Narrow Results

There is no doubt that the investigation of the interaction and propagation of plasma and electromagnetic waves play an important role in understanding these phenomena. The three-dimensional Yu–Toda–Sasa–Fukuyama equation (YTSFE) is a competent mathematical model of waves in plasma, electromagnetics, or fluids. An optimal system of infinitesimal symmetries is constructed to discover extensive and astonishingly exact solutions to the YTSFE. The outstanding solutions include periodic, polynomials, fractional, logarithmic, exponential, hyperbolic, exponential integral, Airy and complex functions. These solutions are significant because they help understand how plasma and electromagnetic applications work at different boundary or initial conditions.

In this paper, the thermophoretic motion equation based on Korteweg–de Vries is utilized to analyze new complexiton and soliton-like solutions. The homogenous balance approach is employed to generate auto-Bäcklund transformation of the concerned problem. This transformation is capitalized to extract abundant explicit and analytic solutions. Moreover, Hirota bilinear form of the concerned equation is taken under consideration to discover complexiton solutions via extended transform rational function approach. 3D visualization of the acquired solutions is also included to discuss its physical behavior.

In this work, we use a novel fractional-order derivative for the fractionally perturbed Chen–Lee–Liu nonlinear equation. The new extended hyperbolic function (EHF) method is applied for obtaining new optical soliton solutions of the mentioned equation. Three-dimensional graphics and projection 3D plots are used for showing the dynamic wave formations of the soliton solutions. Then, we contrast our findings with the earlier existing results for the nonlinearly perturbed CCL. The generated solutions show that the extended hyperbolic function (EHF) method for finding soliton solutions to highly nonlinear equations is productive, suitable, and competent in optical fibers, fractional calculus, and nonlinear sciences.

In this work, the extended Jacobi elliptic function expansion approach is used to analyze a generalized $(3+1)$-dimensional Gross–Pitaevskii equation with distributed time-dependent coefficients because of its use in the Bose–Einstein condensation. The Gross–Pitaevskii equation plays a significant role in Bose–Einstein condensation, where it characterizes the dynamics of the condensate wave function. By using this approach with a homogeneous balance principle, the spatiotemporal soliton solutions and exact extended traveling-wave solutions of governing equation have been successfully obtained. A few double periodic, trigonometric and hyperbolic function solutions from the Jacobi elliptic function solutions have been found under specific constraints on a parameter. It is obvious that the proposed approach is the most straightforward, efficient and useful way to handle numerous nonlinear models that arise in applied physics and mathematics in order to generate various exact solutions. A case with variable gain, constant diffraction and parabolic potential strength has been considered in this study to derive exact solutions. Numerous novel varieties of traveling-wave solutions have been revealed in this work, including the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular and the singular soliton solutions and these newly obtained results differ from those previously investigated for this governing equation. In addition to addressing a scientific explanation of the analytical work, the results have been graphically presented by 3D plots and contour plots for some suitable parameter values to understand the physical meaning of the derived solutions. Due to their applicability to a variety of quantum systems, the acquired solutions are of considerable importance.

In this paper, the generalized Kudryashov (GK) approach and the sine-Gordon expansion approach are used for constructing new specific analytical solutions of the deoxyribonucleic acid model, which include the well-known bell-shaped soliton, kink, singular kink, periodic soliton, contracted bell-shaped soliton and anti-bell-shaped soliton. The efficacy of these strategies demonstrates their utility and efficiency in addressing a wide range of integer and fractional-order nonlinear evolution problems. The physical relevance of the demonstrated results has been proven using three-dimensional forms. It is interesting to mention that the solutions achieved here using the provided methods are extra-extensive and may be used to explain the internal interaction of the deoxyribonucleic acid model originating in mathematical biology. The suggested approach was utilized to get exact traveling wave solutions for fractional nonlinear partial differential equations appearing in nonlinear science.

The coupled nonlinear Schrödinger (CNLS) system is often used to describe physical problems such as wave propagation in a birefringent optical fiber. Under investigation in this paper is the nonlocal ${𝒫}{𝒯}$-symmetric reverse-space type of CNLS system, through constructing two types of Darboux transformation (DT), we will derive a new series of nonlocal analytic solutions: (1) Single-periodic and double-periodic solutions; (2) Four different combinations about dark and anti-dark solitons, as well as the bright and dark breathers on a periodic background. Moreover, the fundamental properties and dynamical behaviors of those solutions will be discussed.

In this work, newly created soliton solutions for ion sound and Langmuir waves with an Atangana–Baleanu fractional (ABF) are given. Symbolic software is used to perform the unified solver method (USM) and Weierstrass elliptic function method (WEFM) in order to solve this model. In terms of hyperbolic functions, extended trigonometric functions, and so forth. Single-wave solutions that were entirely novel and universal were attained. The way the soliton solutions behaved in connection with the two-dimensional (2D) and three-dimensional (3D) graphics was also investigated. The dynamic investigation of the newly created soliton solutions reveals that they have several soliton forms like single-soliton, bell-shaped and mixed-form soliton profiles. This strategy is viewed as promising for handling a range of ABF evolution systems.

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.

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 $(exp(-\chi (\epsilon )))$-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.

In this work, we seek to investigate the dynamics of bright soliton in a chain of coupled pendulum pairs. After deriving the linear dispersion relation from the equation of the model, we find that among the obtained modes, the fast mode is the one on which we are going to be focused. Since the discrete simultaneous equation describing the dynamics of the model has not been extensively studied in the literature, we assume that the two lines of the model are proportional to each other. We use the rotating wave approximation method to derive a NLS equation governing the propagation of waves in the network. Depending on the choice of wave number, we deduce that the system supports bright and hole-soliton solutions. We use the obtained bright soliton as the initial condition for numerical computation, which demonstrates the significant role of the transverse coupling parameter in the system. That is, it affects the behavior of the forward-bright soliton generated in the system. The lattice allows gain and loss phenomena during the propagation of the waves.

This study delves into the realm of optical soliton solutions within the intricate framework of highly dispersive couplers integrated with optical metamaterials featuring a parabolic law nonlinear refractive index. Employing the extended auxiliary equation method, the investigation systematically unveils various soliton solutions, including dark solitons, bright solitons, singular solitons, and a distinctive amalgamation of bright and singular solitons. The results illuminate the diverse dynamics of solitons in these complex systems, offering crucial insights into the interplay among dispersion, nonlinearity, and metamaterial characteristics. Beyond enhancing the theoretical understanding of soliton behavior in optical metamaterial couplers, the findings hold practical significance by potentially influencing the design and optimization of optical communication devices and systems.

This paper explores a specific class of equations that model the propagation of optical pulses in dual-core optical fibers. The decoupled nonlinear Schrödinger equation with properties of M fractional derivatives is considered as the governing equation. The proposed model consists of group-velocity mismatch and dispersion, nonlinear refractive index and linear coupling coefficient. Different types of solutions, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted by using the integration methods known as fractional modified Sardar subequation method and modified F-expansion method. 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. A fractional complex transformation is applied to reduce the governing model into the ordinary differential equation and then by the assistance of balance principle the methods are used, depending upon the balance number. Also, we plot the different graphs with the associated parameter values to visualize the solutions behaviours with different parameter values. The findings of this work will help to identify and clarify some novel soliton solutions and it is expected that the solutions obtained will play a vital role in the fields of physics and engineering.

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

In this work, first it is shown that the Hirota–Ramani equation, which governs the nonlinear propagation of coupled Langmuir and dust-acoustec wave in a multicomponent dusty plasma, possesses kinds of wave profile such as singular periodic profile, periodic profile, singular soliton profile, M-shape rational profile, bright soliton profile, kink profile in the form of the trigonometric, hyperbolic, and rational solutions. With the aid of symbolic computation, we select the Hirota–Ramani equation with a source to investigate the validity and advantage of the improved $(\frac{G^{\prime }}{G})$-expansion method and construct some frames of the 3D profiles and the contour profiles to the equation along with the 2D profiles of some solutions to understand the traveling wave dynamics. Following the selection of appropriate values for the associated parameters, more generalized solutions are provided, along with certain patterns in the solutions that are examined. This improved method is effective, concise, reliable, and can be applied for further futuristic applications.

The Kadomtsev–Petviashvili (KP) equations are nonlinear partial differential equations which are widely used for the modeling of wave propagation in hydrodynamic and plasma systems. This study aims to make a valuable contribution to the literature by providing new solitary waves to the (2+1)- and (3+1)-dimensional potential Kadomtsev–Petviashvili (pKP)-B-type Kadomtsev–Petviashvili (BKP) equations. For this, the auxiliary equation method associated with Bernoulli equation is used and new solutions for the considered equations are obtained. The stability of obtained solutions is demonstrated using nonlinear analysis. It is shown that this method for the considered pKP–BKP equations is an important step forward in an overall mathematical framework for similar equations.

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.

This research explores the intriguing realm of eighth-order embedded solitons in highly dispersive media with cubic–quartic nonlinear susceptibilities ${\chi }^{(2)}$ and ${\chi }^{(3)}$, all within the dynamic context of multiplicative white noise and the framework of Itô calculus. Two different approaches are used in this study. The new auxiliary equation approach produces the bright soliton and singular soliton solutions, while the addendum Kudryashov’s approach produces the bright soliton, singular soliton and combo bright–singular soliton solutions. The system under investigation and the results documented within this work stand as pioneering and original contributions to the field of nonlinear optics. Furthermore, a collection of 2D, 3D and contour plots is produced to visually represent the spatial distribution and progression of different solutions. This not only contributes to the advancement of nonlinear equations in theory but also offers valuable insights for practical applications.

In this paper a new strategy is proposed in order to extend the class of complex and chaotic dynamics which can be generated by a State-Controlled CNN (SC-CNN). It is shown that, with slight modifications to the original paradigm, it is possible to represent a wide class of nonlinearities by using only piecewise linear (PWL) nonlinearities as output functions of the CNN. The adopted circuitry is presented, together with several examples concerning the emulation of the Rässler and Lorenz chaotic systems, and the Partial Differential Equations (PDE) governing the complex phenomena of solitons (Korteweg–de Vries PDE) and the activator–inhibitor reaction for the formation of patterns in the sea shells (Meinhardt–Gierer PDE). The results achieved enforce the role of SC-CNNs as universal paradigm for the generation of nonlinear, complex dynamics.

Based on the study of the dynamics of a dissipation-modified Toda anharmonic (one-dimensional, circular) lattice ring we predict here a new form of electric conduction mediated by dissipative solitons. The electron-ion-like interaction permits the trapping of the electron by soliton excitations in the lattice, thus leading to a soliton-driven current much higher than the Drude-like (linear, Ohmic) current. Besides, as we lower the values of the externally imposed field this new form of current survives, with a field-independent value.

Assuming the quantum mechanical "tight binding" of an electron to a nonlinear lattice with Morse potential interactions we show how electric conduction can be mediated by solitons. For relatively high values of an applied electric field the current follows Ohm's law. As the field strength is lowered the current takes a finite, constant, field-independent value.