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This paper presents an improved Cham method as an efficient technique for obtaining analytical exact solutions to nonlinear partial differential equations. We apply this method to solve the nonlinear Schrödinger equation in conformable operator, a challenging equation frequently used in various scientific fields. The method enables us to derive different traveling wave solutions, such as kink, coindal waves, breather waves, periodic singular solutions, and periodic multi-wave solitons. We also provide graphical representations of some of the obtained solutions to help understand their dynamic characteristics. These results highlight the effectiveness and adaptability of the method and demonstrate its potential to solve other partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Fokas–Lenells equation, which is a derivation form of the nonlinear Schrödinger equation and can be used to describe nonlinearity in the propagation of optical pulses. To seek solutions for this nonlinearity, the Logistic method has been proposed in order to drive acceptable and understandable results regarding physical meaning. This method is defined based on the summation of an ordinary differential equation. Moreover, for further study, the Bernoulli (F ′ / F) -expansion method is used by considering hypothetical solutions as a function of the Bernoulli equation. Subsequently, solutions of the Fokas–Lenells equation are successfully acquired, displaying efficient results in hyperbolic, trigonometric, and exponential equations. The importance of these results appears in the definition of the wave function under the consideration of the appropriate coefficients. Hence, computer simulation is used for better understanding of the generated results. The dynamic behaviors of these solutions are demonstrated in 3D graphs, contour maps, and line plots, under applying various parameters, solutions of the waves show different soliton behaviors, including bright-dark, bright, and breather solitons. The results indicate that the exerted methods are novel, reliable, and effective approaches which can be employed in a wide range of nonlinear differential equations. These methods and their begotten results are far from the complexities of mathematical structures and therefore go beyond previous efforts in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to generate a Liouville integrable Hamiltonian hierarchy by introducing a specific matrix eigenvalue problem with four components. The adopted approach is the zero curvature formulation. A bi-Hamiltonian formulation is furnished through applying the trace identity, which shows the Liouville integrability of the resulting hierarchy. Two examples of generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations are presented. [ABSTRACT FROM AUTHOR]

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations. [ABSTRACT FROM AUTHOR]

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized (2+1)-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots. [ABSTRACT FROM AUTHOR]

Fractional calculus highlights its importance and has been expanded to many fields. This paper focuses on fractional integrable systems and their novel structural solutions to comply with the forefront development of nonlinear waves. Specifically, a system of fractional integrable coupled nonlinear Schrödinger (CNLS) equations are first derived from the associated linear spectral problem equipped with space-time conformable fractional derivatives of different orders. Then, by extending Darboux transformation (DT) and its generalized version to the derived fractional CNLS equations, we obtained fractional single-soliton solutions and double- and triple-semirational solutions that obey the dual power laws of independent variables. Under the dominant influences of space-time fractional orders, some novel solution structures are finally revealed based on the obtained solutions. In terms of characteristics, these spatial structures exhibit inclination, symmetry and spikes. Observing from the perspective of propagation, some of these obtained solutions have time-varying velocities and widths. From the perspective of spatiotemporal structures, some solutions form collapsed quadrangular pyramids. All of these novel solution structures cannot be supported by the integer-order CNLS equations. [ABSTRACT FROM AUTHOR]

This work plumbs the nonlinear dynamics of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation (gKPBe), which is used to describe some interesting physical phenomena in the fields of fluids. The resonance conditions of the soliton molecules on the (x , y), (x , z) and (y , z) planes are investigated and the soliton molecules are obtained on the basis of the N-soliton solutions that are extracted by virtue of the Hirota form. Furthermore, some novel hybrid interactions including the interaction between the soliton and soliton molecule, the interaction between the different soliton molecules are also explored. Finally, the sub-equation approach is exerted to explore the various wave solutions, which include the kinky wave, bright-dark wave and the singular periodic wave solutions. Correspondingly, the graphical descriptions of the attained solutions are drawn to present a better understanding of the physical attributes. The derived solutions can enlarge the exact solutions of the (3 + 1)-dimensional gKPBe and lead us to understand the nonlinear dynamic behaviors better. [ABSTRACT FROM AUTHOR]

Unlike Zabusky and Kruskal who exploited the Korteweg–de Vries traveling solitons to explain the FPU-recurrence phenomenon, we consider a more robust time periodic Kuznetzov–Ma breather to resolve the paradox. The nonlinear Schrödinger equation is derived from the equation of motion of β -FPU chain, by using the method of multiple scales combined with a quasi-discreteness approximation. Modulational instability leads to the generation of a nonlinear wave of finite background, known as the Kuznetzov–Ma breather. The spatial localization and time periodic profile of the Kuznetzov–Ma breathers make it ideal in mimicking the FPU-recurrence phenomenon, as underscored by results of numerical simulations. [ABSTRACT FROM AUTHOR]

Most recently, Baber et al. (Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability, Mod. Phys. Lett. B (2024) 2450341, doi:10.1142/S021798492450341X2450341-1) claimed to have obtained exact solutions of a two-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity that governs beam movement within a multimode optical fiber featuring a parabolic index shape, using the Jacobi Elliptic Function Expansion. Additionally, they have analyzed the modulation instability for the underlying model. It is the aim of this comment to point out that all results found in that work and published in the journal Mod. Phys. Lett. B (doi:10.1142/S021798492450341X2450341-1) are all erroneous, from the exact solutions of the considered equation to the modulation instability of the underlying model. [ABSTRACT FROM AUTHOR]

The study of modified nonlinear Schrödinger equation plays a significant role in the description of wave propagation through optical waveguides and rogue waves in ocean. The main aim of this work is to obtain solitons as well as other types of exact wave solutions to the modified nonlinear Schrödinger equation. The accurate traveling wave solutions are obtained using the (exp (− χ (ε))) -exponential expansion method and the modified auxiliary equation method. Consequently, soliton solutions and periodic wave solutions have been retrieved. The graphical illustrations of the results are provided for suitable values of the parameters involved in the solutions to explain the dynamical nature of the considered equation. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the nonlinear paraxial wave equation in Kerr media, a generalization of the nonlinear Schrödinger equation utilized to characterize the dynamics of optical beam propagation. By using three potent analytical methods, namely, the Sine-Gordon expansion method, the functional variable method, and the Bernoulli (G′/G)-expansion method, numerous novel soliton solutions are derived. These solutions, comprising hyperbolic, trigonometric and exponential functions, represent significant additions to the field of optics. Furthermore, we elucidate the physical characteristics of these novel optical soliton solutions by presenting a series of three-dimensional (3D) and two-dimensional (2D) graphs with appropriate parameter values. [ABSTRACT FROM AUTHOR]

This research paper is dedicated to the investigation of the optical soliton solutions with the power law of self-phase modulation (SPM) and the third-order dispersion (TOD), completely neglecting the chromatic dispersion (CD). The aim is focused on two main points. First, obtaining the straddled soliton solution which permits us to have bright and singular soliton by applying the new Kudryashov (nKM) scheme which was recently introduced as a simple, effective, and reliable method. Second, to observe the influences of the TOD, SPM, and inter-modal dispersion (IMD) on pure-cubic optical soliton dynamics in the absence of the CD. [ABSTRACT FROM AUTHOR]

The perturbed nonlinear Schrödinger equation is employed to characterize the dynamics of optical wave propagation when confronted with dissipation (or gain) and nonlinear dispersion that vary with both time and space. This equation serves as a fundamental model for investigating pulse dynamics within optical fibers and has application to nanofiber applications. This study successfully discovers optical solitons within this framework using the unified solver, Jacobi elliptic function, and simplest equation methods. We extract solutions using hyperbolic, trigonometric, and rational functions, including multi-solitons, dark, singular, bright, and periodic singular solitons. This study thoroughly compares our results with existing literature to provide novelty and significance of our findings. We have incorporated a detailed comparison between the methods employed in our study, which highlights their importance and strength. We have derived soliton solutions for the examined equations and generated 3D contour and 2D visual representations of the resulting solution functions. Alongside obtaining the soliton solutions, we offer a graphical exploration of how the parameters in the considered equations influence the system. [ABSTRACT FROM AUTHOR]

A model condition for the propagation of ultrashort pulses in optical fiber frameworks could be established through the extensive generalization of the nonlinear Schrödinger equation introduced by Triki and Biswas. This study uses two significant and effective trustworthy methods to examine the Triki–Biswas (TB) equation. Sardar Subequation Methods (SSM) and the generalized Kudryashov (KUD) method generate solutions for trigonometric, hyperbolic, exponential, and rational functions. These approaches are specifically designed for handling solitary wave patterns, dark-singular-mixed solitons, combined dark-bright solitons, singletons, bright solitons, exponential and rational functions, as well as periodic solitons. All of these solution classes contribute to the physical dynamics of outcomes. These findings represent an innovative extension of the soliton domain within the TB model and are being reported for the first time in our investigation. In addition, Mathematica simulations are used to display the 3D and 2D graphs to explain the identified solutions' physical dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we address various optical soliton solutions and demonstrate the different dynamics of solitary waves to a (3+1)-dimensional nonlinear Schrödinger equation (NLSE) with parabolic law (NLSE) using a newly created powerful and effective method named as the extended generalized Riccati equation mapping method. This technique presents an organized manner to reveal the essential dynamics. There is great significance of the nonlinear Schrödinger equations and their several formulations in numerous fields of science, particularly in nonlinear optics, optical fibers, quantum electronics, and plasma physics. Through the use of numerical simulations and mathematical analysis, we explore the characteristics and behavior of these solitary wave solutions in a variety of scientific contexts. These results demonstrate the essential complexity of the governing equation and yield novel derived solutions. These solutions contribute to a better understanding of nonlinear wave phenomena by highlighting the fundamental dynamics establishing solitary waves in the NLSE. To enhance our wider knowledge, we provide effective graphic representations of the nonlinear wave structures in the derived solutions utilizing a variety of graphs, including 3D, 2D, and density plots. Moreover, a specific transformation has been applied to transform the system into a planar dynamical system, and several phase portraits have been presented to examine its behavior. Furthermore, upon introducing a perturbed term, chaotic behavior has been observed across different parameter values through both two-dimensional and three-dimensional graphics. [ABSTRACT FROM AUTHOR]

In this work, the new nonlinear (1 + 1) -dimensional Schrödinger (NLS) equation, which describes wave propagation in optical fibers, has been successfully studied using the (G ′ / G) -expansion method and the Unified method (UM). These techniques have been applied for the first time in extracting new analytic solutions for the mentioned NLS equation. The obtained solutions exhibit various types of optical solitons, including singular solitons, solitary waves, periodic solitons, and wave collisions. These solutions have wide-ranging applications in various fields such as fiber optics, telecommunication systems, plasma physics, hydrodynamics, and nonlinear optics. The behavior of the derived solutions has been visually represented using three-dimensional, two-dimensional, and contour plots, considering appropriate choices of the involved parameters within specific time intervals. These plots provide insights into the dynamics and characteristics of the solutions. The effectiveness, reliability, and ease of application of the (G ′ / G) -expansion method and the UM are demonstrated through the obtained solutions. These mathematical tools have proven to be powerful in solving nonlinear partial differential equations, offering valuable contributions to the field of mathematical physics, soliton theory and nonlinear sciences. [ABSTRACT FROM AUTHOR]

Under investigation in this work is a coupled nonlinear Schrödinger equation (CNLSE), which can be used to describe the dynamics of light beams and pulses in P T -symmetric coupled waveguides. Its breather wave (BW) and rogue wave (RW) solutions are presented here. The BW solutions can be converted into various soliton solutions including the Akhmediev breather, Kuznetsov–Ma soliton breather and Peregrine soliton. The dynamics of the solutions are graphically discussed. Moreover, we observe that the two BWs can evolve into an Akhmediev breather with a Peregrine soliton. We hope that the results can be used to enrich dynamical behavior of BWs and rogue waves in the CNLSE. [ABSTRACT FROM AUTHOR]

The (2 + 1) -dimensional coupled nonlinear Schrödinger equation has versatile applications for modeling nonlinear waves in different areas such as optics, atmospheric science, fluid dynamics, and plasma physics. This study focuses on the propagation of optical solitons and their interaction within various mediums such as multi-mode fiber and fiber arrays. The Sardar-subequation method is applied to achieve the bright, dark, combined bright–dark, periodic, combined dark-singular and singular optical solitons solutions. The 3D, contour and 2D profiles for some of the assimilation solutions are also plotted to show the physical behavior of the solutions. The obtained results demonstrate the effectiveness of the adopted methodology in obtaining traveling wave solutions and have many applications in applied mathematics and engineering. The novelty of our work lies in the successful application of Sardar-subequation method in obtaining diverse soliton solutions and exploring their physical behavior. This study goes beyond previous efforts in the literature by presenting a comprehensive analysis of soliton dynamics. This ability to describe and predict the behavior of optical solitons in diverse mediums opens up new opportunities for investigating the existing systems. [ABSTRACT FROM AUTHOR]

The discrete Ablowitz–Ladik (AL) equation is the discrete version of nonlinear Schrödinger equation, which may have potential physical applications in nonlinear optics, polaron motion and anharmonic lattice dynamics. In this paper, a discrete reverse-time nonlocal coupled AL equation is first proposed and studied. First of all, we correspond this new discrete reverse-time nonlocal equation to continuous nonlocal coupled equation by use of the continuous limit technique. Second, we build the generalized (k , N − k) -fold Darboux transformation for this new discrete equation. As an application, we obtain some novel position controllable nonlocal singular rogue wave (RW) and period wave solutions on constant seed backgrounds, whose structures and positions are controlled by some special parameters. Moreover, we also study dynamical behaviors of some RW solutions via numerical simulations and large asymptotic analysis. These new results and phenomena may be helpful to comprehend some physical phenomena. [ABSTRACT FROM AUTHOR]

The nonlinear coupled Schrödinger equation in fiber Bragg gratings is studied in this paper. The existence of soliton solutions and periodic solutions are proved by qualitative analysis, and exact solutions are given, as well as the parameter condition of each solution is described. Then the modulation instability (MI) analysis is carried out and the linear stability criterion is given. In particular, external perturbation terms are introduced to prove that the equation exists chaotic behaviors. [ABSTRACT FROM AUTHOR]

In this paper, the nonparaxial nonlinear Schrödinger (NNLS) equation by considering its integrability which enables the propagation of ultra-broad nonparaxial beams in a planar optical waveguide is studied. The plenty numbers of solitary wave solutions by using Hirota's bilinear scheme are found, in addition, the bilinear transformation and also the related theorem for getting to the bilinear form of nonlinear system are considered. Two new simple approaches are implemented to recover periodic wave, bright soliton, singular, and singular soliton for this model. Because of the significance of the NNLS in modeling the propagation of solitons through an optical fiber, the recovered solitons are vital for describing and understanding a variety of fundamental physical processes. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions by providing the feasibility and reliability of the used procedure was also discussed. For more physical illustration and knowledge of the physical characteristics of this equation, some important solutions are discussed graphically in the form of 2D and 3D plots by selecting suitable parameters. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct a six-component integrable hierarchy associated with a matrix spatial spectral problem of arbitrary order. The adopted method is the zero curvature formulation. The corresponding Hamiltonian formulation is furnished by using the trace identity, which guarantees the Liouville integrability for the resulting hierarchy. Two illustrative examples of integrable equations of lower orders are six-component coupled nonlinear Schrödinger equations and modified Korteweg–de Vries equations. [ABSTRACT FROM AUTHOR]

Apply style for article title, author, affiliation and email as per stylesheet. Several decades ago, antennas had simple shapes that were described in Euclidean geometry. Nowadays, scientists try to make the structure of fractal geometry for applications in the field of electromagnetism, which has led to the development of new innovative antenna devices. Non-integer dimensional space (NDS) is useful to describe the concept of fractional space in fractal structure for real phenomenon of electromagnetic wave propagation. In this work, we investigate effects of NDS and normalized frequency on modulational instability (MI) gain in lossless left-handed metamaterials (LHM). We derive the nonlinear Schrödindiger equation (NLSE) with non-integer transverse laplacian. By means of linear stability analysis method, MI gain expression is also determined. Different forms of figures are obtained due to the signs of group velocity dispersion (GVD) and defocusing/focusing nonlinearity. We show how the increasing value of the normalized frequency enhances the amplitude as well as the bandwidth of MI gain, and waves are more unstable due to non-integer dimension. The obtained results are new and have a relatively newer application in telecommunication by constructing the fractal-shaped antennas operating in multi-frequency bands. [ABSTRACT FROM AUTHOR]

In this paper, we consider coupled variable coefficients nonlinear Schrödinger equations (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. With the help of a gauge transformation of the spectral problem, the generalized Darboux transformation (DT) for the VCNLS is constructed. As an application of the generalized DT, we first derive the second-order and third-order semi-rational solutions of the VCNLS. In addition, we investigate the dynamics of the second-order and third-order semi-rational solutions with different coefficients. Finally we find that the coefficients D (t) and R (t) mainly affect the shape of the above solutions, while the amplification or absorption coefficient (t) will change the amplitude. The obtained results will enrich the study of theory of the nonlinear localized waves. [ABSTRACT FROM AUTHOR]

The optical soliton solutions to the fractional nonlinear Schrödinger (NLS) equation in the presence of nonlinear oscillating coefficient with Beta and M-truncated derivatives are studied by applying a complex wave transformation that converts the fractional NLS equation to an ordinary differential equation. The optical solution structures are attained with the use of the Sardar sub-equation (SSE) method. The NLS equation is an important nonlinear complex model which governs the propagation of an optical pulse in a birefringent optical fiber. The fractional NLS equation is used in optical telecommunication, high-energy physics, gas dynamics, electrodynamics and ocean engineering. The graphical presentation of the attained results is also discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, higher-order localized waves for a coupled fourth-order nonlinear Schrödinger equation are investigated via a generalized Darboux transformation. The N th-order localized wave solutions of this equation are derived via Lax pair and Darboux matrix. Evolution plots are made and dynamical characteristics of the obtained higher-order localized waves are analyzed through numerical simulation. It is observed that rogue waves coexist with dark–bright solitons and breathers. The presented results also show that different values of the involved parameters have diverse effects on the higher-order localized waves. [ABSTRACT FROM AUTHOR]

With an increasing demand for larger bandwidth and solving the problem of internet bottleneck, the formation of solitons with fractional temporal evolution has prompted substantial attention in the field of optics. In this paper, we have used three methods namely, the new mapping method, the unified Riccati equation method, and the new modified sub-ODE method to construct the new soliton solutions to the perturbed nonlinear Schrödinger equation (NLSE) with fractional temporal evolution. Two types of nonlinearities are considered namely, the Kerr law and the power law for all the three utilized methods. Existence criteria for all soliton solutions are presented. We provide two- and three-dimensional graphical illustrations of some acquired solutions. This work also proves the reliability, power, and effectiveness of these new analytical techniques for solving other nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, exact solutions of nonlinear Schrödinger equation with third-order dispersion in constant potential are achieved. This equation plays an important role in mathematical physics such as nonlinear optics. To deal with the mentioned equation, Bernoulli (G / G) -expansion method has been proposed by exerting some transformation and using homogenous balance. On the other hand, (G / G) -expansion method has also been used to obtain the addition results and find out the differences between the proposed method and the applied method. As a result of this study, the solutions of these two methods are expressed by hyperbolic function solutions and trigonometric function solutions. To have acceptable concept of dynamic structures of begotten results and find relation between these solutions and certain parameters, some figures are illustrated when the potential is taken as different values, which show us bright soliton and periodic wave. To complete our study, the Lagrangian and Hamiltonian of nonlinear Schrödinger equation with third-order dispersion are also established. [ABSTRACT FROM AUTHOR]

Nonlinear evolution equations play enormous significant roles to work with complicated physical phenomena located across the nature world. The Schrödinger type equations bearing nonlinearity are important models that flourished with the wide-ranging arena concerning plasma physics, nonlinear optics, fluid flow and the theory of deep-water waves. In this exploration, we retrieve the soliton and other solutions in an appropriate form to the coupled nonlinear Schrödinger equations by means of the improved tanh method and the rational (G ′ / G) -expansion method. The suggested system of nonlinear Schrödinger equations is turned into a differential equation of a single variable through executing some operations. Thereupon, successful implementation of the advised techniques regains the abundant exact traveling wave solutions. The obtained solutions are figured out in the profiles of three-dimensional (3D), two-dimensional (2D) and contour by assigning suitable values of the involved unknown constants. These diverse graphical appearances enable the researchers to understand the underlying mechanisms of intricate phenomena of the leading equations. The individual performances of the employed methods are praiseworthy which deserve further application to unravel any other nonlinear partial differential equations arising in various branches of science. [ABSTRACT FROM AUTHOR]

The sub-equation method is implemented to construct exact solutions for the conformable perturbed nonlinear Schrödinger equation. In this paper, we consider three different types of nonlinear perturbations: The quadratic–cubic law, the quadratic–quartic–quintic law, and the cubic–quintic–septic law. The properties of the conformable derivative are discussed and applied with the help of a suitable wave transform that converts the governing model to a nonlinear ordinary differential equation. Furthermore, the order of the expected polynomial-type solution is obtained using the homogeneous balancing approach. Dark and singular soliton solutions are derived. [ABSTRACT FROM AUTHOR]

The generalized mixed nonlinear Schrödinger equation is the core of this thesis study and it plays a pivotal position in many physical applications. In this paper, on the whole, in the framework of the Riemann–Hilbert approach, by analyzing the spectral problem of the Lax pair and applying the transformation of the matrix, the Riemann–Hilbert problem of the generalized mixed nonlinear Schrödinger equation for specific values of the parameters is constructed. After that, in the case of irregularity, the N -soliton solutions of this equation at specific values of the parameters can be presented by the scattering transformation. In particular, some three-dimensional images of this nonlinear system can be graphically depicted. [ABSTRACT FROM AUTHOR]

This paper aims to compute solitary wave solutions and soliton wave solutions based on the ansatz methods to the perturbed nonlinear Schrödinger equation (NLSE) arising in nano-fibers. The improved tan (/ 2) -expansion method and the rational extended sinh–Gordon equation expansion method are used for the first time to obtain the new optical solitons of this equation. The obtained results give an accuracy interpretation of the propagation of solitons. We held a comparison between our results and those are in the previous work. The outcome indicates that perturbed NLSE arising nano-fibers is used in optical problems. Finally, via symbolic computation, their dynamic structure and physical properties were vividly shown by three-dimensional, density, and t -curves plots. These solutions have greatly enriched the exact solutions of (2+1)-dimensional perturbed nonlinear Schrödinger equation in the existing literatures. [ABSTRACT FROM AUTHOR]

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved tan (ϕ (ξ) / 2) expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions. [ABSTRACT FROM AUTHOR]

We present the complete analysis of traveling wave solutions to a special kind of nonlinear Schrödinger equation with logarithmic nonlinearity, and obtain all traveling wave solutions. As a result, we prove this equation does not have any Gaussian traveling wave solution. However, by modifying this equation into another form, we can actually obtain a Gaussian traveling wave solution, which verifies the conclusion that existing Gaussian traveling solution requires two restrictions: (1) balance between the dispersion terms and logarithmic nonlinearity; and (2) balance of the parameters. [ABSTRACT FROM AUTHOR]

In this work, we present a collective variable (CV) approach to establish dispersive solitary wave solutions for the Kaup–Newell Equation (KNE). The full CV theory has been utilized to enunciate the soliton molecules through its ground-laying parameters including the power of each pulse, phase and center-of-mass. Additionally, the dynamics of an ultra short pulse has been analyzed by using CV. This work may be utilized for various dynamics of solitons as well as the influence the amplitude, temporal position, frequency, phase and chirp on the solitons' nonlinear parameters. Moreover, the numerical simulations have been designed by means of appropriate parameter values to explain more on the obtained results. [ABSTRACT FROM AUTHOR]

This paper comprises the different types of optical soliton solutions of an important Triki–Biswas model equation with beta-time derivative. The beta derivative is considered as a generalized version of the classical derivative. The aforesaid model equation is the generalization of the derivative nonlinear Schrödinger equation that describes the ultrashort pulse propagation with non-Kerr dispersion. The study is carried out by means of a novel beta derivative operator and three efficient integration schemes. During this work, a sequence of new optical solitons is produced that may have an importance in optical fiber systems. These solutions are verified and numerically simulated through soft computation. [ABSTRACT FROM AUTHOR]

This paper presents the modulation instability (MI) analysis and the different types of optical soliton solutions of the Triki–Biswas model equation. The aforesaid model equation is the generalization of the derivative nonlinear Schrödinger equation which describes the ultrashort pulse propagation with non-Kerr dispersion. The study is carried out by means of a novel efficient integration scheme. During this work, a sequence of optical solitons is produced that may have an important in optical fiber systems. The results show that the studied model hypothetically has incredibly rich optical soliton solutions. The constraint conditions for valid soliton solutions are also reported. The gained results show that the applied method is efficient, powerful and can be applied to various complex models with the help of representative calculations. [ABSTRACT FROM AUTHOR]

In this study, the auxiliary equation method is applied successfully to the Lonngren wave equation. Bright soliton, bright–dark soliton solutions are produced, which play an important role in the distribution and distribution of electric charge. In the conclusion and discussion section, the effect of nonlinearity term on wave behavior in bright soliton traveling wave solution is examined. The advantages and disadvantages of the method are discussed. While graphs representing the stationary wave are obtained, special values are given to the constants in the solutions. These graphs are presented as 3D, 2D and contour. [ABSTRACT FROM AUTHOR]

This paper presents the (1+1)-dimensional chiral nonlinear Schrödinger equation (1DCNLSE) and (2+1)-dimensional chiral nonlinear Schrödinger equation (2DCNLSE) that define the edge states of the fractional quantum hall effect. In this paper, we implement the Riccati–Bernoulli sub-ODE method in reporting the solutions of these two nonlinear physical models. As a result of this, some singular periodic waves, dark and singular optical soliton solutions are generated for these models. Some of the acquired solutions are illustrated by three-dimensional (3D) and two-dimensional (2D) graphs utilizing suitable values of the parameters with the help of the MAPLE software to demonstrate the importance in the real-world of the presented equations. [ABSTRACT FROM AUTHOR]

Nonlinear evolution equations of arbitrary order bearing a significantly broad range of capability to illustrate the underlying behavior of naturalistic structures relating to the real world, have become a major source of attraction of scientists and scholars. In quantum mechanics, the nonlinear dynamical system is most reasonably modeled through the Schrödinger-type partial differential equations. In this paper, we discuss the (2+1)-dimensional time-fractional nonlinear Schrödinger equation and the (1+1)-dimensional space–time fractional nonlinear Schrödinger equation for appropriate solutions by means of the recommended enhanced rational (G ′ / G) -expansion technique adopting Cole–Hopf transformation and Riccati equation. The considered equations are turned into ordinary differential equations by implementing a composite wave variable replacement alongside the conformable fractional derivative. Then a successful execution of the proposed method has been made, which brought out supplementary innovative outcomes of the considered equations compared with the existing results found so far. The well-generated solutions are presented graphically in 3D views for numerous wave structures. The high performance of the employed technique shows the acceptability which might provide a new guideline for research hereafter. [ABSTRACT FROM AUTHOR]

In this paper, we derive solutions to the derivative nonlinear Schrödinger equation, which are associated to real and complex discrete eigenvalues of the Kaup–Newell spectral problem. These solutions are obtained by investigating double Wronskian solutions of the coupled Kaup–Newell equations and their reductions by means of bilinear method and a reduction technique. The reduced equations include the derivative nonlinear Schrödinger equation and its nonlocal version. Some obtained solutions allow not only periodic behavior, but also solitons on periodic background. Dynamics are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, the coupled mixed derivative nonlinear Schrödinger equations are investigated, which govern the propagation of the femtosecond optical pulse in optical fibers. First of all, based on the soliton solutions in bilinear form, the breathers are constructed by choosing a pair of complex conjugate wave numbers. Then, the interactions between a breather and either an anti-dark soliton or a dark soliton are studied according to the existence conditions of dark and anti-dark solitons. The double-pole solution can also be obtained by a coalescence of two wave numbers. In addition, the influence of physical parameters on the phases and propagation direction of the breathers and double-pole solitons is studied by the qualitative analysis and graphical illustration. [ABSTRACT FROM AUTHOR]

Spin excitations for the magnetic materials are used in the nonlinear signal processing devices and microwave communication systems. Under consideration in this paper is a (2 + 1) -dimensional nonlinear Schrödinger (NLS) equation which describes the spin dynamics for a Heisenberg ferromagnetic spin chain. Through a reduced transformation, we convert such an equation into the (1 + 1) -dimensional focusing NLS equation. Via the rogue-periodic solutions associated with two types of the Lie symmetry transformations of the NLS equation, we present the lump- and rogue-periodic solutions. Besides, the lump and mixed lump-soliton solutions are deduced. We graphically investigate the lump- and rogue-periodic waves and find that the amplitudes of the lumps and rogue waves are negatively related to | A | and | γ | ; the distances between two valleys of the lumps and widths of the rogue waves are affected by J and J 1 , where A is the uniaxial crystal field anisotropy parameter, J and J 1 are related to the bilinear exchange interaction, γ is the lattice parameter. [ABSTRACT FROM AUTHOR]

In this work, we investigated one-dimensional and two-dimensional quantum systems with higher-order dispersions and higher-order nonlinear interactions. Based on the high-order nonlinear Schrödinger equation (NLSE) and via the F -expansion method, we derived the analytical dark soliton solution for the one-dimensional system first. By applying the self-similar method and using the results of the one-dimensional case, the analytical dark soliton solution of the system in the two-dimensional case was derived. The dynamic evolution pattern of the two-dimensional dark soliton is pictorially demonstrated. The theoretical results of our work can be used to guide the detection and experimental study of dark soliton in a two-dimensional quantum system, using high-order dispersion and higher-order nonlinear interactions. [ABSTRACT FROM AUTHOR]

The new types of nonlinear localized state of generalized order parameter in the three-layered structure of band antiferromagnets with congruent sections of the Fermi surface are obtained. The nonlinear effects inside the narrow plane defects between the layers representing the local magnetization in the direction of polarization of the spin density wave are described by the nonlinear source function. The variation of the Ginsburg–Landau functional near the Lifshitz point leads to nonlinear Schrödinger equation (NSE) with cubic nonlinear terms and self-consistent nonlinear potential modeling the plane defects. The dispersion equations of two type stationary nonlinear localized states are derived. The solutions of dispersion equations in explicit analytical form in dependence on characteristics of plane defects and the distance between them are found and analyzed. The localized states of order parameter existing only near the plane defects with nonlinear response are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we examine the Fokas–Lenells equation (FLE) that depicts the promulgation of ultra-short pulsation in visual fibers while confirming the terms of the following asymptotic arrangement beyond those indispensable for the nonlinear Schrödinger equation. In addition the model includes both spatio–temporal dispersal and self-steepening terms. Then, we discuss deep visual solutions of the FLE via taking the modified Kudryashov method and the extended tanh expansion method. [ABSTRACT FROM AUTHOR]

Based on inverse scattering transformation, a variable-coefficient fifth-order nonlinear Schrödinger equation is studied through the Riemann–Hilbert (RH) approach with zero boundary conditions at infinity, and its multi-soliton solutions with N distinct arbitrary-order poles are successfully derived. By deriving the eigenfunction and scattering matrix, and revealing their properties, a RH problem (RHP) is constructed based on inverse scattering transformation. Via solving the RHP, the formulae of multi-soliton solutions are displayed when the reflection coefficient possesses N distinct arbitrary-order poles. Finally, some appropriate parameters are selected to analyze the interaction of multi-soliton solutions graphically. [ABSTRACT FROM AUTHOR]

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple. [ABSTRACT FROM AUTHOR]

In this work, the soliton solutions of a (3 + 1)-dimensional coupled nonlinear Schrödinger system with time-dependent coefficients arising in optical fiber has been studied analytically. A complex traveling wave ansatz is used to transform the proposed coupled partial differential equations into ordinary differential equations (ODEs). Then, we solved the obtained nonlinear ODEs for the envelope function in the traveling wave parameter. The obtained waves envelope have the forms of bright and dark soliton-like solutions which are considered as new solutions of the studied system. As a kind of completing the analysis, the modulation instability is discussed based on the linear stability analysis. [ABSTRACT FROM AUTHOR]