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In this paper, we study the Radhakrishnan–Kundu–Lakshmanan equation (RKLE) that plays a key role in modeling the nonlinear propagation of the light pulses. Two novel methods namely Wang's Bäcklund transformation-based method and Wang's direct mapping method are employed to construct the diverse exact optical soliton solutions. Abundant exact solutions like the bright soliton solution, dark soliton solutions, singular soliton solutions, algebraic solitary wave solutions and the singular periodic wave solutions expressed in the form of the rational function type, double-exp function type, Sin-Cos function type, Tan function type, Cot function type, Sinh-Cosh function type, Tanh function type, Coth function type, Sech function type, Csch function type, Sec function type and Csc function type are obtained. The dynamics of the different exact solutions are presented through the 3D plot, contour and 2D curve, and the corresponding physical interpretations are elaborated in detail. It is confirmed that the proposed methods are effective and powerful, and can be adopted to construct the optical solitons of the other PDEs arising in the optical physics. [ABSTRACT FROM AUTHOR]

This paper recovers stationary optical solitons for the perturbed Radhakrishnan–Kundu–Lakshmanan equation with nonlinear chromatic dispersion. There are eight approaches that have been implemented to recover these solitons. A full spectrum of solitons is recovered including straddled solitons. The numerical simulations portray a complete picture that supplements the analytical results.

In this article, stochastic solutions of Radhakrishnan–Kundu–Lakshmanan equation (RKLE) with multiplicative noise are investigated. Studying is conducted with the aid of the modified extended direct algebraic integration technique which depended on the extended Riccati equation. To aid in our evaluation, a graphic representation of noise’s impact on the retrieved solutions is presented using MATLAB software packages.

The nonlinear perturbed Radhakrishnan-Kundu-Lakshmanan (RKL) equation is a significant model in the engineering field for analyzing data transmission through nonlinear optical fibers. The two-variable (ψ′/ψ,1/ψ)-expansion method is a reliable scheme to compute self-controlled solitary wave solutions. In this study, we have constructed a wide range of geometric shapes and further inclusive soliton solutions comprising rational, trigonometric, and hyperbolic functions and their integration to the stated equation, devising the acknowledged method. The physical details of the ensued solitary wave solutions and the effects of fractional parameters are explained by sketching three- and two-dimensional graphs. For the appropriate values of the parameters, the periodic, V-shaped, bell-shaped, compacton, singular periodic, flat bell-shaped, plane-shaped, and some other types of solitons are formulated. The effect of the fractional derivative has been demonstrated by plotting the two-dimensional graphs for different values of the fractional order. It is important to note that changes in physical and auxiliary parameters cause variations in the wave profile. The formerly described physical phenomena may be better understood using the standard and new types of solitons. The results obtained exhibit the effectiveness, compatibility, and reliability of the two-variable (ψ′/ψ,1/ψ)-expansion approach for extracting soliton solutions of fractional-order nonlinear evolution equations in science, technology, and engineering.

The main objective of this paper is to study the soliton solutions and dynamics analysis of the fractional Radhakrishnan–Kundu–Lakshmanan equation with multiplicative noise in the Stratonovich sense. Firstly, the wave transformation is used to obtain the nonlinear ordinary differential equation, and then the nonlinear ordinary differential equation is transformed into a two-dimensional plane dynamic system with a Hamiltonian system. Secondly, the phase portrait and sensitivity of the plane dynamic system and its perturbed system are studied using Maple software. Thirdly, the soliton solutions of the stochastic fractional Radhakrishnan–Kundu–Lakshmanan equation can be constructed, and the Jacobian function solutions and hyperbolic function solutions are obtained. Finally, some three-dimensional and two-dimensional diagrams of the obtained solutions are also drawn. Moreover, the modulation stability of the equation under consideration is also given.

This paper retrieves different types of solutions such as M -shaped rational solutions with one and two kinks, homoclinic breather, multi-waves, periodic cross-kink and kink-cross rational solutions for Radhakrishnan–Kundu–Lakshmanan equation (RKLE). We will study the M -shaped interaction with rogue and kink, we will also obtain the M -shaped interaction with periodic and kink, periodic cross-rational solutions. At the end, we will express the graphical representation for our newly achieved solutions. [ABSTRACT FROM AUTHOR]

The main purpose of this study was to produce abundant new types of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation that represents unstable optical solitons that emerge from optical propagations through the use of birefringent fibers. These new types of soliton solutions have behaviors that are bright, dark, W-shaped, M-shaped, periodic trigonometric, and hyperbolic and were not realized before by any other method. These new forms have been detected by using four different techniques, which are, the extended simple equation method, the Paul-Painlevé approach method, the Ricatti-Bernoulli-sub ODE, and the solitary wave ansatz method. These new solitons will be arranged to create a soliton catalog with new impressive behaviors and they will contribute to future studies not only for this model but also for the optical propagations through birefringent fiber.

The nonlinear perturbed Radhakrishnan-Kundu-Lakshmanan (RKL) equation is a significant model in the engineering field for analyzing data transmission through nonlinear optical fibers. The two-variable (ψ ′ / ψ , 1 / ψ)-expansion method is a reliable scheme to compute self-controlled solitary wave solutions. In this study, we have constructed a wide range of geometric shapes and further inclusive soliton solutions comprising rational, trigonometric, and hyperbolic functions and their integration to the stated equation, devising the acknowledged method. The physical details of the ensued solitary wave solutions and the effects of fractional parameters are explained by sketching three- and two-dimensional graphs. For the appropriate values of the parameters, the periodic, V-shaped, bell-shaped, compacton, singular periodic, flat bell-shaped, plane-shaped, and some other types of solitons are formulated. The effect of the fractional derivative has been demonstrated by plotting the two-dimensional graphs for different values of the fractional order. It is important to note that changes in physical and auxiliary parameters cause variations in the wave profile. The formerly described physical phenomena may be better understood using the standard and new types of solitons. The results obtained exhibit the effectiveness, compatibility, and reliability of the two-variable (ψ ′ / ψ , 1 / ψ)-expansion approach for extracting soliton solutions of fractional-order nonlinear evolution equations in science, technology, and engineering. [ABSTRACT FROM AUTHOR]

The main purpose of this study was to produce abundant new types of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation that represents unstable optical solitons that emerge from optical propagations through the use of birefringent fibers. These new types of soliton solutions have behaviors that are bright, dark, W-shaped, M-shaped, periodic trigonometric, and hyperbolic and were not realized before by any other method. These new forms have been detected by using four different techniques, which are, the extended simple equation method, the Paul-Painlevé approach method, the Ricatti-Bernoulli-sub ODE, and the solitary wave ansatz method. These new solitons will be arranged to create a soliton catalog with new impressive behaviors and they will contribute to future studies not only for this model but also for the optical propagations through birefringent fiber. [ABSTRACT FROM AUTHOR]

In this article, we are attracted to discover the multiple-optical soiltons in birefringent fibers for Radhakrishnan–Kundu–Lakshmanan equation (RKLE) by applying the Sardar-subequation method (SSM) and the new extended hyperbolic function method (EHFM). We construct the solutions in the form of exponential, trigonometric, and hyperbolic functions solitons solutions like mixed complex solitons and multiple-optical solitons solutions. In addition, singular periodic wave solutions are constructed, and the restraint conditions for the presence of soliton solutions are also defined. Moreover, the physical interpretation of the obtained solutions is disclosed in forms of 3D and 2D plots for different suitable parameters. The attained results indicate that the implemented computational scheme is straight, proficient, and brief and can be applied in more complex phenomena with the associate of representative computations. We have obtained several sorts of new solutions.

In this study, we express the Radhakrishnan–Kundu–Lakshmanan equation with an arbitrary index of n ∈ Q . We investigated the solitary wave solutions of the Radhakrishnan–Kundu–Lakshmanan equation by mean of the Jacobi elliptic function expansion technique. As a result, we constructed several distinct solutions include dark, bright, singular, periodic, hyperbolic, trigonometric, and Jacobi elliptic function types solutions. To highlight the dynamic behavior of the generated solutions, specific values for the parameters are also assigned. The above techniques could also be employed to get a variety of exact solutions for other nonlinear models in physics, applied mathematics, and engineering. [ABSTRACT FROM AUTHOR]

In this work, we examine optical solitons to the Radhakrishnan–Kundu–Lakshmanan equation (RKL) equation which describes the optical pulses in birefringent fiber (Raza and Javid in J Appl Anal Comput 10:1375–1395, 2020; Seadawy et al. in Opt Quant Electron 53:324, 2021) by using the New Generalized Auxiliary Equation Method (NGAEM). After some mathematical transformations, owing some constraint relations it is obtained two categories of soliton solutions. The first includes bright and dark optical solitons, while in the second class it is divulged the combined bright-dark and bright-bright optical solitons. Taking some suitable parameters of the model and the NGAEM, it is put up W-shaped optical solitons and diverse other solutions. Thereafter, we use the continuous waves as solutions of the model with small perturbations, to show the effects of the TOD, ellipticity angle and XPM on the Modulation Instability (MI) gain in normal and anomalous dispersion regime. It has been indicated that the third-order dispersion in normal/aanomalous dispersion regime can generate MI growth rate. At the same time, the ellipticity angle and such others parameters of the model play an important role during the MI growth rate (gain). Compared the obtained appropriate results in terms of analytical results and dynamics of the MI to Refs. (Raza and Javid 2020; Seadawy et al. 2021; Yepez-Martinez et al. in Chin J Phys 58:137–150, 2019; Drummond et al. in Opt Commun 78:137–142, 1990; Li et al. in Commun Theor Phys 65:231–236, 2016), they are new in our knowledge. [ABSTRACT FROM AUTHOR]

In this article, stochastic solutions of Radhakrishnan–Kundu–Lakshmanan equation (RKLE) with multiplicative noise are investigated. Studying is conducted with the aid of the modified extended direct algebraic integration technique which depended on the extended Riccati equation. To aid in our evaluation, a graphic representation of noise's impact on the retrieved solutions is presented using MATLAB software packages. • Stochastic Radhakrishnan–Kundu–Lakshmanan equation. • Modified extended direct algebraic technique. • Stochastic solitons. • Noise's impact on the exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the generalized exponential rational function method is applied to obtain analytical solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. We obtain novel soliton, traveling waves and kink-type solutions with complex structures. We also present the two- and three-dimensional shapes for the real and imaginary part of the solutions obtained. It is illustrated that generalized exponential rational function method (GERFM) is simple and efficient method to reach the various type of the soliton solutions. [ABSTRACT FROM AUTHOR]

Abstract This paper applies Laplace–Adomian decomposition scheme to display numerical dispersive bright and dark optical solitons that are modeled by Radhakrishnan–Kundu–Lakshmanan equation. The computational results are achieved using MATHEMATICA. The maximum error for these plots is also displayed. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the soliton solutions and dynamics analysis of the fractional Radhakrishnan–Kundu–Lakshmanan equation with multiplicative noise in the Stratonovich sense. Firstly, the wave transformation is used to obtain the nonlinear ordinary differential equation, and then the nonlinear ordinary differential equation is transformed into a two-dimensional plane dynamic system with a Hamiltonian system. Secondly, the phase portrait and sensitivity of the plane dynamic system and its perturbed system are studied using Maple software. Thirdly, the soliton solutions of the stochastic fractional Radhakrishnan–Kundu–Lakshmanan equation can be constructed, and the Jacobian function solutions and hyperbolic function solutions are obtained. Finally, some three-dimensional and two-dimensional diagrams of the obtained solutions are also drawn. Moreover, the modulation stability of the equation under consideration is also given. • This paper studies the stochastic space-fractional Radhakrishnan–Kundu–Lakshmanan equation. • The exact solutions of this mode are obtained by using the bifurcation method and qualitative theory of dynamical systems. • Phase portrait, bifurcation, chaotic pattern and modulation stability of the equation are studied. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The current work handles dispersive optical solitons in fiber Bragg gratings with Radhakrishnan–Kundu–Lakshmanan equation. Bright, dark and singular solitons are retrieved by the extended auxiliary equation scheme and unified Riccati equation methodology. • The current paper handles dispersive optical solitons in fiber Bragg gratings by the aid of two integration technologies. • The governing model is the Radhakrishnan–Kundu–Lakshmanan (RKL) equation. [ABSTRACT FROM AUTHOR]

The generalized Radhakrishnan–Kundu–Lakshmanan equation has been considered and its general solutions in terms of the Jacobi elliptic functions have been obtained. Two possible strategies have been discussed for obtaining the solutions and the subtle difference between them has also been highlighted to explain the general nature of the obtained solutions. [ABSTRACT FROM AUTHOR]

This paper studies the perturbed Radhakrishnan–Kundu–Lakshmanan (RKL) equation and its special form, the generalized RKL equation. Kerr law nonlinearity is considered for both equations. The Riccati-Bernoulli sub-ODE technique is implemented to these nonlinear equations so that their optical solitons and complex wave solutions are retrieved. In order to derive a sequence of solutions of considered equations, Bäcklund transformation can be carried out. 3D and 2D graphics of some solutions are depicted for chosen suitable parameters. [ABSTRACT FROM AUTHOR]

The solution of the variable coefficient Radhakrishnan–Kundu–Lakshmanan equation is obtained by using the similarity reduction method. It is found that the variable coefficients appearing in the equation are not independent and certain conditions need to be imposed on them in order to find analytical solutions. The phase and the envelope of the travelling wave solution are obtained in terms of Jacobi elliptic cosine function. [ABSTRACT FROM AUTHOR]

The motivation for this paper comes after reading recent paper by Khalida Bibi (Bibi, 2021). In this paper K. Bibi tried to find the exact solutions of the Radhakrishnan–Kundu–Lakshmanan equation. We illustrate that the method by K. Bibi is absurd. We also show that his approach should not be applied to this type of equations. We also demonstrate that the solutions on the Radhakrishnan–Kundu–Lakshmanan equation by K. Bibi are wrong. In this paper we also give the correct solution of the Radhakrishnan–Kundu–Lakshmanan equation using the traveling wave reduction. [ABSTRACT FROM AUTHOR]

The Radhakrishnan–Kundu–Lakshmanan equation is studied with arbitrary refractive index. The partial differential equation is not integrable and we look for exact solutions using the traveling wave reduction. The overdetermined system of equations is presented for the packet wave envelope. Two compatibility conditions for existence of exact solutions to the overdetermined system of equations are used to construct solutions of nonlinear ordinary differential equations. Exact solutions in the form of periodic and solitary waves of the generalized Radhakrishnan–Kundu–Lakshmannan equation are given. [ABSTRACT FROM AUTHOR]

In this paper, the Φ 6 -model expansion method is applied to obtain the new exact solutions of the perturbed Radhakrishnan-Kundu-Lakshmanan equation that appears with Kerr law nonlinearity. The obtained exact solutions are based on the Jacobi elliptic function, hyperbolic function and trigonometric function solutions. Specific values of the parameters consistent with the constraint conditions are also assigned to reveal the dynamic behavior of the solutions. [ABSTRACT FROM AUTHOR]