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Preparation of atomic Bose–Einstein condensate (BEC) with tunable interactions in a quasi-one-dimensional (quasi-1D) optical trap is essential for both the observation of bright matter-wave solitons and the quantum simulation based on the discrete atomic momentum states. However, the quasi-1D BEC has been obtained by a complex process, which includes the creation of a three-dimensional BEC and its adiabatic transform into a quasi-1D trap. Here, we report the direct creation of a quasi-1D BEC of 133 Cs atoms by the fast evaporative cooling of ultracold atoms prepared by the degenerated Raman sideband cooling. We produce the pure BEC of up to 5.5 × 10 4 atoms in a quasi-1D optical trap with an evaporative time of 6 s. We demonstrate the anisotropic expansion of the atomic cloud after the release from the quasi-1D trap and study the dependence of both the phase space density and the temperature on the number of atoms in the trap during the evaporative cooling. Our study facilitates the promising applications of quasi-1D interacting atomic gases. [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]

This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, we use symbolic computation and ansatz function schemes, to investigate the soliton solutions for the nonlinear Schrödinger equation (NLSE) along with Kudryashov's quintuple self-phase modulation system (KQSPMS) including dual-form of nonlocal nonlinearity (DFNLN). We initially determine the ordinary differential (OD) form for this model through a variable transformation. Then we introduce numerous new dynamical soliton types: the M-shaped rational soliton, the M-shaped interaction between one and two stripe solitons, the periodic cross-M-shaped rational (PCMR) soliton, the periodic cross-kink (PCK) soliton, multi-waves, and the homoclinic breather soliton. Secondly, we determine the partial differential (PD) form for this model through a variable transformation. Moreover, a lump soliton, a periodic wave, a rogue wave, a lump interaction with a periodic and kink wave, and three different types of breather soliton will obtain. We'll demonstrate these solutions' unique structure and extremely interesting interaction behavior. We'll also use graphs (3-D and contour plots) to discuss the dynamics of the results after setting the parameters to the proper values. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities is considered. A series of direct transformations is performed for the traveling wave reduction of the original equation. The implicit function method is used for nonlinear ordinary differential equation with some constraints on the parameters. Some analytical solutions are found in the form of periodic and solitary waves. Three conservation laws, corresponding to the generalized nonlinear Schrödinger equation are obtained and conserved quantities corresponding to the solution in implicit form are found [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of optical pulses of the fractional extended nonlinear Schrödinger equation and the effect of the third-order dispersion parameter are studied. The equation physically defines the propagation of femtosecond in nonlinear optical fiber and plasma physics. These types of equations are applicable in many real-world scenarios, such as physical sciences and optical engineering. One of the most important applications of such equations is the study of solitons, which are self-sustaining solitary waves that retain their shape and momentum even after collision with other solitons. Solitons are employed in many different fields, including fluid dynamics (where they simulate long-lasting ocean waves) and optical communications (where they transmit information over large distances without distortion). Different types of optical pulses, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted by using the recently integration method known as modified Sardar subequation method. The technique used in this analysis is effective and robust, as demonstrated by the computational results. Different sets and values of the physical parameters are used to study the optical soliton solutions of the resulting system. Moreover, different graphs with the associated parameter values are sketched to visualize the solutions behaviour. [ABSTRACT FROM AUTHOR]

The optical solitons represent a fascinating intersection of nonlinear physics and optical technology. In this study, we examine the three-component Dirac–Manakov system through the application of a unified Darboux transformation. We elaborate on the construction of the n-fold Darboux transformation and elucidate the correlation between optical solitons and seed solutions. Our research successfully yields both single-soliton and double-soliton solutions derived from zero-seed solutions. Additionally, we derive periodic solutions from non-zero seed solutions. To visualize these solutions, we employ Maple to generate three-dimensional and density diagrams, thereby facilitating a comprehensive understanding of the solution structures within the three-component Dirac–Manakov system. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to study various optical soliton profiles of the Schrödinger–Hirota (SH) equation having the parabolic law nonlinearity by utilizing the Sardar-Subequation (SSe) method. First, the general method of the suggested SSe method is presented. Then using traveling wave transformation the suggested equation is converted into the nonlinear ordinary differential equation, where the real and imaginary parts are separated. Furthermore, the proposed method is utilized and novel optical solitons of the suggested SH equation with parabolic law are obtained. Furthermore, the solutions are validated through stability analysis. The exact analytical solutions are graphically simulated where the effects of different parameters on the soliton wave profiles are studied. [ABSTRACT FROM AUTHOR]

This manuscript elaborate a novel characteristic of stochastic multi breather type, periodic, hybrid periodic and different type of waves solutions for the Shrödinger–Hirota (SH) equation. The mentioned equation is studied by a Wiener process and white noise which is used for the sudden and large-scale fluctuation. By applying the wave transformation technique the considered equation is transformed into ordinary differential equations, gives meaningful analysis. From the solution of the SH equation various type of results, including breather like waves, hybrid periodic waves, and singular solitons. Breather waves are applied in optical fiber communications, oceanography, and plasma physics. Singular waves find use in medical imaging, seismic research, and nonlinear optics. Hybrid singular-periodic waves have applications in photonics, acoustics, and material science, enabling advancements in diverse fields. To visually capture these intricate behaviors, we used Mathematica software to generate 2D and 3D graphs of the analytical stochastic solutions. This combination of mathematical rigor and numerical simulations provides a comprehensive understanding of the phenomena under study, making a valuable contribution to the field of nonlinear wave equations and stochastic processes. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger–Poisson equation with gravitational potential field is examined, along with its many soliton interactions and travelling wave solutions. By applying Hirota Bilinear scheme, we will give a brief study on transformation of solitons, also by Sine–Gordon expansion scheme solitary travelling wave solutions will be obtained in terms of trigonometric and hyperbolic functions. The innovative Hirota bilinear method is a powerful and standard technique, play a crucial role in producing soliton solutions as well as lump solutions. With the assistance of this method one can transformed integrable equation into Hirota bilinear form under dependent variable transformation. Soliton solutions obtained by Hirota bilinear method are significant in mathematical physics, may be superimposed in fibers. These solitons are applicable in optical communications, which enable to produce faster, richer, more secure and more flexible communication systems. Also we will investigate the gravitational potential's behaviour on soliton solutions, such as parabolic, bright, dark, anti-dark, combination and S-shaped solitons. Contour plots and three-dimensional soliton solutions are provided. These solitons are important in nonlinear processes, magnetic devices, and lasers beams because of variations in gravitational field. [ABSTRACT FROM AUTHOR]

In this work, we investigate the (2+1)-dimensional nonlinear electrical transmission line model (NETLM) and propose a modified generalized Riccati equation mapping (mGREM) approach for obtaining analytical soliton solutions. The NETLM characterizes wave distributions on network lines. We introduce the mGREM technique, which involves a traveling wave transformation to convert the equation into nonlinear ordinary differential equations. By applying this method, we extract various soliton solutions, including traveling waves, kink-type solitons, bell-shape solitons, and multi-solitons. These findings contribute to understanding the (2+1)-dimensional NETLM and provide valuable insights into wave distributions on network lines. To gain a comprehensive understanding of the solutions' dynamics, we conduct numerical simulations and present two-dimensional, three-dimensional, and contour graphs. These graphical illustrations offer valuable insights into voltage propagation patterns within the system. Our results contribute to the existing literature and open avenues for further investigations in the field of NETLM. The significance of this work lies in the successful application of the mGREM method to obtain diverse traveling wave solutions for the (2+1)-dimensional NETLM. By employing symbolic computation in Mathematica, we extract analytical soliton solutions, which go beyond previous efforts in the literature. The proposed approach provides a novel perspective on the NETLM equation and its soliton solutions. The obtained results enhance our understanding of the system's behavior and pave the way for future research in this domain. [ABSTRACT FROM AUTHOR]

The damped Korteweg-de Vries (D-KdV) equation is a significant extension of the Korteweg–de Vries equation, which plays an important role in understanding the complex dissipative wave system. This study explores the historical evolution and mathematical properties of the damped KdV equation. The auxiliary equation approach is utilized on nonlinear damped KdV equation to extract the newly exact solitary wave results. New solutions are extracted in bright solitons, kink wave solitons, mixed dark-bright solitons, anti-kink wave solitons, periodic solitons, dark solitons, and solitary wave structures. The physical behavior of secured solutions are visualizing in contour, 3-D, and 2-D plots based on numerical simulation with the computational software Mathematica. The proposed approach is employed, offering a powerful mathematical tool for analyzing the effects of damping on wave dynamics. Through this investigation, the study sheds light on the captivating interplay between physics and mathematics in the realm of wave phenomena. These newly constructed solutions are important in nonlinear acoustics, optical fiber, nonlinear optics, soliton wave theory, plasma physics, and ion acoustic wave phenomena. As a result, our proposed method is directed, concise, and practical for various kinds of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this article, the modified extended direct algebraic method is applied for the perturbed highly dispersive nonlinear Schrödinger equation with conformable fractional derivative and sextic-power law refractive index. Various types of solutions are extracted such as bright solitons, dark solitons, combo bright-dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions and rational solutions. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

In this research work, we explored the unexpected optical soliton and solitary wave to the nonlinear (2+1)-dimensional Zakharov–Kuzetsov equation by utilizing the auxiliary equation method. The explored optical soliton and solitary wave solutions obtained in variety of soliton including bright solitons, anti-kink wave solitons, periodic optical solitons, kink wave solitons, dark solitons, anti-kink dark solitons, combined solitons, crest-turf form solitons, peakon solitons and periodic solitary waves. The physical structure of extracted optical solitons visualized in three-dim, two-dim and contour plotting by utilizing the numerical simulations with computational software Mathematica. The secured optical solitons will be play important role in the investigation of nonlinear (2+1)-dim Zakharov–Kuzetsov equation and will be develop interest in the researchers for further investigation on this nonlinear model. The physical structure of explored soliton results show that, these solitons will be play important role in the areas of sciences and engineering such as nonlinear optics, transmission system, communication system, nonlinear dynamics, soliton wave theory and ocean engineering. The successful investigation is the witness that the proposed approach is effective, efficient and powerful for the exploration of solitary waves special verities of optical solitons to the various kinds of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper dedicates to study the optical soliton solutions for the time-fractional perturbed Fokas-Lenells equation using the novel modified Sardar sub-equation approach. Several novel optical solutions include a class of categories, comprising mixed dark-bright, bell-shape, singular, and wave optical solutions are constructed. Further, the magnitude of the time-fractional perturbed Fokas-Lenells model by investigating the impact of the conformable fractional parameter and the effect of the time parameter on the novel optical solutions is analyzed. The current type of Sardar sub-equation approach is a general form of several approachs, such as the tanh-function extension approach, the modified Kudryashov approach, the improved Sardar sub-equation method, and others. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics. Time-fractional perturbed Fokas–Lenells soliton solutions could be applied to model and understand the behavior of signals in optical communication systems and potentially improving data transmission. [ABSTRACT FROM AUTHOR]

In this research work, the nonlinear Landau–Ginzburg–Higgs model under investigation on the base of auxiliary equation method. The nonlinear Landau–Ginzburg–Higgs equation mainly describe nonlinear wave propagation, categorizes wave velocity, and materializes several phenomena through the dispersive system. The secured optical soliton solutions are interested, more general and some novel in kink solitons, bright solitons, periodic singular solitons, anti-kink solitons and dark solitons. The physical structure of some explored solitons solutions are represented graphically as contour, 2D and 3D plots by utilizing the symbolic computational tool Mathematica. The secured soliton will be play important role to the investigation of nonlinear phenomena in the various domains of nonlinear sciences and engineering such as optical fibers, nonlinear dynamics, communication system, solitons wave theory, nonlinear optics, transmission system, electronic engineering and ocean engineering. The auxiliary equation method is a basic analytical, powerful, efficient approach for the investigation of optical soliton inside the nonlinear Landau–Ginzburg–Higgs equation and also other nonlinear equations. [ABSTRACT FROM AUTHOR]

Variable-coefficient equations can be used to describe certain phenomena when the inhomogeneous media and nonuniform boundaries are taken into consideration. It is meaningful to solve the exact solution of variable-coefficient equations. In this paper, a (2+1)-dimensional variable-coefficient Boussinesq equation is investigated. The integrability is firstly examined by the Painlevé analysis method. Secondly, the Bäcklund transformations, one- and two-soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation are studied by virtue of the Hirota bilinear method. Propagation characteristics and interaction behaviors of the solitons are discussed: (i) soliton shapes and interaction behaviors are affected by the variable coefficients, and (ii) the two-soliton interaction is elastic, and the shape and velocity does not change after the collision, and only the shift changes. [ABSTRACT FROM AUTHOR]

In a remarkable paper, Cox and Mortell (1986) (A.A. Cox, M.P. Mortell 1986. J. Fluid Mech. 162, pp. 99-116) showed that for an oscillating water tank, the evolution of small-amplitude, long-wavelength, resonantly forced waves follow a forced Korteweg–de Vries (fKdV) equation. The solutions of this model agree well with experimental results by Chester and Bones (1968) (W. Chester and J.A. Bones 1968. Proc. Roy. Soc. A, 306, 23 (Part II)). We compare the fKdV solutions with a number of channel flows with different geometry that have been studied experimentally and numerically. When sweeping the selected wide parameter range, extreme cases of the fKdV equation are covered: single soliton solutions as well as multiple solitons with a rather short wavelength challenging the long-wave fKdV assumption. The transition of solutions with a different number of solitons is rather abrupt and we show that the parameter values for transitions from single soliton towards multi-soliton solutions can be predicted and follow a simple exponential relation. In particular, we compare the fKdV model with solutions from a fully nonlinear Navier–Stokes model. We further consider a case for which the 2D assumption of the fKdV equation is strictly speaking violated. [Display omitted] • Nonlinear waves in a water channel. • Development of solitons and bores in an oscillating channel. • Resonances in an oscillating channel. [ABSTRACT FROM AUTHOR]

We show that the mean curvature flow of generic closed surfaces in R 3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R 4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. [ABSTRACT FROM AUTHOR]

The Schrödinger–Hirota equation is one of the most important models of contemporary physics which is popular throughout the broad fields of fluid movement as well as in the study of thick-water crests, liquid science, refractive optical components and so on. In this paper, we utilize the Hirota bilinear technique and the unified technique to attain various soliton solutions of the governing model analytically. These approaches are robust, powerful and unique also have many applications in different fields of mathematical physics. The solutions attained from these techniques are highly valuable and useful in various fields of sciences specially in the transmissions of optical fibers, also they give different behaviors including V-shaped and periodic soliton solution behavior. Further, the approaches applied here are not applied in this model previously. Therefore, ours is a new work, which summarizes its novelty. The 3D, 2D and contour plots are included to grasp the understanding of solutions' behavior. These findings are valuable in electronic communications such as elliptical circuits and in investigation of solitude controlling. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schrödinger equation on the product space R × T . We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the "auxiliary equation" and then on the application of the Crandall–Rabinowitz argument. [ABSTRACT FROM AUTHOR]

We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special generalized flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special generalized flag manifold with a homogeneous bounded domain, (ii) a special generalized flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special generalized flag manifold. Our theorems strongly extend the results of Cheng and Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180], Cheng, Di Scala, and Yuan [Internat. J. Math. 28 (2017), p. 1750027], Loi and Mossa [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941], Loi and Mossa [Proc. Amer. Math. Soc. 151 (2023), pp. 3975–3984] and Umehara [Tokyo J. Math. 10 (1987), pp. 203–214]. [ABSTRACT FROM AUTHOR]

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of 1 0 0 photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system. [ABSTRACT FROM AUTHOR]

Soliton dynamics and nonlinear phenomena in quantum deformation has been investigated through conformal time differential generalized form of q deformed Sinh-Gordon equation. The underlying equation has recently undergone substantial amount of research. In Phase 1, we employed modified auxiliary and new direct extended algebraic methods. Trigonometric, hyperbolic, exponential and rational solutions are successfully extracted using these techniques, coupled with the best possible constraint requirements implemented on parameters to ensure the existence of solutions. The findings, then, are represented by 2D, 3D and contour plots to highlight the various solitons' propagation patterns such as kink-bright, bright, dark, bright-dark, kink, and kink-peakon solitons and solitary wave solutions. It is worth emphasizing that kink dark, dark peakon, dark and dark bright solitons have not been found earlier in literature. In phase 2, the underlying model is examined under various chaos detecting tools for example lyapunov exponents, multistability and time series analysis and bifurcation diagram. Chaotic behavior is investigated using various initial condition and novel results are obtained. [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]

The concern of this study is to investigate the optical solitons pluses. The (2 + 1) -dimensional Kundu–Mukherjee–Naskar (KMN) mathematical model in birefringent fibers is taken under consideration for this sake because this model has great importance in optics and delineate the propagation of soliton dynamics in optical fiber communication system and the rogue waves in the oceans and the bending of the light beam. Two newly developed approaches first extended rational sinh-Gordon method which is formulated from the well-known sinh-Gordon equation and second exp (− ϕ (ϖ)) -expansion function method are manipulated for the construction of optical solitons in distinct formats. Bright, dark, and singular along their combined forms, periodic and plane wave solutions are extracted with the aid of proposed methods. Moreover, the modulation instability of investigated model is also carried out via linear stability theory. To endorse the physical compatibility of the results, the 2D, 3D, contour, and density plots have been delineated using appropriate parametric values. According to our literature research, these two methods that we are working on have not been applied to the KMN equation in birefringent fibers before, and we believe that the new solutions we have obtained will be useful to researchers working in modeling in this field. The evaluated results suggested that the techniques employed in this research to recover inclusive and standard solutions are approachable, efficient, and speedier in computing and can be considered a handy tool in solving more complex phenomena with the assistance of symbolic computation. [ABSTRACT FROM AUTHOR]

This paper explores the truncated M-fractional three-component coupled nonlinear Schrödinger (tc-CNLS) equation, that regulates the behavior of optical pulses in optical fibers. These equations are utilized in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. The study of multi-component NLS equations has gained significant attention due to their ability to elucidate various complex physical phenomena and exhibit more dynamic structures of localized wave solutions. The freshly invented integration tools, known as the fractional modified Sardar subequation method (MSSEM) and fractional enhanced modified extended tanh-expansion method (eMETEM), are employed to ensure the solutions. The study focuses on extracting various types of optical solitons, including bright, dark, singular, bright-dark, complex, and combined solitons. Optical soliton propagation in optical fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. Furthermore, hyperbolic, periodic and exponential solutions are generated. The utilized methodology is effective in explaining fractional nonlinear partial differential equations (FNLPDEs) as it offers pre-existing solutions and additionally derives novel exact solutions by mixing outcomes from various procedures. Furthermore, we plot the visualizations of solutions by plotting 3D, 2D, and contour graphs with the corresponding parameter values. The findings of this paper can improve the understanding of the nonlinear dynamical behavior of a specific system and demonstrate the efficacy of the methodology used. We anticipate that our study will provide substantial benefits to a considerable group of engineering model experts. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems. [ABSTRACT FROM AUTHOR]

In this work, we establish novel solitary wave solutions of three nonlinear evolution dynamical models, namely, Boussinesq model, (3 + 1) -dimensional generalized shallow water wave (SWW) and (4 + 1) -dimensional Fokas dynamical models by applying two-variable (G ′ ∕ G , 1 ∕ G) - expansion technique. The SWW equations are usually appropriate when the fluid is in a state where the vertical length scale is significantly smaller than the length of horizontal scale. In computer simulations for ocean engineering, Boussinesq-type models are frequently used to represent water waves in harbors and shallow seas. In order to understand the physical processes of waves inside and on the surface of water, the Fokas dynamical model plays a crucial part in wave theory. Analytical solutions in different forms such as solitons, solitary waves, trigonometric, hyperbolic, rational function solutions, breathers-type waves and more wave solutions are devised through using the proposed method. The exact solutions are also presented in graphical form having applications in engineering and other areas of applied sciences. The obtained results show that the given technique is universal and efficient. In addition, this technique can also be applied on lots of other nonlinear dynamical wave models occurring in many scientific real-world application domains, including engineering sciences, mathematics physics, and many more. [ABSTRACT FROM AUTHOR]

The nonlinear Kadoma equation with M-truncated derivatives (NLKE-MTD) is taken into consideration here. By using generalized Riccati equation method (GRE method) and Jacobi elliptic function method, new hyperbolic, rational, trigonometric and elliptic solutions are discovered. Because the NLKE is widely employed in optics, fluid dynamics and plasma physics, the resulting solutions may be used to analyze a wide variety of important physical phenomena. The dynamic behaviors of the different derived solutions are interpreted using 3D and 2D graphs to explain the effects of M-truncated derivatives. We may conclude that the surface moves to the right as the order of M-truncated derivatives increases. [ABSTRACT FROM AUTHOR]

This study aims to examine the properties of the nonplanar (cylindrical and spherical) ion-acoustic solitary waves (SWs) and cnoidal waves (CWs) in a collisionless, unmagnetized electron-ion (EI) plasma having a Cairns–Tsallis distribution for the electrons. This study is structured around two main lines. The first trend involves deriving the nonplanar Korteweg-de Vries (KdV) equation by utilizing the method of reductive perturbation (MRP). This equation describes small-amplitude (non)planar acoustic waves (AWs). Furthermore, the nonplanar Kawahara equation (KE) is formulated to examine the significant magnitude of planar and nonplanar SWs and CWs. The current plasma model supports compressive and rarefactive IA solitary and cnoidal structures, depending upon the associated physical factors such as the nonextensive parameter (nonextensivity) and nonthermal parameter (nonthermality). When the plasma compositions reach some critical values, such as the critical value of nonthermality, the coefficient of the quadratic nonlinear term vanishes. Hence, both nonplanar modified KdV (mKdV) equation and modified KE (mKE) with cubic nonlinearity are derived to accurately depict the dynamics of both small and large amplitudes of nonplanar SWs and CWs and any other structures related to this family of evolution equations. The influences of the nonextensivity and nonthermality on the profile of (non)planar KdV soliton and the (non)planar Kawahara SWs and CWs are numerically examined using some semi-analytical and numerical approximations. Also, the impact of the nonextensivity on the profile of (non)planar mKdV soliton and the (non)planar modified Kawahara SWs and CWs is reported. It is tracked down that the variety of different plasma parameters significantly alters the characteristic properties of the small and large amplitude ion-acoustic waves (IAWs) discussed by the nonplanar KdV-type equations. [ABSTRACT FROM AUTHOR]

Exact sub and supersonic soliton solutions of the one-dimensional Westervelt equation are presented, describing the propagation of pressure waves with high amplitude or frequency in nonlinear acoustic media. The sub and super-sonic waves only exist on a pedestal, in the absence of which the excitations propagate at the same speed as the small amplitude waves. For nonzero background, the soliton velocity is dependent on the linear acoustic properties such as, ambient density and sound speed, as also the nonlinearity coefficient. The nonlinearity coefficient is known to vary significantly with pathological conditions of tissue, often modeled by Westervelt equation, and hence these nonlinear characteristic excitations may find application in tissue tomography. [ABSTRACT FROM AUTHOR]

The ion-acoustic Gardner solitons in negative ion plasma have been studied. The Gardner equation is deduced by employing the reductive perturbation technique. We have investigated the ion-acoustic Gardner solitons in negative ion plasma in the parametric regime where the KdV equation is no longer valid to study soliton solution. It is found that for a given set of parameter values with negative ion concentration ( $ \alpha $ α) less than the critical value ( $ {\alpha _c} $ α c ), i.e. $ \alpha $ α < $ {\alpha _c} $ α c ; the system supports compressive Gardner soliton, whereas for $ \alpha $ α > $ {\alpha _c}, $ α c , rarefactive Gardner soliton exists. It is also investigated that Gardner solitons exist only for the fast mode, whereas no such structure is found in the case of the slow mode. The present study may be helpful to understand the nonlinear ion-acoustic solitary waves in space plasma and laboratory experiment plasmas also can be helpful for the study of Ionospheric auroral and magnetospheric plasmas where negative ions are present. [ABSTRACT FROM AUTHOR]

In this research article, the dynamics of optical solitons in cascaded system with spatio-temporal dispersion is discussed. The Kerr medium is considered for this investigation. Two versatile integration tools have been used to extricate different traveling wave solutions. Rapid convergent approximation method has been employed to extract bright, singular and periodic solitons whereas polynomial function solutions are obtained by unified method. The constraints on the parameters for the validity of these solutions are also provided. Galilean transformation is prosecuted to construct the planer dynamical system of the said model. Runge–Kutta fourth-order technique is used to extract the non-linear periodic waves of the considered problem and consequences are depicted through different portraits. Further, by employing an extrinsic periodic force, chaotic and quasiperiodic pattern of discussed model is analyzed for different values of the parameters. While chaotic and quasiperiodic behaviors are observed for some values of parameters of considered system by keeping the force and frequency of the perturbed dynamical system fixed. Sensitive anatomy is carried out for diverse initial value problems. It is investigated that the chaotic and quasiperiodic pattern is present in the perturbed dynamical system. [ABSTRACT FROM AUTHOR]

Cosmic plasmas are considered as the most abundant form of ordinary matter in the Universe while observations of the cosmic dust in different regions provide an insight into the Universe's recycling processes. For different types of the cosmic dusty plasmas, we hereby, with the symbolic computation and observational/experimental supports, study a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation, which can describe the electron-acoustic, dust-acoustic, positron-acoustic, dust-magneto-acoustic, ion-acoustic, magneto-acoustic, ion, quantum-dust-ion-acoustic or dust-ion-acoustic waves in one of the cosmic/laboratory dusty plasmas. With respect to the fluctuation of the electron or ion density, or perturbation of the magnitude of the magnetic field, or electrostatic wave potential, or radial-direction component of the velocity of ions or dust particles, a set of the auto-Bäcklund transformations, several soliton families and a set of the similarity reductions are symbolically computed out, depending on the variable coefficients which represent the dispersion, nonlinearity, geometric effect, Burgers/dusty-fluid-viscosity dissipation and diffraction/transverse perturbation. Variable-coefficient constraints on the soliton solutions are presented. Our analytic results are in agreement with those dusty-plasma-experimentally reported. Future dusty-plasma experiments and observations might justify some other effects hereby offered. [ABSTRACT FROM AUTHOR]

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an η – ρ -Einstein soliton, then the integral curves generated by the velocity vector field u are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient η – ρ -Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an η – ρ -Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient η – ρ -Einstein soliton in f (ℛ) -gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function ψ remains invariant under the velocity vector u. Also, for two models f (ℛ) = e α ℛ + α log ℛ and f (ℛ) = ℛ n (α = constant and ℛ is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system nonlinear (2 + 1)-dim Broer–Kaup–Kupershmit (BKK) equation under consideration is based on extension of modified rational expansion method. The various kinds of multiple soliton solutions named anti-kink soliton, traveling wave solutions, periodic solution, dark soliton, kink soliton, bright soliton, anti-kink bright solitons, kink bright solitons for nonlinear BKK system are successfully constructed under symbolic computation Mathematica. The calculated results are very interesting, different and novel which have not been investigated in past literature. The graphical demonstration of constructed solutions show by contour, 2D and 3D shape by using computer software Mathematica. The constructed multiple soliton solutions prove that EMRE approach is very straightforward, efficient, reliable and powerful for the investigation of other nonlinear partial differential equations. The investigated results will be described and play a keen role in the study of nonlinear physical phenomena in the area of physics and engineering. [ABSTRACT FROM AUTHOR]

In this study, we study two generalized stochastic Korteweg-de Vries (KdV) equations. The Painlevé property of these nonlinear models is tested using Kruksal's method, which establishes the model's integrability. As a result, using Hirota's bilinear approach and symbolic computation, the N-soliton solutions are constructed. In addition, the extended hyperbolic function method (EHFM), the modified Kudryashov method (MKM), and the sub-equation method (SEM) are used to acquire the bright soliton, dark soliton, singular soliton, periodic, rational, and exponential solutions. To help understand the dynamic features of the derived soliton solutions, we present a number of 2D, 3D, and contour graphs using appropriate parametric values. [ABSTRACT FROM AUTHOR]

Spatially overlapped coupled matter‐wave bright solitons are considered in binary Bose‐Einstein condensates with nonlinear optical lattices(NOLs) and study the properties of scattering patterns due to a localized defect. It is shown that spatially overlapped solitons become separated and exhibit different scattering patterns with the variation of inter‐species interaction. The NOL interplays with the defect and tends to influence the scattering properties. However, the effect of NOL varies with strength of inter‐component interaction. It is found that, if this interaction is weak then the scattering center gets shifted outside the defect. However, for a stronger inter‐component interaction, the scattering takes place from the defect and leads to different scattering patterns. [ABSTRACT FROM AUTHOR]

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave variables is investigated using the Painlevé test. Periodic and solitary wave solutions of the studied equation are presented. The investigated equation belongs to the class of generalized nonlinear Schrödinger equations and may be used for the description of optical solitons in a nonlinear medium. [ABSTRACT FROM AUTHOR]

We derive a divergence formula for an m-quasi Yamabe gradient soliton with finite m, and use it to give a short proof of the result "A compact m-quasi Yamabe gradient soliton (M n , g) , n ≥ 3 , with finite m, has constant scalar curvature". We also show that an m-quasi Yamabe gradient soliton with finite m and positive scalar curvature, whose soliton vector field leaves the Ricci tensor invariant, is shrinking and has constant scalar curvature. [ABSTRACT FROM AUTHOR]

Soliton "molecules", i.e., bound states of two or several solitons, represent a fundamental concept, which manifests itself in various nonlinear systems. They are dynamically similar to chemical molecules, attracting great interest to fundamental studies and offering potential applications (such as multilevel encoding of optical data). Here, the study demonstrates a novel dichromatic soliton‐molecule compounds (DSMC) built as a hybrid bound state of multiple bound soliton pulses carried by two wavelengths in a fiber laser. The DSMCs are maintained by two different binding mechanisms, viz., the self‐phase modulation (SPM) interaction between temporal solitons at the same wavelength, mediated by their tails, and the cross‐phase modulation (XPM) interaction between solitons at different wavelengths. They also exhibit unique temporal and spectral vibration profiles. Both static DSMCs and ones with robust internal vibrations are generated experimentally in the fiber laser, and numerically as solutions of the corresponding dissipative nonlinear model. The findings reported here expand the concept of soliton molecules and further promote their similarity to chemical molecules. [ABSTRACT FROM AUTHOR]

This paper aims to study a generalized extended (2 + 1) -dimensional Kadomstev–Petviashvili (KP) equation. The KP equation models several physical phenomena such as shallow water waves with weakly nonlinear restoring forces. We will use a variety of wave ansatz methods so as to extract bright, singular, shock waves also referred to as dark or topological or kink soliton solutions. In addition to soliton solutions, we will also derive periodic wave solutions and other analytical solutions based on the invariance surface condition. Moreover, we will establish the multiplier method to derive low-order conservation laws. In order to have a better understanding of the results, graphical structures of the derived solutions will be discussed in detail based on some selected appropriate parametric values in 2-dimensions, 3-dimensions and contour plots. The findings can well mimic complex waves and their underlying properties in fluids. [ABSTRACT FROM AUTHOR]

This work focuses on the study of the paraxial wave model with a space–time fractional form. This model has more importance for describing light propagation in nonlinear optical fibers and telecommunication lines. The main aim of this work is to observe the effect of the fractional parameters and compare the truncated M -fraction with the beta-fraction, conformable fraction form, and classical form of the PW model. For this observation, we applied the Simplest equation technique to acquire analytical solutions to the space–time (spatiotemporal) M -fractional paraxial wave model. We are able to acquire several new optical soliton solutions, including periodic waves, kink-type waves, rogue-type waves, and several novel periodic waves, by providing the appropriate fractional parametric values. These solutions have significance for shedding light on a number of physical phenomena in the realms of optical fiber and communication sciences. The diverse values of fractional parameters and the three-dimensional and contour plot graphs of certain chosen solutions are depicted, which are the most accurate physical characterizations of the outcomes. We also sketch the comparative graph of diverse fractional forms and the classical form of the paraxial wave equation in two-dimensional plots. Consequently, our findings represent an important breakthrough in this complex area and help further develop our comprehension of the behavior of solitons. [ABSTRACT FROM AUTHOR]

In this work, we use two newly powerful and efficient techniques, the unified method and the unified Riccati equation expansion (UREE) method, to derive solitary wave solutions for the (1 + 1)-dimensional longitudinal wave equation (LWE), presenting a systematic approach to reveal the underlying dynamics. By incorporating mathematical analysis and numerical simulations, we investigate the behavior and properties of these solitary wave solutions in various fields of science. The derived solutions are entirely new, and these findings show the essential complexities of the longitudinal wave equation. These obtained solutions shed light on the fundamental mechanisms governing solitary waves in the longitudinal wave equation, contributing to a deeper understanding of nonlinear wave phenomena. We present effective visualizations of the dynamical wave structures using various graphs, such as 3D, 2D, and contour plots in the obtained solutions to improve our comprehensive understanding. Consequently, numerous types of wave profiles, including periodic wave, interaction between periodic wave and kink wave, W-shape wave, kink wave, interaction between breather wave and kink wave, traveling wave, mixed periodic, singular soliton, and some new types of soliton profiles were found. Consequently, this work presents a comprehensive study of solitary wave solutions using the unified method and UREE method for the (1 + 1)-dimensional LWE equation. The outcomes of the current study manifest that the considered methods are significant and systematic in solving nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This article studies high dispersion solitons in optical couplers that are composed of metamaterials using the modified extended direct algebraic method (MEDAM). Different dynamic behaviors of unique soliton waves that include singular and combo-soliton solutions are raised using the MEDAM. The solutions that are produced include singular periodic solutions, bright and dark soliton solutions, Jacobi elliptic functions, and singular soliton solutions. Furthermore, the physical depiction of the found solutions is given through contour graphs, 2D, and 3D graphics. The main objective of this paper is to incorporate meta-materials into a governing model for highly dispersive solitons in optical couplers. The modified extended direct algebraic technique, reveals several solitons, where the uniqueness is found. Moreover, by the end of this work, we present 2D, 3D, and contour charts of the various obtained solutions to further highlight our findings. [ABSTRACT FROM AUTHOR]

The supercontinuum spectrum is generated through a wide range of wavelengths by sending a short and strong pulse to the nonlinear medium and sputtering at the output and is used in optical measurements, spectroscopy, biological imaging optical coherence photography, etc. Integrated photonics is an idea to realize low-cost and microscale communication, sensing, and fast computing methods. In addition, the miniaturization and integration of photonic structures make possible new designs and applications that are inaccessible in their large volumes. Lithium niobate is one of the most widely used and attractive materials in the field of photonics due to its extraordinary electro-optical, acoustic-optical, nonlinear optics, wide transparency window, and relatively high refractive index. In this work, a lithium niobate waveguide is designed by choosing basic solitons as the input pulse and considering various effects such as high-order scattering, self-phase modulation, second harmonic generation, Raman effect, self-downward effect, etc. The proposed waveguide has super-sustainable production. This supercontinuum spectrum is designed in the waveguide, for the 10th order bright soliton at a distance of 35 mm, 4 times the initial width, for the 20th order bright soliton at 9 mm, 4 times the initial width, and for the 30th order bright soliton at 4.5 mm, 5 times the initial pulse width in the frequency domain are created. Such dimensions are suitable for use in photonic integrated circuits. [ABSTRACT FROM AUTHOR]

This study delves into a complex mathematical model, namely the fractional Chen–Lee–Liu (FCLL) model in birefringent fibers which has two components in the form of vector solitons in optical fiber. The FCLL model has remained a uniform fascination for scientists due to its energetic significance in the study of circuit design, signal processing, secure communications, encryption and decryption of chaotic signals, as well as in plasma physics. For the intellectual curiosity of new precise solutions of the selected model, two innovative norms namely, the modified Sardar-sub equation scheme and the improved F -expansion method have been suggested in the sense of a M-truncated derivative. By employing the aforementioned strategies the bright, dark, bright-dark, periodic, combo, singular, mixed trigonometric, hyperbolic, exponential as well as rational soliton solutions are segregated. Moreover, we examined the sensitivity analysis of selected model. By selecting appropriate parameters, numerical simulations of the attained outcomes are sketched as interesting figures that present the meaning of the obtained results. The generated outcomes are optimistic, demonstrating that the selected approaches are robust, categorical, and efficient in discovering novel solutions to the variety of complex nonlinear models that have arisen in the recent phase of applied sciences. [ABSTRACT FROM AUTHOR]

In this work, we take into account the (2+1)-Davey Stewartson equation (DSE) and the (2+1)-complex coupled Maccari system (CCMS) and their analytical solutions. Besides, we tackle the role of the problem parameters on the soliton behavior produced by the presented DSE. Exact traveling wave solutions are highly useful in numerical and analytical theories for such equations. While numerical methods are widely used, improving analytical approaches for obtaining analytical solutions is necessary for a deeper understanding of dynamics. This study marks a significant milestone by implementing the efficient analytical approach, enhanced modified extended tanh expansion method, for the first time to the (2+1)-DSE and (2+1)-CCMS equations, thereby making a notable contribution to the existing literature. We have shown that features of the soliton solutions can represent the spread of propagation on the wavefronts and show a reasonable dependency on parameter values. Some of the solutions discovered in three- and two-dimensional arrangements can also be described in graphic representations of their behavior. With the help of the graphical depictions, bright, singular, and periodic singular soliton characters for the (2 + 1) -DSE and singular, dark, bright, and periodic singular soliton characters for the (2 + 1) -CCMS are acquired. The results show that the utilized analytical technique is easily applicable, efficient, reliable, robust, and categorical when it comes to finding analytical solutions for different nonlinear models. Moreover, the problem parameters and the coupling coefficients have significant influences on the behavior of the solitons of the DSE, and this examination is studied for the first time in this article. [ABSTRACT FROM AUTHOR]