Jacobi Elliptic Function Solutions Research Articles

Modulated wave and modulation instability gain brought by the cross-phase modulation in birefringent fibers having anti-cubic nonlinearity

In this paper, we investigate the modulated wave and W-shaped profile in birefringent fibers having the anti-cubic nonlinearity terms. We use the traveling wave hypothesis to show out the velocity of the soliton and the constraint relation on the anti-cubic nonlinear terms. We use the Jacobi elliptic function solutions to point out two types of combined solutions. After some assumption on the modulus of the Jacobi elliptic function, we have shown out the combined bright-bright soliton and dark-dark soliton-like solutions. We use the linearizing algorithm to analyze the modulation instability (MI) growth rate. We have shown that the anti-cubic nonlinear terms and cross-phase modulation (XPM) can increase MI bands and the amplitude of the MI growth rate. To corroborate the prediction made on analytical results, we use the numerical investigation to show the propagation of the modulated wave and W-shaped profile in terms of cell index. We exhibited through the numerical results that the modulated wave can conserve high energy during its propagation in birefringent fibers. The obtained results will certainly open new perspectives in optical fibers during the transmission of huge data.

Ion-acoustic wave structures in the fluid ions modeled by higher dimensional generalized Korteweg-de Vries–Zakharov–Kuznetsov equation

In this paper, the higher dimensional generalized Korteweg-de-Varies–Zakharov–Kuznetsov (gKdV-ZK) equation is under investigation. This model is used in the field of plasma physics which describes the effects of magnetic field on the weak ion-acoustic wave. We have applied two techniques, called as ϕ6-model expansion method and the Hirota bilinear method (HBM) to explore the diversity of wave structures. The solutions are expressed in the form of hyperbolic, periodic and Jacobi elliptic function (JEF) solutions. Moreover, the solitary wave solutions are also extracted. A comparison of our results to well-known results is made, and the study concludes that the solutions achieved here are novel. Additionally, 3-dimensional and contour profiles of achieved outcomes are drawn in order to study their dynamics as a function of parameter selection. On the basis of the obtained results, we can assert that the proposed computational methods are straightforward, dynamic, and well-organized, and will be useful for solving more complicated nonlinear problems in a variety of fields, particularly in nonlinear sciences, in conjunction with symbolic computations. Additionally, our discoveries provide an important milestone in comprehending the structure and physical behavior of complex structures. We hope that our findings will contribute significantly to our understanding of ocean waves. This study, we hope, is appropriate and will be of significance to a broad range of experts involved in ocean engineering models.

Optical solitons in polarization preserving fibers for perturbed resonant NLSE with Kerr law nonlinearity and Bohm potential having multiplicative white noise via Itô calculus

Object:We discuss for the first time, the perturbed resonant nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity and Bohm potential having a multiplicative white noise. Methods:Three integration schemes via modified simple equation method, sine–cosine method and Jacobi elliptic equation method are used. Results:We retrieve soliton solutions to the model, which are described using parametric constraints. Jacobi elliptic function solutions, bright, dark, singular soliton solutions are presented.

The study of new optical soliton solutions to the time-space fractional nonlinear dynamical model with novel mechanisms

In this work, we successfully employ two constructive mathematical tools such as the extended rational sine-cosine/sinh-cosh and novel Φ6-expansion model techniques, for obtaining different types of optical solitons and other solutions to the (2+1)-dimensional time-space fractional nonlinear Schrodinger (NLS) equation within the modified Riemann–Liouville derivatives. Solitary wave, dark, singular, and their combined solutions are retrieved. Moreover, hyperbolic, trigonometric, plane wave periodic wave, and Jacobi elliptic function solutions are also derived. The novelty of the obtained solutions is revealed by comparing them with the previous finding. By selecting suitable parametric values, numerical simulation and physical interpretations of the achieved outcomes are manifested with interesting figures presenting the meaning of the acquired results. The novel approaches are used through the symbolic computations impart a dynamical and potent mathematical implement for solving various benevolent nonlinear wave problems. The results suggest that these constructing techniques are concise, direct, efficient, and straightforward. The resulting solutions are novel, intriguing, and potentially helpful for understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including telecommunication, engineering, mathematical biology, mathematical physics, and nonlinear optics etc.

New Exact Solutions to the Lakshmanan–Porsezian–Daniel Equation with Kerr Law of Nonlinearity

In this study, some new exact travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) equation with Kerr law of nonlinearity are retrieved by the complete discrimination system for the polynomial method. Under the travelling wave transformation, the LPD equation is reduced to an ordinary differential equation. The new exact travelling wave solutions including rational solutions, triangle function solutions, solitary wave solutions, and Jacobian elliptic function solutions are obtained and graphically illustrated using three-dimensional and two-dimensional graphs. Comparing with the previous results for LPD equation, some of new solutions in this work such as elliptical solution are not studied, which shows the complete discrimination system method is efficient.

Application of the F-Expansion Method for Solving the Fokas-Lenells Equation

By the aid of traveling wave hypothesis, the F-expansion method has been implemented in this paper to obtain Jacobian-Elliptic function solutions for the optical Fokas-Lenells model. The hyperbolic-function solutions are derived as special cases from the Jacobian-Elliptic function solutions. The present approach is straightforward to determine the exact solutions for the Fokas-Lenells equation. The existence criteria of the obtained solutions are also reported.

Bifurcation Analysis and Travelling Wave Solution for a Generalized Time Fractional KdV Equation

Abstract By using bifurcation theory of planar dynamic system to a generalized time fractional KdV equation, a large number of solutions including irrational function solution, Jacobian elliptic function solution, trigonometric function solution, solitary solution as well as compacton solution are derived. Under different parametric selection, various kinds of sufficient conditions and numerical simulations are demonstrated to visualize their mechanism.

New exact solutions to the coupled (n + 1)-dimensional Klein–Gordon–Zakharov equation with power law nonlinearities

In this paper, a special class of exact solutions of the coupled higher-dimensional Klein–Gordon–Zakharov (KGZ) equation with power law nonlinearities is constructed explicitly by separation transformation method. One merit of this work is that when the dimension is larger than one, there is an arbitrary function [Formula: see text], which is used to search for more general exact solutions of the coupled higher-dimensional KGZ equation. As a result, the Jacobi elliptic function solutions and soliton solutions with free parameters are derived, which lead to rich localized nonlinear structures of the coupled higher-dimensional KGZ equation with power law nonlinearities.

Optical solitons and single traveling wave solutions for the Triki–Biswas equation describing monomode optical fibers

In this paper, the polynomial complete discriminant system method is introduced to construct optical solitons and single traveling wave solutions for the Triki–Biswas equation, which is one of the most important mathematical and physical equation usually used to simulate phenomena in nonlinear optics, quantum mechanics and plasma physics. The Jacobi elliptic function solutions, dark solitons solutions, bright solitons solutions, singular solitons solutions and other solutions have been presented.

Optical solutions and conservation laws of the Chen–Lee–Liu equation with Kudryashov's refractive index via two integrable techniques

The fundamental objective of this study is to find optical solutions to the perturbed Chen–Lee–Liu equation with Kudryashov's arbitrary refractive index model. Therefore, we applied the enhanced Kudryashov's and improved extended tanh-function techniques to the model. We obtained bright soliton, singular soliton, singular periodic wave, Jacobi elliptic function, exponential function, Weierstrass function, and dark soliton solutions. The obtained results are different from the other solutions available in the literature. As a result of the study, we can say that the proposed methods are effective and powerful for securing exact solutions of the nonlinear partial differential equations. Finally, using a new conservation theorem, we were able to establish the conservation laws for the current model.

Closed-form solutions and conserved quantities of a new integrable (2 + 1)-dimensional Boussinesq equation of nonlinear sciences

Abstract In this paper, we investigate a newly introduced integrable (2 + 1)-dimensional Boussinesq equation. Solutions of this equation are obtained by Lie symmetry reductions and direct integration. We achieve diverse solitary wave solutions of the equation among which are non-topological soliton as well as Jacobi elliptic function solutions. Moreover, we generate some closed-form solutions of the equation which are in the form of bright, singular and non-singular periodic solitons. Power series solution of the equation is also generated. In a bid to have a sound understanding of the physical phenomena of the underlying model, we exhibited graphically the motion of the secured results. Besides, we discuss the obtained results as well as their respective graphs. Conclusively, we construct conservation laws of the aforementioned equation by employing the general multiplier approach.

Travelling wave solutions for the doubly dispersive equation using improved modified extended tanh-function method

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation.

The classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation

The main purpose of this paper is to study the single traveling wave solutions of the fractional coupled nonlinear Schrödinger equation. By using the complete discriminant system method and computer algebra with symbolic computation, a series of new single traveling wave solutions are obtained, which include trigonometric function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, solitary wave solutions and rational function solutions. As you can see, we give all the classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation. The obtained results substantially improve or complement the corresponding conditions in the literature (Esen and Sulaiman in Optik 167:150-156, 2018), (Eslami in Appl. Math. Comput. 258:141-148, 2016), (Han et al. in Phys. Lett. 395:127217, 2021). Finally, in order to further explain the propagation of the fractional coupled nonlinear Schrödinger equation in nonlinear optics, two-dimensional and three-dimensional graphs are drawn.

Exact Solutions to Space-Time Fractional Fokas-Lenells Equations With Parameters

The exact solutions of the space-time fractional Fokas-Lenells equation with parameters in nonlinear optics are obtained by using the complete discrimination system for polynomial method, including rational function solutions, periodic solutions, solitary wave solutions, Jacobi elliptic function solutions and hyperbolic function solutions. The relevant images of the exact solution are drawn, and the influence of parameters on the structure of the solution is analyzed.

New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.

Soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation

<abstract><p>In this paper, our main purpose is to study the soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. New soliton solutions have been obtained by the extended $ (G'/G) $-expansion method, first integral method and complete discrimination system for the polynomial method, respectively. The solutions we obtained mainly include hyperbolic function solutions, solitary wave solutions, Jacobi elliptic function solutions, trigonometric function solutions and rational function solutions. Moreover, we draw its three-dimensional graph.</p></abstract>

Optical solitons and single traveling wave solutions of Biswas-Arshed equation in birefringent fibers with the beta-time derivative

<abstract><p>This article describes the construction of optical solitons and single traveling wave solutions of Biswas-Arshed equation with the beta time derivative. By using the polynomial complete discriminant system method, a series of traveling wave solutions are constructed, including the rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and inverse trigonometric function solutions. The conclusions of this paper comprise some new and different solutions that cannot be found in existing literature. Using the mathematic software Maple, the 3D and 2D graphs of the obtained traveling wave solutions were also developed. It is worth noting that these traveling wave solutions may motivate us to explore new phenomena which may be appear in optical fiber propagation theory.</p></abstract>

Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.

Quantum oscillations between weakly coupled Bose–Einstein condensates: Evolution in a non-degenerate double well

In this paper, we discuss coherent atomic oscillations between two weakly coupled Bose–Einstein condensates that are energetically different. The weak link is notionally provided by a laser barrier in a (possibly asymmetric) multi-well trap or by Raman coupling between condensates in different hyperfine levels. The resultant boson Josephson junction dynamics is described by a double-well nonlinear Gross–Pitaevskii equation. On the basis of a new set of Jacobian elliptic function solutions, we describe the period of the oscillations as well as associated quantities and predict novel observable consequences of the interplay of the energy difference and initial phase difference between the two condensate populations.

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.