Extended Auxiliary Equation Method Research Articles

Optical soliton solution of the perturbed Biswas-Milovic equation having cubic-quintic-septic law nonlinearity in the presence of spatio-temporal and chromatic dispersion

In this manuscript, we investigate the analytical and soliton solutions of the cubic-quintic-septic law for the perturbed Biswas-Milovic equation, considering spatio-temporal and chromatic dispersions. The perturbed Biswas-Milovic equation with the spatio-temporal and chromatic dispersion terms provides a comprehensive study for describing nonlinear optical wave propagation in optical fiber. We use the wave transformation to reduce the main equation to a nonlinear ordinary differential equation. The transformation of the original equation into a more simplified form aims to attain a more profound comprehension of the fundamental dynamics of the system. We retrieve the analytical solutions of the presented model by implementing the new Kudryashov technique and a subversion of the new extended auxiliary equation approach. Besides, bright, singular, and V-shape soliton structures are represented. By employing powerful analytical techniques, we systematically derive a wide range of soliton solutions. This approach successfully captures diverse soliton types highlighting the novelty of applying the new Kudryashov technique and a subversion of the new extended auxiliary equation method to this complex model. Moreover, we analyze the soliton behavior influenced by various parameters. The analysis of the parameter influences reveals the complicated relationship governing the dynamics of the perturbed Biswas-Milovic model. Furthermore, this manuscript includes the modulation instability analysis for the presented model. Conducting modulation instability analysis for the presented equation enhances our understanding of the system’s stability and dynamics.

On optical soliton solutions of the higher-order Lakshmanan-Porsezian-Daniel model having the cubic-quintic-septic law in the presence of spatio-temporal and chromatic dispersion

The higher-order Lakshmanan-Porsezian-Daniel equation (LPDE) with the cubic-quintic-septic (CQS) law having spatiotemporal and chromatic dispersion terms (STD-CD) is examined to derive new optical soliton solutions. To accomplish this aim, we operated on a simple version of the new extended auxiliary equation method (SAEM26). The optical soliton solutions of the LPDE with CQS as well as STD-CD are constructed in detail. Moreover, 3D-surface, contour, and 2D plots are presented for the bright and periodic singular soliton solutions. Additionally, the effects of diverse model parameters on the bright soliton structure are surveyed, and these effects are displayed with 2D graphics. The findings established in this work can positively contribute to research in nonlinear optics, while the SAEM26 can be effectively applied to similar nonlinear models.

Bright soliton of Stochastic perturbed Biswas-Milovic equation with cubic-quintic-septic law having multiplicative white noise

For the first time, the adopted stochastic form of the perturbed Biswas-Milovic equation with cubic-quintic-septic law having spatio-temporal and chromatic dispersion in the presence of multiplicative white noise in Ito sense was presented and examined. The Biswas-Milovic equation ˆ models numerous physical phenomena occurring in optical fiber. We analyzed the optical soliton solutions of the stochastic model with the aid of a subversion of the new extended auxiliary equation method. Furthermore, we investigated the evaluation of the noise impacts and the effects of some model parameters on the dynamics of the generated soliton. Finally, graphical depictions of the derived soliton types were represented for some solution functions. The stochastic model and the derived results will contribute to the comprehension of the nonlinear dynamics of pulse propagation in optical fibers which has great importance for the advancement of optical communication engineering.

Pure-Cubic Optical Solitons and Stability Analysis with Kerr Law Nonlinearity

In this research paper, we investigate the effects of third-order dispersion and nonlinear dispersion terms on soliton behavior for pure-cubic solitons in the absence of chromatic dispersion. The research proceeds in several stages. First, we derive the nonlinear ordinary differential equation form by utilizing the complex wave transform. In the second stage, we employ a simplified version of the new extended auxiliary equation method to derive both bright and singular optical solitons. Subsequently, we examine the influence of model parameters on these bright and singular solitons in the third stage. To support our findings, we present solution functions accompanied by effective graphical simulations. We report observations regarding the effects of parameters in the relevant sections. The validity of our results is confirmed through their satisfaction of the model equation. Furthermore, we apply the Vakhitov-Kolokolov stability criterion to ensure the stability of the obtained bright soliton solution. Notably, the novelty of this paper lies in its application of a simplified version of the extended auxiliary equation approach to recover optical solitons. This study stands apart from previously published works that utilized various expansion approaches, yielding a distinct spectrum of results.

Optical soliton, linear stability analysis and conservation laws via multipliers to the integrable Kuralay equation

This paper studies the integrable Kuralay equation which is a kind of integrable spin system in (1＋1) dimensions. Several new exact solutions of different kinds such as optical solitons, solitary waves and elliptic function solutions of this equation are retrieved by utilizing two effective techniques, namely, the modified F-expansion method and the new extended auxiliary equation method. The modulation instability (MI) analysis of the model is studied by using the linear-stability analysis and the MI gain spectrum is acquired. Moreover, the multiplier approach is applied to retrieve the conservation laws. At last, physical interpretations of the obtained results are demonstrated.

Chirped optical solitons and stability analysis of the nonlinear Schrödinger equation with nonlinear chromatic dispersion

The present paper aims to investigate the chirped optical soliton solutions of the nonlinear Schrödinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index. The exquisite balance between the chromatic dispersion and the nonlinearity associated with the refractive index of a fiber gives rise to optical solitons, which can travel down the fiber for intercontinental distances. The effective technique, namely, the new extended auxiliary equation method is implemented as a solution method. Different types of chirped soliton solutions including dark, bright, singular and periodic soliton solutions are extracted from the Jacobi elliptic function solutions when the modulus of the Jacobi elliptic function approaches to one or zero. These obtained chirped optical soliton solutions might play an important role in optical communication links and optical signal processing systems. The stability of the system is examined in the framework of modulational instability analysis.

Comment on “Optical solitons for Radhakrishnan–Kundu–Lakshmanan equation in the presence of perturbation term and having Kerr law”

We analyzed a recent paper (Ozdemir, 2022). The author used three approaches to construct solutions of Radhakrishnan–Kundu–Lakshmanan equation — the generalized projective Riccati equations method, the simple version of the new extended auxiliary equation method, and unified Riccati equation expansion method. We demonstrate that the first approach used by the author is erroneous. The second approach, which is, in fact, a modification of the Kudryashov method was used in an inefficient version with unnecessary restrictions. The third approach leads to several solutions that are equivalent to one. It is shown that the solutions presented in paper are wrong. The correct solutions of the RKL equation are presented.

Optical solitons for Radhakrishnan–Kundu–Lakshmanan equation in the presence of perturbation term and having Kerr law

Purpose:In this research study, obtaining the analytical and soliton solutions to the perturbed Radhakrishnan–Kundu–Lakshmanan (RKL) equation with Kerr law nonlinearity is aimed via the generalized projective Riccati equations method (GPREM), a simple version of the new extended auxiliary equation method (SAEM26), and unified Riccati equation expansion method (UREEM). At the same time, the roles of some parameters included in the perturbed RKL equation on soliton dynamics are analyzed. Methodology:The presented methods are successfully employed to the perturbed RKL equation. In the application of the presented methods, to convert the perturbed RKL equation into a nonlinear ordinary differential equation, we choose suitable complex wave transformation for the proposed model. Later, a linear equation system is derived using the GPREM, SAEM26, and UREEM, the system is solved, the appropriate solution sets are obtained, and the soliton solutions are achieved, respectively. Findings:The singular, bright and dark soliton solutions are generated by choosing the suitable set and parameter values. To comprehend the physical dynamics of some solutions, 3D, contour, and 2D graphs are demonstrated. In addition, 2D graphs are drawn to show how some parameters in the main equation have an effect on soliton behaviors. The examination indicates that the model parameters have a substantial effect on the soliton dynamics. Depending on the soliton forms, the effect can be varied. The results presented in this paper will be useful for future works in soliton theory and the presented methods can be effectively implemented to such equations. Originality:The effects of the model parameters included in the perturbed RKL equation on soliton dynamics are analyzed for the first time in this study.

A variety of new soliton structures and various dynamical behaviors of a discrete electrical lattice with nonlinear dispersion via variety of analytical architectures

Nonlinear electrical lattices are powerful experimental tool for studying nonlinear dispersive media and creating opportunities for the realistic modeling of electrical solitons. In this work, we adopt the generalized auxiliary equation methods, the first integral method, and the Möbius transformation approach to extract a variety of traveling and solitary wave solutions of the nonlinear Salerno equation describing the nonlinear discrete electrical lattice with nonlinear dispersion. The kink, antikink, dark, bright, peakons, antipeakons, and periodic wave solutions are all derived. The existence criteria of solitons are described. It has been shown that for the selective values of arbitrary constants in an auxiliary equation, the generalized auxiliary equation method II is reduced to the new ‐model expansion method as well as the new extended auxiliary equation method. The impact of free parameters on the obtained solutions is investigated and graphically depicted using physical descriptions. The acquired results are important for the validity of numerical and experimental results and further understanding of the wave propagation in the nonlinear discrete electrical lattice. In addition, the stability analysis of the obtained solitary wave solutions is studied. Furthermore, using the maple software, all derived solutions were checked by re‐entering them into the considered equation.

Stochastic optical solitons with multiplicative white noise via Itô calculus

Purpose:This study has been carried out to examine the stochastic optical soliton solutions of the nonlinear Schrödinger’s equation (NLSE) with Kerr law nonlinearity by multiplicative noise in Itô sense and the behavioral changes on soliton dynamics. Methodology:In order to see the noise effect better, bright and dark soliton shapes belonging to the NLSE have been obtained using the subversion of the new extended auxiliary equation method (SAEM246) and verified with the computer algorithm that developed for this study. Findings:For the NLSE, the distortion of the soliton solutions obtained by the SAEM246 method under different noise effects is clearly shown and simulated. Originality:This method has never been applied before to a NLSE containing a stochastic function to analyze the noise effect. The novelty of the problem and applied method has resulted in many new and original soliton solutions and their behaviors under the noise effects.

Propagation of diverse exact solitary wave solutions in separation phase of iron (Fe-Cr − X(X = Mo, Cu)) for the ternary alloys

In this paper, a specific consideration is paid to the nonlinear dynamics of solitary waves to kinetics of phase separation in iron [Formula: see text] based on ternary alloys. The convective–diffusive Cahn–Hilliard (CH) equation is used as a mathematical model to describe the dynamics of the separation phase for the ternary alloys of iron. A variety of solitary wave solutions with unknown parameters are extracted in different shapes like kink-type, bell-shape, shock-type, combine soliton, trigonometric, hyperbolic and Jacobi’s elliptic function solutions with the assistant of recently computational tools, namely, extended Fan-sub equation method (EFSEM) and extended auxiliary equation method (EAEM). In addition, 3D, 2D, and their corresponding contour profiles of earned results are sketched in order to observe their dynamics with the choices of involved parameters. On the bases of achieved results, we may claim that the proposed computational methods are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

Construction of new traveling and solitary wave solutions of a nonlinear PDE characterizing the nonlinear low-pass electrical transmission lines

In this study, we intend to analyze the traveling and several other solitary wave solutions in the nonlinear low-pass electrical transmission line model using the new mapping method, the new extended auxiliary equation method, and the extended Kudryashov method. A type of traveling and solitary wave solutions emerge, consisting of hyperbolic function, trigonometric, rational, periodic, and doubly periodic solutions that reflect kink, anti-kink wave solitons, bright-dark optical solitons, singular solitons, and other traveling waves. The three integration techniques applied are efficient, effective, and versatile for the creation of new bright, dark, singular, and non-singular periodic and solitary wave propagation solutions in nonlinear low-pass electrical transmission lines. To see the extant physical significance of the considered equation, we present some 2D and 3D figures for some solutions. We compare the obtained results with those obtained in the literature. We investigate and demonstrate the stability of the soliton solutions.

Optical solitons in fiber Bragg gratings having Kerr law of refractive index with extended Kudryashov’s method and new extended auxiliary equation approach

This paper reveals optical solitons and other solutions to fiber Bragg gratings with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are indeed a technological marvel that supplements chromatic dispersion when its count runs low. The extended Kudryashov’s method and new extended auxiliary equation method have been implemented. Chirped and chirp–free bright, dark and singular solitons, with dispersive reflectivity, are presented.

Construction of bright-dark solitary waves and elliptic function solutions of space-time fractional partial differential equations and their applications

In this article, several kinds of wave solutions such as solitons, solitary waves and Jacobi elliptic function solutions in which many are novel of nonlinear fractional partial differential equations (NLFPDEs) in mathematical physics, has obtained with the aid of improved extended auxiliary equation method. The efficiency of the current technique is demonstrated by applications to three NLFPDEs, namely, (2+1)-dimensional space-time fractional Zoomeron equation, space-time fractional Benjamin–Bona–Mahony and space-time fractional modified third-order KdV equations. Several kinds of exact solutions are constructed which have key applications in applied sciences. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The resulting numerous solutions will also be helpful in applied mechanics and various fields of sciences. This powerful technique can be applied to other wave models that can arise in mathematical physics.

Optical solitons for Biswas–Milovic equation by new extended auxiliary equation method

Biswas–Milovic equation is used to narrate the nonlinear dynamics of optical solitons in nonlinear optics. With the aid of new extended auxiliary method, we find Jacobi elliptic functions solutions which degenerate to bright, dark, singular and other solitary wave solutions with Kerr law and dual power law nonlinearities.

Optical solitons and other solutions to the dual-mode nonlinear Schrödinger equation with Kerr law and dual power law nonlinearities

In this article, we apply four insightful integration algorithms, namely, the tanh-coth method, the unified Riccati equation method, the modified simple equation method and the new extended auxiliary equation method for constructing new optical soliton solutions and other solutions to the dual-mode nonlinear Schrödinger equation with Kerr law and dual power law nonlinearities related to the optics. Many families of Jacobi elliptic solutions, dark, bright, singular soliton solutions and other solutions have been found.

Comment on “the solitons and periodic travelling wave solutions for Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations in quantum field theory, Optik 168 (2018) 807-816”

In the recently published article, S.Saha Ray investigated Dodd-Bullough-Mikhailov (DBM) and Tzitzeica-Dodd-Bullough (TDB) equations via his new extended auxiliary equation method. And he claimed that new types of Jacobi elliptic function solutions had been obtained and meanwhile a new family of explicit travelling wave solutions were derived. In this comment, we show that all of exact traveling solutions presented by S.Saha Ray do not satisfy the new extended auxiliary equation method and therefore are incorrect. In addition, general traveling wave solutions of the two differential equation are presented by making full use of certain results published previously.

Dispersive optical solitons of time-fractional Schrödinger–Hirota equation in nonlinear optical fibers

In this article dispersive optical soliton solutions that are governed by the time-fractional Schrödinger–Hirota equation have been presented. Some new soliton solutions of the time-fractional SH equations are obtained in this work. Using extended auxiliary equation method, optical soliton solutions are sought for the fractional SH equations with power law nonlinearity as well as Kerr law nonlinearity. Using fractional complex transform the time-fractional SH equation is converted into the nonlinear ordinary differential equation, and then, the resulting equation is solved using a novel analytical method viz. extended auxiliary equation method. As a result of this, some new optical soliton solutions have been successfully obtained. The obtained results show that the proposed method is a straightforward technique to find out new optical solutions of time-fractional SH equation.

Solitons and other solutions for coupled nonlinear Schrödinger equations using three different techniques

In this paper, we apply three different techniques, namely, the sine–cosine method, the new extended auxiliary equation method and the modified simple equation method for constructing many new exact solutions with parameters as well as bright–dark, singular and other soliton solutions of the coupled nonlinear Schrodinger equations. The solutions of these coupled nonlinear equations are compared with the well-known results.

New exact solutions for the Wick-type stochastic nonlinear Schrödinger equation in nonlinear optics

The exact solutions of Wick-type stochastic nonlinear Schrödinger equation (NLSE) using extended auxiliary equation method (EAEM) has been obtained in this paper. Hermite transform has been used for transforming the stochastic NLSE into deterministic PDE. Also, the stochastic solutions in the white noise space have been obtained by applying inverse Hermite transform on the deterministic set of solutions.