a more exact solution. This becomes more important when extra nonlinearity due to nonlinear rotations is included in the calculations.

Discrete localized excitations for discrete conformable fractional cubic–quintic Ginzburg–Landau model possessing the non-local quintic term

Considering conformable fractional derivative to modify the Jacobian elliptic function method, rich exact solutions of discrete conformable fractional complex cubic–quintic Ginzburg–Landau model possessing non-local quintic term are derived. Solutions with alternating phases have different properties from their continuous analogs. The effects of filter and linear dissipation parameters on explosion excitation and wave amplitude modulation are studied.

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.

New periodic solutions for Ginzburg-Landau in three different derivatives via extended Jacobian elliptic function method

Here, we analyze the new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. The new periodic solutions are obtained by the use of the extended Jacobian elliptic function (JEF) expansion method for the CGLE equation arising in physics. These solutions are practiced for three suggested definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Graphic depictions of the solutions obtained are also displayed. Some recent solitary waves have been found to be periodic, solitons-shock, conical, soliton-shock-like, super-soliton, localized soliton waves and other solutions can be directly evaluated.

A variety of wave amplitudes for the conformable fractional (2 + 1)-dimensional Ito equation

The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed.

Bubbles interactions in fluidized granular medium for the van der Waals hydrodynamic regime

This paper investigates the fluidized granular materials (FGM) with the van der Waals normal form (VDWF) under the effects of friction and viscosity. The system of macroscopic balance is presented, including the mass, momentum, and energy equations of local densities. For two different types of collisions, elastic and inelastic collisions, analytical solutions of the nonlinear PDEs governing the granular model are investigated using the hydrodynamic equations for granular matter motion. The integrability of the proposed model is analyzed by applying the Painleve analysis. Moreover, the Backlund transformation (BT) is established using the Painleve truncation expansion. New traveling wave solutions of the VDWF within FGM are obtained by using the BT, tanh function, Jacobi elliptic function methods to study the phase separation phenomenon. As two pairs of rarefaction and shock waves emerge and travel away giving the appearance of bubbles, the resulting solutions of the proposed model show a behavior similar to those found in the molecular dynamic simulations. The dispersion relation and their properties to the model equation are investigated. Besides, stability analysis of the VDWF in its ODE form is demonstrated using the phase portrait classifications. Finally, using two- and three-dimensional graphics for seeking model solutions under the influence of friction and viscosity, qualitative agreements with previous related works are shown.

Exact wave solutions of the fourth order non-linear partial differential equation of optical fiber pulses by using different methods

In this work, we utilize the modified Jacobi elliptic function method and the modified(G′∕G) expansion method to get explicit exact solutions for the fourth order non-linear partial differential equation that describe the pulses dynamics in the optical fibers. With the aid of these methods, we get many exact solutions like bright and dark solitons, periodic, hyperbolic and rational type solutions.

Combined soliton solutions of a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrödinger equation in the cubic–quintic nonlinear material

We concentrate on a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrodinger equation with cubic–quintic nonlinearities, and get some exact combined soliton solutions and periodic solutions based on the Jacobian elliptic function method and double function method. We discuss dynamical behaviors of bright soliton, dark soliton, periodic bright soliton train and periodic M-shaped soliton train, and find that shift distances along the x-direction of these solitons all add if the value of the fractional order increases.

The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method.

How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to learn the same initial data by selecting the different numbers of initial points sampled, residual collocation points sampled, network layers, and neurons per hidden layer, respectively. The result shows that the PINNs well reconstruct the dynamical behaviors of the quasiperiodic wave, periodic wave, and soliton solutions for the KdV-mKdV equation, which gives a good way to simulate the solutions of nonlinear partial differential equations via one deep learning 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.

On nonlinear vibration behavior of piezo-magnetic doubly-curved nanoshells

In this paper, nonlinear vibration behaviors of multi-phase Magneto-Electro-Elastic (MEE) doubly-curved nanoshells have been studied employing Jacobi elliptic function method. The doubly-curved nanoshell has been modeled by using nonlocal elasticity and classic shell theory. An exact estimation of nonlinear vibrational behavior of smart doubly-curved nanoshell has been obtained via Jacobi elliptic function method. This method can incorporate the influences of higher order harmonics leading to an exact estimation of nonlinear vibration frequency. It will be indicated that nonlinear vibrational frequency of doubly-curved nanoshell relies on nonlocal effect, material composition, curvature radius, center deflection and electro-magnetic field.

Photovoltaic spatial solitons and periodic waves in a photovoltaic crystal

A coupled nonlinear Schrödinger equation describes the propagation of a spatial-phase-modulated photovoltaic soliton and a non-phase-modulated beam in a photovoltaic crystal. With the help of the Jacobian elliptic function method and the leading term analysis, three kinds of Jacobian elliptic function solutions including coupled sn–cn type solution, coupled sn–dn type solution and solution constructed by products of elliptic functions are reported. In the long wave limit, these solutions become dark-bright soliton pair solution and coupled twin-peaked type solution. From these solutions, the amplitude, width and frequencies of solitons are determined by the Glass constants of the spatial-phase-modulated, non-phase-modulated and background beams.

Investigating nonlinear vibrations of multi-scale truncated conical shell segments with carbon nanotube/fiberglass reinforcement using a higher order conical shell theory

This research deals with the nonlinear vibration analysis of functionally graded carbon nanotubes and fiber-reinforced composite truncated conical shell segments based upon third-order shear deformation theory. A detailed procedure for obtaining material properties of the multi-scale carbon nanotube/fiber-reinforced composite based on the three-dimensional Mori–Tanaka scheme has been provided. The truncated conical shell segments have been reinforced by distributed carbon nanotubes in the thickness direction according to uniform, linear, and nonlinear functions. The nonlinear equations have been solved via both Galerkin’s technique and Jacobi elliptic function method. Based on the numerical results, the effects of diverse carbon nanotube distribution, fiber volume, fiber orientation, and semi-vertex and open angles of the segment on vibrational frequencies of the truncated conical shell have been studied.

Nonlinear dynamic characteristics of nonlocal multi-phase magneto-electro-elastic nano-tubes with different piezoelectric constituents

Analysis of exact nonlinear dynamic behavior of multi-phase magneto-electro-elastic (MEE) nanoshells has been presented in this paper using Jacobi elliptic functions. Multi-phase MEE material is constructed form piezoelectric and piezo-magnetic constituents for which the material properties can be controlled based on the percentages of the constituents. Nonlinear governing equations are established for MEE nanoshell based on nonlocal elasticity theory and an exact solution is provided using Jacobi elliptic function method. This method gives an exact value of vibration frequency in nonlinear regime and overcomes the shortcomings of several approximate solutions applied in the studies on nanostructures. It will be shown that nonlinear vibration behavior of MEE nanoshell in electro-magnetic field depends on the constituent’s percentages. Influences of nonlocal scale factor, piezoelectric reinforcement, magnetic field intensity, and electrical voltage on vibration frequencies of the nanoshell are also investigated.