Riccati Equation Expansion Research Articles

Investigation of optical soliton solutions for the cubic-quartic derivative nonlinear Schrödinger equation using advanced integration techniques

This paper aims to investigate optical soliton solutions in the context of the cubic-quartic derivative nonlinear Schrödinger equation with differential group delay, incorporating perturbation terms for the first time. Motivated by the need to better understand soliton dynamics in advanced optical communication systems, we employ three integration techniques: the direct algebraic approach, Kudryashov’s method with an addendum, and the unified Riccati equation expansion method. Our study reveals that, by appropriately selecting parameter values, the resulting solutions include Jacobi elliptic functions that describe straddle solitons, bright, dark, and singular solitons. We also identify the conditions under which these soliton pulses can exist. Furthermore, we provide numerical simulations to illustrate these solutions under specific parameter settings, highlighting their potential applications in optical fiber systems.

Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics

Optical solitons and traveling wave solutions for the higher-order dispersive extended nonlinear Schrödinger equation are studied. Ultrashort pulse propagation in optical communication networks is described by this equation. To find exact solutions to the model, the unified Riccati equation expansion method and the Jacobi elliptic function expansion method are successfully applied. The optical solutions includes various solitary wave solutions, such as dark, bright, combined dark-bright, singular, combined periodic, periodic, Jacobian elliptic, and rational functions. Three-dimensional and two-dimensional graphs of solutions are presented. Also, the dynamical behavior of waves and the impact of time on solutions by selecting appropriate parameters are illustrated.

A Lie symmetry approach to traveling wave solutions, bifurcation, chaos and sensitivity analysis of the geophysical Korteweg–de Vries equation

Strong waves known as tsunamis are caused by earthquakes, landslides, or volcanic eruptions that traverse oceans. This article examines the geophysical Korteweg–de Vries (GPKdV) equation, which controls the propagation of tsunami waves in seas. The study involves exploring symmetry diminution exerted Lie group analysis, examining the properties of the dynamical structure with the help of bifurcation phase pictures, and researching the dynamic demeanor of the perturbed dynamical system utilizing chaos theory. Techniques such as 3D and 2D phase portraits, time series analysis, poincaré maps, looking into the existence of multistability in the autonomous structure under diverse beginning circumstances, lyapunov exponent, and bifurcation diagram are applied to identify chaotic demeanor. Furthermore, the study establishes general forms of solitary wave results, containing periodic, trigonometric, and singular soliton results, by using the unified Riccati equation expansion approach to address the examined problem analytically. These results are visually represented as 2D and 3D graphs with cautiously chosen parameters, along with their accompanying constraint conditions. Moreover, the sensitivity evaluation of the investigated equation is discussed and demonstrated pictorially. The discoveries revealed are intriguing, novel, and potentially helpful in understanding a wide range of physical events in engineering and science.

The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system’s phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

Investigation of soliton solutions to the Peyrard-Bishop-Deoxyribo-Nucleic-Acid dynamic model with beta-derivative

This study purposes to extract some fractional analytical solutions of the Peyrard-Bishop-Deoxyribo-Nucleic-Acid ([Formula: see text]-PBDNA) dynamic model with the beta-derivative by the unified Riccati equation expansion method (UREEM). Furthermore, we examine the role of various parameters of the fractional model on the soliton dynamic. The research focuses on computational biophysics and materials science, examining the impact of various parameters on the fractional model. This paper contributes to understanding soliton solutions and the [Formula: see text]-PBDNA dynamic model, demonstrating the applicability of the UREEM method to various fractional models. Some soliton solutions of the model are successfully generated by applying the UREEM. Implementing the UREEM, we take a fractional wave transformation to convert the model into a nonlinear ordinary differential equation. So, a linear equation system is generated. After the system is solved, the soliton solutions are gained for the appropriate solution sets. Finally, 3D, 2D and contour graphs of diverse solutions are depicted at suitable values of parameters. In addition, this paper presents 3D, 2D and contour graphs of various solutions with suitable parameter values. The results are beneficial for interpreting the model in future work and confirm that UREEM is effectively applicable to diverse fractional models, coupled with a comprehensive graphical analysis of how different parameters influence these solutions.

Symmetry reductions and conservation laws of a modified-mixed KdV equation: exploring new interaction solutions

<abstract><p>This article represented the investigation of the modified mixed Korteweg-de Vries equation using different versatile approaches. First, the Lie point symmetry approach was used to determine all possible symmetry generators. With the help of these generators, we reduced the dimension of the proposed equation which leads to the ordinary differential equation. Second, we employed the unified Riccati equation expansion technique to construct the abundance of soliton dynamics. A group of kink solitons and other solitons related to hyperbolic functions were among these solutions. To give the physical meaning of these theoretical results, we plotted these solutions in 3D, contour, and 2D graphs using suitable physical parameters. The comprehend outcomes were reported, which can be useful and beneficial in the future investigation of the studied equation. The results showed that applied techniques are very useful to study the other nonlinear physical problems in nonlinear sciences.</p></abstract>

On obtaining analytical soliton solutions of Drinfeld-Sokolov-Satsuma-Hirota equation via two efficient methods

In this work, we use the enhanced modified extended tanh method (eMETEM) and the unified Riccati equation expansion (UREEM) to find analytical solutions to the Drinfeld-Sokolov-Satsuma-Hirota equation (DSSH). The use of coupled nonlinear partial differential equations in the modeling of many physical phenomena, the origin of the Drinfeld-Sokolov-Satsuma-Hirota equation being the formation of the coupled system, and the fact that this model also constitutes a fundamental model in representing numerous physical events, primarily shallow water and coastal regions, have been the driving force behind the study. To visualize the obtained solutions, contour, two and three-dimensional plots are presented. The proposed methods have effectively generated a range of solitons, such as kink, singular, and periodic singular types. The graphic presentations complete the interpretation of the physical significance of the obtained kink and singular soliton types, and the interpretation of the obtained graphs within this framework. In this sense, the findings of the study will help to shape future research in this field.

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

In this paper, we explore analytical solutions for the (3+1)-dimensional time-fractional modified Korteweg–de Vries Zakharov-Kuznetsov equation, which incorporates a conformable derivative. Our interest in this model is driven by its significant role in simulating ion-acoustic waves in magnetized plasma. We adopt the unified Riccati equation expansion method and the new Kudrashov method to discover soliton solutions. Our approach uncovers various soliton types, such as kink, singular, periodic-singular, and bright solitons. We conduct a thorough analysis of how different parameters affect wave propagation, enhancing our study with descriptive figures and insightful observations. Furthermore, we delve into the modulation instability characteristic of this model. The influence of specific parameters, like wave number and the order of the conformable derivative, on wave dynamics is demonstrated through detailed visualizations. We also present 2D and 3D graphical representations of these solutions.

Symmetry and complexity: A Lie symmetry approach to bifurcation, chaos, stability and travelling wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvili equation

This work delves into the investigation of the nonlinear dynamics pertaining to the (3+1)-dimensional Kadomtsev-Petviashvili equation, which describes the propagation of long-wave with dissipation and dispersion in nonlinear media. The research entails an exploration of symmetry reductions using Lie group analysis, an analysis of the dynamical system’s characteristics through bifurcation phase portraits, and a study of the perturbed dynamical system’s dynamic behavior through chaos theory. Chaotic behavior is identified using various tools for detecting chaos, including the Lyapunov exponent, 3D phase portrait, Poincare map, time series analysis, and an exploration of the presence of multistability in the autonomous system under different initial conditions. Additionally, the research applies the unified Riccati equation expansion method to solve the considered equation analytically and constructs the general solutions of solitary wave solutions such as trigonometric function solutions, periodic and singular soliton solutions. These solutions come with their associated constraint conditions and are demonstrated through visual representations in the form of 2D, 3D, and density plots with carefully selected parameters. Furthermore, the stability analysis of the considered equation is also discussed and shown graphically. The results of this work are relevant and have applications in describing the propagation of long-wave with dissipation and dispersion in nonlinear media.

Dispersive optical soliton solutions in birefringent fibers with stochastic Kaup–Newell equation having multiplicative white noise

In this article, we study for the first time the stochastic Kaup–Newell equation in birefringent fibers. Three integration algorithms are applied, namely, the unified Riccati equation expansion method, the addendum to Kudryashov's method, and the addendum to modified sub‐ODE method. Many types of optical soliton solutions such as Jacobi‐elliptic function solutions, Weierstrass elliptic function solutions, bright solitons, dark solitons, singular solitons, and periodic function solutions are obtained. Numerical schemes give a visual perspective to the solutions derived analytically.

Optical Solitons for the Dispersive Concatenation Model

The study undertakes a comprehensive exploration of optical solitons within the context of the dispersive concatenation model, utilizing three distinct integration algorithms. These approaches, namely the enhanced Kudryashov' s method, the Riccati equation expansion approach, and the Weierstrass' expansion scheme, offer distinct perspectives and insights into the behavior of optical solitons. By employing the enhanced Kudryashov' s approach, the research uncovers a spectrum of soliton solutions, including straddled, bright, and singular optical solitons. This algorithm not only provides a nuanced understanding of the various soliton types but also highlights the occurrence of singular solitons that exhibit unique characteristics. The Riccati equation expansion approach, on the other hand, yields dark solitons in addition to singular solitons. This particular method expands our comprehension of soliton behavior by encompassing the presence of dark solitons alongside singular ones. This diversification contributes to a more comprehensive grasp of soliton phenomena. Furthermore, the application of the Weierstrass' expansion scheme extends the analysis to encompass bright, singular, and other variations of straddled solitons. This method introduces further complexity and diversity to the optical soliton. Importantly, the study meticulously addresses the parameter constraints that govern the behavior of these solitons. By providing a comprehensive presentation of these constraints, the research enhances the practical applicability of the findings, offering insights into the conditions under which these soliton solutions emerge.

New Dynamic Multiwave Solutions of the Fractional Peyrard–Bishop DNA Model

Abstract In this paper, we have studied the new solitary wave solutions of the beta-fractional derivative form of the Peyrard–Bishop DNA model (PB-DNAM). These solutions are responsible for analyzing the nonlinear interaction between the adjacent displacements of the DNA strand. To get these solutions, we have applied the generalized Riccati equation expansion method. Under different parametric conditions and fractional values, the obtained solutions show different wave patterns including w-shape, bright, combined dark-bright, periodic wave solutions, bell shape, m-shape, w-shape along with two bright solutions, and m-shape along with two dark solutions. These physical characteristics are analyzed thoroughly by graphical representations. The solutions show the successful application of the proposed method which will be helpful in finding analytical solutions to other nonlinear problems.

Highly dispersive optical solitons in fiber Bragg gratings with stochastic perturbed Fokas–Lenells model having spatio-temporal dispersion and multiplicative white noise

In this paper, we derive optical soliton solutions with highly dispersive optical solitons in fiber Bragg gratings with stochastic perturbed Fokas–Lenells model having spatio-temporal dispersion and multiplicative white noise. In this regard, we have applied two effective methods of distinct analytical mathematical methods to find explicit solutions to many physical and engineering problems, namely addendum Kudryashov’s (AK’s) method and the unified Riccati equation expansion method (UREEM). Bright soliton, Straddled solitary solutions, singular solitons, dark soliton are acquired.

Soliton Solutions of the Generalized Dullin-Gottwald-Holm Equation with Parabolic Law Nonlinearity

In this paper, soliton solutions of the generalized Dullin-Gottwald-Holm (gDGH) equation with parabolic law nonlinearity are investigated. The gDGH describes the behavior of waves in shallow water with surface tension. There are only a few studies in the literature regarding gDGH equation with parabolic law nonlinearity, and to our best knowledge, the unified Riccati equation expansion method (UREEM) has not been applied to this equation before. Many soliton solutions of the considered gDGH equation are successfully attained using the UREEM, which is a powerful technique for solving nonlinear partial differential equations. We verify that the obtained analytical solutions satisfy the gDGH equation using Mathematica. Furthermore, some plots of the acquired solitons are demonstrated with the aid of Matlab to examine the properties of the soliton solutions. The obtained results show that the considered gDGH equation admits dark, bright, singular, and periodic solutions. This study may contribute to a comprehensive investigation of the soliton solutions of the gDGH equation, which has practical applications in fields such as oceanography and nonlinear optics.

Stochastic dispersive Schrödinger–Hirota equation having parabolic law nonlinearity with multiplicative white noise via Ito calculus

Purpose:The aim of this study is to introduce the stochastic dispersive Schrödinger–Hirota equation with parabolic law nonlinearity (SHEPL) with multiplicative white noise via Ito calculus and investigate its stochastic optical soliton solutions. For this purpose, we used unified Riccati equation expansion (UREEM) and a sub-version of an auxiliary methods to construct analytical solutions. Methodology:Firstly, we implemented traveling wave transform into SHEPL with multiplicative white noise via Ito calculus and constructed imaginary and real parts of its nonlinear ordinary differential equation (NODE) form. Then, we presented solution algorithms of UREEM and a sub-version of an auxiliary methods and successfully implemented to obtained NODE. We created analytical solutions of SHEPL with using the appropriate solution sets which involve unknown parameters and necessary transformations. Then we presented graphical representations of these solution functions by investigating the noise effect on the soliton structure and presented the gained results. Findings:We proposed the stochastic SHEPL equation for the first time in this article. We obtained stochastic optical soliton solutions, reflected their graphical representations and also observed effect of noise factor on the obtained soliton structure. Originality:The stochastic model of SHEPL has not been studied and the results have not been reported.

Highly Dispersive Optical Solitons in Birefringent Fibers of Complex Ginzburg–Landau Equation of Sixth Order with Kerr Law Nonlinear Refractive Index

In this paper, we derived optical soliton solutions with a highly dispersive nonlinear complex Ginzburg–Landau (CGL) equation in birefringent fibers that have Kerr law nonlinearity. We applied two mathematical methods, namely the addendum Kudryashov’s method and the unified Riccati equation expansion method. Straddled solitary solutions, bright soliton, dark soliton and singular soliton solutions were obtained.This model represents the propagation of a dispersive optical soliton through a birefringent fiber. This happens when pulses propagating through an optical fiber split into two pulses.

(3+1)-dimensional Sasa–Satsuma equation under the effect of group velocity dispersion, self-frequency shift and self-steepening

Purpose:The aim of this study is to examine the effects of the coefficients of group velocity dispersion, self-frequency shift and self-steepening terms on the optical solitons of the (3+1)-dimensional Sasa–Satsuma equation which is important for modeling many physical events and it has recently been introduced to the literature. Methodology:To get desired results we have used the solution approach as following. The first step is defining the wave transform and applying it to the nonlinear partial differential equation (NLPDE). Then we have obtained the ordinary differential equation (ODE). Afterwards it is introduced and applied the new Kudryashov method (nKM) and unified Riccati equation expansion method (UREEM) to obtain the algebraic equation system. Finally, by determining the parameters from the solution of algebraic system we have substituted them in the solution function. Findings:By using the solution function we have obtained interesting soliton graphics under different parameter values. In addition to the basic solitons of the (3+1)-dimensional Sasa–Satsuma equation (SSE), V-shaped and periodic singular type solitons have also been obtained. Originality:The findings of the soliton behaviors under the parameter values effect has not been studied before. The results presented in the article are reported for the first time.

Cubic-quartic embedded solitons with [formula omitted] and [formula omitted] nonlinear susceptibilities having multiplicative white noise via Itô calculus

The objective of this paper is to study the cubic-quartic embedded solitons with χ(2) and χ(3) nonlinear susceptibilities having multiplicative white noise in the Itô sense. The unified Riccati equation expansion method and the addendum Kudryashov’s method are employed to accomplish this task. The straddled soliton, the combo bright-singular soliton, the bright soliton, the singular soliton solutions as long as periodic and rational solutions are obtained. The numerical schemes give a visual perspective to the solutions derived analytically.

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.

On the investigation of optical soliton solutions of cubic–quartic Fokas–Lenells and Schrödinger–Hirota equations

Purpose:When it comes to third and higher-order dispersion, the Schrödinger–Hirota equation is one of the main models developed outside the classical NLSE management models for optical soliton transmission. The cubic–quartic Fokas–Lenells equation is also one of the recently developed equations, which has importance in the field of telecommunications regarding optical soliton transmission in the absence of chromatic dispersion. In this study, in order to examine the optical solitons, the Schrödinger–Hirota equation in the presence of the chromatic dispersion and the cubic–quartic Fokas–Lenells equation discarding the chromatic dispersion were investigated. For this intent, by obtaining certain soliton types using the unified Riccati equation expansion method (UREEM), optical soliton solutions were obtained for both models and graphical representations and comments were made. Methodology:By developing appropriate computer algorithms and applying UREEM in the following ways, symbolic calculation software was made and analytical optical soliton solutions were obtained. Findings:Through computer algebra software, we plotted the obtained results via 3D, 2D views and we also illustrated the investigation of wave behavior caused by parameter change on 2D graphics. Originality:Different soliton behavior under the parameters effect of the Schrödinger–Hirota equation having chromatic dispersion and the cubic–quartic Fokas-Lenells equation is investigated and the obtained results are reported.