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In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ 6 model in dimension 1 + 1. We construct a sequence of approximate solutions (ϕ k (v , t , x)) k ∈ N ⩾ 2 for this model to understand the effects of the collision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ 6 model. [ABSTRACT FROM AUTHOR]

The properties of small‐amplitude solitary electron acoustic waves in a plasma where the hot electrons take regularized kappa distribution are investigated via the reductive perturbation method. It is found there only exists the sup‐electron‐acoustic rarefactive soliton, the electron acoustic velocity is modified by the regularized kappa distribution. The amplitude and width of the electron acoustic soliton decrease monotonously with the increase of the exponential cutoff parameter of hot electrons. But they vary with respect to the spectral index non‐monotonously. As to the observed parameters in the Earth's inner magnetosphere, the speed, electric field strength, and the width of solitary electron acoustic wave are comparable with the observations. [ABSTRACT FROM AUTHOR]

Korteweg-de Vries-type equations have occurred in the fields of planetary oceans, atmospheres, cosmic plasmas and so on, while nonlinear evolution equations with the variable coefficients have provided a realistic perspective on the inhomogeneities of media and non-uniformities of boundaries. In this paper, we investigate a (3+1)-dimensional Korteweg-de Vries equation with the time-dependent coefficients in a fluid. Based on the Hirota method, we obtain a bilinear form via the binary Bell polynomial approach. Based on the bilinear form, we derive the N-soliton, breather and periodic-wave solutions, where N is a positive integer. Besides, we investigate the asymptotic behaviors of the breather and periodic-wave solutions. Breather waves and periodic waves are graphically displayed. Finally, relation between the periodic-wave solutions and one-soliton solutions is discussed. This paper provides an intuitive understanding for the nonlinear phenomena of those obtained solutions, and those nonlinear phenomena have potential application value in fluid dynamics and other fields. [ABSTRACT FROM AUTHOR]

Complex by DDMC/sncRNAs(a-miR-155, piR-30074, and miR-125b) have shown the full recovery of mice from virus-induced sarcoma after treatment by changing to normal cells from cancer cells. With kinetic of inhibiting tumor growth, the difference from control (m m 2) of one intravenous injection and two intravenous injections is the same curvature and rate so that this is a soliton wave having a permanent form (velocity, shape) such as one faster soliton overcoming another one without any changing shape in the case of two intravenous injections. Signal transduction for this DDS must be Hill-type sigmoids following the nonlinear Schrödinger model by transfer of energy, sine-Gordon soliton model by momentum, and Fisher KPP soliton model by mass transfer. We find out that the cell outlet/inlet response of DDMC/sncRNAs is more dependent on the soliton signal not losing energy and shape without jamming communication. The endoplasmic reticulum-mitochondrial C a 2 + fluxes induced by soliton to Chromosome in nuclear will take place epigenetic modifications on 5-position of cytosine (5mC) by TET enzymes and thymine DNA glycosylase (TDG) as an overall intracellular reaction. By soliton flow, the result shows the equations of quantum mechanics can be related to the epigenetic control with C a 2 + fluxes followed by "induced fit model" Hill equation S-shaped. [ABSTRACT FROM AUTHOR]

In this research, we delve into the exploration of the Kuralay-IIA equation by employing two power full approaches, namely extended hyperbolic function method (EHFM) and the improved modified Sardar-subequation method (IMSSEM). Applications of the Kuralay-IIA equation can be observed in a variety of physical phenomena, particularly in optical fibers, magnetic materials, and nonlinear optics. This equation performs an essential part in understanding gas behavior. Specifically, it offers an image of the relationship between volume, temperature and pressure in a gas system. We proceed through the matter in detail through the use of space curves which demonstrate integrable motion. Using the proposed techniques, our investigation provides an extensive variety of soliton solutions, such as periodic, dark–bright, bright, dark, solitary, and some other solutions. These solutions are in good alignment with previous study findings in this area. Analytical wave solutions are vital as they provide a basic comprehension of the physics or mathematics at play and establish a framework for further studies. The findings obtained from this study may help design models to come. The techniques used in this study are very effective, straightforward, and capable of handling more nonlinear models with high reliability. We use the Mathematica software to validate the accuracy and reliability of our results. Using carefully constructed 2D and 3D graphs generated, the resulting solutions are visually shown. [ABSTRACT FROM AUTHOR]

We begin by deriving the nonlinear Schrödinger equation (NLSE) in the presence of nonlinear microstructures, where the refractive index undergoes periodic modulation in the transverse direction for photonic crystal containing circular shape. Our investigation delves deeply into the intricate mechanisms behind transverse and longitudinal modulation of the refractive indexes, which aligns with the design of numerous manufactured slab microstructure waveguides. Theory of solitary waves in nonlinear microstructure, here unconventional fiber has been studied and examined in detail. In this work, composite methods leading both transverse and longitudinal modulation of refractive indexes have been presented in detail. The work also presents the diverse characteristics of this NLSE for both homogeneous and nonhomogeneous medium, encompassing various orders of nonlinearity specific to the nonlinear microstructure under consideration. Additionally, we analyze the wave propagation profiles for wide signals, which are wider than the periodicity of micro-structured photonic crystal. We also conduct a perturbation analysis for narrow signals that are even narrower than the periodicity of photonic crystal in transverse direction. [ABSTRACT FROM AUTHOR]

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. [ABSTRACT FROM AUTHOR]

Optical fibers are used in the communications, biological sensors and chemical sensors. We investigate a variable-coefficient modified Hirota equation for the amplification or absorption of pulses propagating in an inhomogeneous optical fiber. With respect to the complex envelope of the optical field, we construct the infinitely-many conservation laws based on the existing Lax pair. According to the existing Darboux transformation, we derive the three-soliton solutions, the higher-order breather solutions and breather-to-soliton transition condition. Amplitudes of the two solitons change after the interaction, while velocities of them are unchanged via asymptotic analysis. When $ P(z)=0 $ P (z) = 0 , interactions among the three parabolic or wavy solitons, interaction between the two parabolic or wavy or crooked breathers, and interactions among the three parabolic and wavy breathers are presented, where $ P(z) $ P (z) is related to the nonlinear focus length. Velocities of three solitons or two crooked breathers with $ P(z)\neq 0 $ P (z) ≠ 0 are different from those with $ P(z)=0 $ P (z) = 0. Based on the breather-to-soliton transition condition, when $ P(z)=0 $ P (z) = 0 , parabolic or wavy multi-peak and M-shaped solitons are presented; when $ P(z)\neq 0 $ P (z) ≠ 0 , the crooked periodic wave and anti-dark soliton are shown. [ABSTRACT FROM AUTHOR]

The existence of small-amplitude ion-acoustic solitary waves is investigated in a cometary plasma with five components: hot solar electrons, cometary electrons that are slightly colder, kappa-described hydrogen ions, and positively and negatively charged oxygen ions. The reductive perturbation method is implemented on the basic set of equations to derive the generalized Gardner equation and the solitary wave solution is obtained. Employing the traveling wave transformation, the Gardner equation is deduced for the dynamical system and the bifurcation of ion-acoustic waves (IAWs) is examined. The existence of several nonlinear trajectories, like homoclinic trajectories, quasi-periodic trajectories, and chaotic trajectories are confirmed by the notion of phase plane analysis, Poincare mapping, and Lyapunov exponents for IAWs. Finally, the numerical solutions are presented graphically, which provides a clear insight into the influence of physical parameters on wave propagation. This study could aid in the understanding of the propagation dynamics of ion-acoustic waves in cometary plasmas. [ABSTRACT FROM AUTHOR]

We propose a simple optical model, which consists of two Kerr nonlinear parallel planar waveguides sandwiching a one-dimensional photonic crystal for demonstrating the phenomenon of spontaneous symmetry breaking in optics. We study the shape and stability of soliton pairs propagating in two planar waveguides and the spontaneous symmetry breaking of the power distribution of solitons in the two waveguides. The results of numerical simulation show that there is supercritical type of spontaneous symmetry breaking in this system. More optical power P and smaller refractive index of photonic crystals C 0 make soliton solutions more prone to spontaneous symmetry breaking bifurcation. Asymmetric soliton solutions and symmetric soliton solutions correspond to different dispersion relations. Under the given total power P, for the symmetric soliton solutions, the propagation constant k increases linearly with C 0 , and for asymmetric soliton solutions, the propagation constant k decreases nonlinearly with C 0 . The minimum point of k (C 0) is just the demarkation between the symmetric soliton solutions and the asymmetric soliton solutions. If C 0 is fixed, the minimum point of the Hamiltonian of the system, E (P) , is also the critical point of the two different kinds of soliton solutions. We also study stable and unstable distributions of two kinds of soliton solutions. [ABSTRACT FROM AUTHOR]

The inverse scattering transform is a tool for solving the initial value problem of nonlinear equations, and the solution of the initial-value problem by inverse scattering transform proceeds in three steps: direct scattering, time evolution and inverse problem. In this paper, we discuss the higher-order integrable discrete nonlinear Schrödinger equation which is subjected to inverse scattering transform under nonzero boundary conditions, and the corresponding soliton solutions are obtained and illustrated. [ABSTRACT FROM AUTHOR]

In this paper, two different meshfree method with radial basis functions (RBFs) is proposed to solve Boussinesq-type (Bq) equations. The basic conservative properties of the equation are investigated by computing the numerical values of the motion's invariants. The accuracy of the method is tested using computational tests to simulate solitary waves in terms of L_∞ error norm. The outcomes are contrasted with analytical solution and a few other earlier studies in the literature. The results show that meshless methods are very effective and accurate. [ABSTRACT FROM AUTHOR]

The interaction between two Airy pulses propagating at different wavelengths is numerically investigated. The periodically varying peak intensity of the soliton that emerges from stronger Airy pulse (pump pulse) leads to the formation of periodic temporal boundary. The relatively weaker Airy pulse (probe pulse) on interaction with this boundary gets partially reflected as well as transmitted. As a result, the probe pulse spectrum splits into two parts‐ the reflected pulse spectrum undergoes redshift while transmitted pulse exhibits blueshift. The probe pulse witnesses maximum reflection when point of interaction lies on the intensity maxima of the emergent soliton from pump Airy pulse. On the other hand, maximum transmission occurs when probe Airy pulse interacts at the intensity minima of the soliton. The reflection and transmission processes can be manipulated by tuning the time delay between pump and probe Airy pulses. In the case of a sufficiently intense pump pulse, the temporal boundary mimics the artificial optical event horizon, and the weak probe Airy pulse is completely reflected. This phenomenon is equivalent to the temporal version of total internal reflection. The results of the study hold potential applications in optical manipulation and temporal waveguiding. [ABSTRACT FROM AUTHOR]

These days, watching the shallow water waves, people think about the nonlinear Broer-type models, e.g., a (2+1)-dimensional generalized Broer-Kaup system modeling, e.g., certain nonlinear long waves in the shallow water. For that system, with reference to, e.g., the wave height and wave horizontal velocity, this paper avails of symbolic computation to obtain (A) an auto-Bäcklund transformation with some solitons; (B) a group of the scaling transformations and (C) a group of the hetero-Bäcklund transformations, to a known linear partial differential equation, from that system. Results rely on the coefficients in that system [ABSTRACT FROM AUTHOR]

Studies on the shallow water waves belong to the cutting-edge issues in sciences and engineering. In this paper, introducing symbolic computation, for a generalized nonlinear shallow water wave equation, with respect to the displacement and velocity of the water, we establish an auto-Bäcklund transformation with some solitonic solutions, as well as a set of the similarity reductions, the latter of which ought to be focused towards a known ordinary differential equation. Our results are seen to tie to the gravitational force and wave height. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS. [ABSTRACT FROM AUTHOR]

The presence of solitary wave excitation in biopolymers offers a framework for understanding processes such as energy transport, localization, transcription, and translation. While numerous models have been proposed to explain these biomolecular processes, we consider the helicoidal Peyrard-Bishop (HPB) model, incorporating the combined effects of viscosity and solvent interaction. By applying a semidiscrete approximation to the obtained Hamiltonian and performing further transformations, we derive a nonlinear Schrödinger equation with a perturbation term that depends on the solvent interaction potential. Using the Hirota bilinearization method, we obtain one- and two-soliton solutions. Analysis of the collision profiles reveals inelastic interactions, resulting in an uneven exchange of energy between the solitary waves. This uneven energy exchange leads to the suppression of one of the solitary waves. [ABSTRACT FROM AUTHOR]

This study reveals the dynamics of short optical pulses in nonlinear media. Then, for the survival of different types of solitary waves in the nonlinear medium, the constraint relations to the physical parameters and chirps associated with each of the solitary waves are explicitly shown. We authenticate numerically all those obtained from the analytical method and show the chirping effect is saturative in nature and can be managed by changing coefficients of inter-modal dispersion, stimulated Raman scattering (SRS), and self-steepening effect. We also verify the more operative condition of intermodal dispersion. [ABSTRACT FROM AUTHOR]

In this paper, the optical soliton and solitary wave solutions of the (2 + 1) -dimensional Mel'nikov equation are investigated using the Kudryashov R function technique. The Kudryashov R function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the (2 + 1) -dimensional Mel'nikov equation have been displayed in 3D, 2D and contour graphs. [ABSTRACT FROM AUTHOR]

In the all-optical domain, the reflective semiconductor optical amplifier (RSOA) creates due attention in the research community because the RSOA has high gain, low-power consumption, high-speed signal processing ability as well as low noise performance. In this work, we have proposed dibit-based OR and NOR gates using RSOA. Dibit-based logic is utilized in this design to get the reduced bit-error problems and high degree of parallelism. To verify the functionality of the aforementioned designs, we have used MATLAB software and also the quality factor (Q), extinction ratio (ER), contrast ratio (CR), and bit-error-rate (BER) have been calculated. [ABSTRACT FROM AUTHOR]

For the systems modeled by a generalized reaction–diffusion equation with the higher‐order quintic nonlinearity, we explore the soliton dynamics via the F‐expansion method. By the novel F‐base function ansatz, we first derived the bright soliton and kink soliton solutions for the one‐dimensional case of the reaction–diffusion equation with quintic nonlinearity. Furthermore, we employed self‐similar techniques to analyze the higher‐dimensional dynamics of bright soliton and kink soliton solutions supported by the (2+1)‐dimensional reaction–diffusion equation system with quintic nonlinearity. Additionally, we conducted stability analysis of derived soliton solutions. Our theoretical results demonstrate that under certain parametric setting, the reaction–diffusion equation model with higher‐order nonlinearity supports bright soliton and kink soliton in higher‐dimensional as well as lower dimensional setting, which provides guidance for observing and investigating soliton behavior in systems modeled by the reaction–diffusion equation with higher‐order quintic nonlinearity. [ABSTRACT FROM AUTHOR]

A new generalised non-local nonlinear Schrödinger (NLS) equation is introduced which possesses a Lax pair and is parity–time (PT)-symmetric. Thus, it is confirmed that the generalised non-local NLS equation is integrable. The inverse scattering transform for the generalised non-local NLS equation is developed using a Riemann–Hilbert problem for rapidly decaying initial data and an approach for finding pure soliton solutions is described. The analytical characteristics of the eigenfunctions, scattering data and their symmetries are discussed. Finally, using Mathematica some important two-dimensional plots of the wave solutions are shown to illustrate the dynamics of the model. [ABSTRACT FROM AUTHOR]

In ultracold Fermi gases, by adjusting the strength of spin orbit coupling to match Fermi energy, many novel quantum effec ts can be genera ted. In the pas t few decades, scholars have conducted extensive theoretical and experimental research on Fermi gases induced by one-dimensional spin orbit coupling. Compared with high-dimensional spin orbit coupling, one-dimensional spin orbit coupling, although relatively simple, is the most reliable and feasible tool for exploring basic quantum physical phenomena in experiments. This paper systematically summarizes the interesting physical phenomena of Fermi gas under one-dimensional spin orbit coupling in theoretical work. Including theoretical research on dynamic oscillation and soliton effect, topological superfluid, Majorana edge state, ferromagnetic phase transit ion, and quantum phase. How to achieve spin orbi t coupling and observe singular phenomena in exp erime nts is a hot and difficult research topic. We summarize several common experimentai schemes and detection met hods. Finally, we look forward to the research on Fermi gas induced by onedimensional spin orbit coupling. One dimensional spin orbit coupling can provide reference for abecedarians and contribute to the study of multi body system regulated by spin orbit coupling. This paper aims to provide a reference for abecedarians in cold atomic physics to gain a deeper understanding of the physical mechanisms of multi-body systems under spin orbit coupling. [ABSTRACT FROM AUTHOR]

For systems modeled by the resonant nonlinear Schrödinger equation, we aim to investigate the soliton dynamics and the stability of the system. We extend the equation to a generalized form (GRD-NLSE) and utilize the modified F-expansion method to derive various soliton solutions, including bright, kink, and singular solitons for both (1 + 1)-dimensional and (2 + 1)-dimensional models. Additionally, we graphically illustrate soliton behavior. To analyze system stability, we perform modulation instability and soliton stability analyses. Our findings provide valuable guidance for experimental studies and observations of solitons in GRD-NLSE systems. [ABSTRACT FROM AUTHOR]

We present a characterization for the rotational soliton for the curve shortening flow (CSF) on the revolution surfaces of R 3 . Furthermore, we describe the behavior of such curves by showing that the two ends of each open curve are asymptotic to a parallel geodesic. [ABSTRACT FROM AUTHOR]

In this research, the modified extended tanh-function (METF) and the extended Jacobi elliptic function expansion (EJEFE) techniques are used to investigate the generation and detection of soliton structures in the Hirota–Maccari (HM) model. Consequently, we obtain soliton solutions with advanced structures, including singular bright soliton, dark soliton, periodic waves, breather waves, periodic breather waves, and multiple bright and dark breather waves. In addition, a lump-type breather wave is also included in the presented solutions. Stability analysis of the obtained solutions is addressed by employing the Hamiltonian technique. 3D surfaces and 2D visuals of the outcomes are represented with the help of a computer application. These findings contribute to understanding nonlinear wave phenomena with potential applications in optics, fluid dynamics, and plasma physics. [ABSTRACT FROM AUTHOR]

The Sharma–Tasso–Olver (STO) equation is a nonlinear, double-dispersive, partial differential equation that is physically important because it provides insights into the behavior of nonlinear waves and solitons in various physical areas, including fluid dynamics, optical fibers, and plasma physics. In this paper, the STO equation is generalized to a fractional equation by using Atangana (or Atangana–Baleanu) fractional space and time beta-derivatives since they have been found to be useful as a model for a variety of traveling-wave phenomena. Exact solutions are obtained for the integer-order and fractional-order equations by using the Sardar subequation method and an appropriate traveling-wave transformation. The exact solutions are obtained in terms of generalized trigonometric and hyperbolic functions. The exact solutions are derived for the integer-order STO and for a range of values of fractional orders. Numerical solutions are also obtained for a range of parameter values for both the fractional and integer orders to show some of the types of solutions that can occur. As examples, the solutions are obtained showing the physical behavior, such as the solitary wave solutions of the singular kink-type and periodic wave solutions. The results show that the Sardar subequation method provides a straightforward and efficient method for deriving new exact solutions for fractional nonlinear partial differential equations of the STO type. [ABSTRACT FROM AUTHOR]

In this paper, the dust acoustic wave is studied in conditions where the dust temperature is not constant. This model includes negatively charged dust particles and thermal ions. The reductive perturbation technique is used and the new point in this study is the nonuniformity of the plasma temperature. The nonconstancy of the temperature is entered as a first-order perturbation in the calculations, and finally the modified Korteweg–de Vries (mKdV) equation is derived. The solution of the modified KdV equation clarifies the change in soliton shape of the wave when it moves in the perturbation plasma. We show how the soliton wave undergoes deformation and amplitude reduction during propagation when it encounters a region with a different temperature. The output of this research can be effective to better understand the wave behavior in a real plasma with nonuniform temperature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a model for determining the thickness of the surface layer of metals and the quantum structure of this layer. An atomically smooth metal is represented as a diagram: nanolayer → mesolayer → bulk phase, which differ from each other in the nature of size effects. There is no size effect in the bulk phase. The thickness of the surface layer of metals R(I) has a size from 1 nm to 10 nm, except for cesium, i.e. they represent a nanostructure. It is shown that the energy levels En of the nanolayer are determined by one fundamental parameter - the lattice constant of the metal à. As soon as the parameter à stops changing, the spectrum of quantum states passes into a continuous spectrum. The nanolayer R(I) is a step function in En, which can be easily reduced to the Lebesgue integral, which plays an important role in quantum theory. Quantum threads (quantum planes) in the nanolayer R(I) can be interpreted as solitons, crowdions, discrete breathers turning into nanocracks. Quantum threads in the R(I) nanolayer and the nanolayer itself can be represented as a nanoparticle with a diameter of R(I) and pre-melting occurs in it in a stepwise manner. It is finally shown that all metal monolayers differ significantly from each other. [ABSTRACT FROM AUTHOR]

We extend the result of Kowalczyk, Martel, and Mu\~noz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133--2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein--Gordon equation with p > 3, to the case 2 ≥ p > 5/3. . The result is attained performing new and refined estimates that allow us to close the argument for power law in the range 2 ≥ p > 5/3. . [ABSTRACT FROM AUTHOR]

Nonlinear waves are pivotal in analyzing the propagation of electromagnetic waves and the dynamics of oceanic systems. These waves are indispensable for modeling the long-term impacts of wave energy in coastal regions, addressing critical issues such as climate change adaptation, erosion and coastal flooding. The Bogoyavlensky–Konoplechenko system describes the interaction of the Riemann wave and the long wave propagating in two dimensions. The present work aims to elaborate symmetry reductions and derive invariant solutions of the proposed system. Meanwhile, the infinitesimal generators under one-parameter transformation are constructed, which render the system invariant. Therefore, a repeated process of reductions results in an equivalent system of ordinary differential equations and hence, leads to exact solutions. The solutions have rich physical significance and are efficient in defining several phenomena due to existing arbitrary functions and constants. To examine the physical nature of these solutions, numerical simulation is performed and thus, wave structures like bright and dark lumps, multisoliton, line multisoliton, periodic and annihilation profiles are analyzed. The bifurcation theory has been used to investigate the stability of dynamical system and examine corresponding phase portraits. Furthermore, the conserved vectors with underlying symmetries are constructed using Noether’s theorem. [ABSTRACT FROM AUTHOR]

Nonlinear phenomena, characterized by behaviors that cannot be explained by linear systems and has significant challenges in understanding and modeling. To address this, the mathematical description of such phenomena relies on differential equations. In this study, we investigate the cubic-quintic-septimal nonlinear (7th order nonlinear media) Schrödinger wave equation, which governs the evolution of light beams in a weak non-local medium. The novelty of our study lies in the application of the improved generalized Riccati equation mapping method to obtain exact solutions for the governed equation. This scheme offers a systematic and reliable approach to exploring nonlinear phenomena, contributing to the advancement of nonlinear science and its practical applications. By applying the proposed scheme, a range of exact solutions encompassing trigonometric, rational, exponential, and hyperbolic functions are derived which offer insights into the dynamics of the light beams. Additionally, 2D and 3D graphical illustration are presented to provide a comprehensive demonstration of their dynamical behavior. Furthermore, it is important to highlight the significance of studying the cubic-quintic-septimal nonlinear Schrödinger wave equation which has application in various domains such as quantum mechanics, optics, and nonlinear wave propagation. Understanding of its solutions facilitates the design and optimization of systems involving weak non-local media. [ABSTRACT FROM AUTHOR]

In this article, the extended fan method and the positive quadratic function technique are used to solve the (2 + 1) -dimensional Chiral nonlinear Schrödinger equation. Several new outcomes are obtained. Several new results are obtained. We can offer soliton-type solutions, triangular-type solutions, simple and combined solutions using Jacobi's elliptic functions. Using suitable parameters, some graphics are given to observe the evolution of these analytical solutions. Many other nonlinear differential equations can benefit from using the methods proposed in this work. [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]

Ion acoustic solitary waves with arbitrary amplitude have been investigated using the Sagdeev potential (SP) method in a collisionless plasma consisting of non‐isothermal electrons, positrons, and positive ions. The SP has been obtained and the dependence of large amplitude solitary structures on various plasma parameters such as ion drift velocity (v0) $({v}_{0})$, non‐thermal parameter (β) $(\beta)$, electron to positron temperature ratio (σ) $(\sigma)$, positron density (p) $(p)$, and Mach number (M) $(M)$ have been studied. It is shown that, for the same values of plasma parameters, accounting for ion drift velocity greatly corrects the solitary wave profile. [ABSTRACT FROM AUTHOR]

This paper examines the complex Ginzburg Landau equation, which describes pulse propagation inside a fiber with the triple power law of self-phase modulation. Since the effect of parameter selection has become very important in relevant model studies recently, self-phase modulation has been added to the complex Ginzburg Landau equation, which has been studied in the literature, and it is aimed at investigating the analytical solutions of the presented equation. Adding the triple power law of the self-phase modulation parameter to the model, in addition to existing studies in the literature, emphasizes the innovative aspect and importance of the study. The first aim is to reveal bright and singular solitons using the new Kudryashov method. The new Kudryashov method is a technique that is frequently used in the literature, is effective for generating analytical solutions, provides ease of operation, and can be applied to a wide class of nonlinear partial differential equations. The second goal is to show that the obtained solutions have modulation stability. By using modulation instability analysis, the gain spectrum is formed for different parameter values. Graphic presentations support the findings. Moreover, bright and singular soliton portraits are demonstrated with 3D and 2D graphs. The novelty of the study lies in the fact that the relevant model has not been studied before with an effective method such as the new Kudryashov method, and the modulation instability has been studied for the first time. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the inverse scattering transform for a nonlinear lattice equation, which can be used to study the fluctuation of nonlinear optics and dynamics of anharmonic lattices. Symmetries, analyticities and asymptotic behaviors of eigenfunctions will be obtained in the direct scattering analysis to establish a suitable Riemann-Hilbert problem. The Riemann-Hilbert problem of the scattering data with simple poles will be constructed. In particular, by using the Laurent expansion and the generalized residue condition to solve the Riemann-Hilbert problem, the determinant representation of N-soliton solution for the equation will be presented. One-dark-soliton under non-vanishing boundary conditions will be displayed through some representative reflectionless potentials. [ABSTRACT FROM AUTHOR]

The inner characteristics of numerous nonlinear phenomena that arise in real-life problems are stated through nonlinear partial differential equations. This exploration is conducted with fractional stochastic Kraenkel–Manna–Merle model in ferromagnetic materials and achieved ample soliton solutions by adapting enhanced rational (G ′ / G) -expansion and improved tanh schemes. Ferromagnetic materials play a vital role in telecommunications applications, data storage, and manipulation. A new wave variable in the sense of confirmable fractional derivative is utilized to convert the governing model into the ordinary system. The motions of ultra-short-wave pulses in ferrite's materials are analyzed by showing the effects of fractional derivatives and noise terms on the Brownian motion through multiple diverse 3D, 2D and contour plots. Periodic, singular periodic, kink, anti-kink etc. are visualized under the different parameter's values involved in the obtained solitary wave solutions. The outcomes made available in the current study might play vital role to depict relevant intricate nonlinear phenomena and inspire the researchers to consider the utilized techniques in further studies. [ABSTRACT FROM AUTHOR]

Quasi-periodic solutions of the KP hierarchy acted by vertex operators are studied. We show, with the aid of the Sato Grassmannian, that solutions thus constructed correspond to torsion free rank one sheaves on some singular algebraic curves whose normalizations are the non-singular curves corresponding to the seed quasi-periodic solutions. It means that the action of the vertex operator has an effect of creating singular points on an algebraic curve. We further check, by examples, that solutions obtained here can be considered as solitons on quasi-periodic backgrounds, where the soliton matrices are determined by parameters in the vertex operators. [ABSTRACT FROM AUTHOR]

Fluid mechanics concerns the mechanisms of liquids, gases and plasmas and the forces on them. We aim to investigate a (2 + 1) -dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics in this paper. With the help of the Painlevé analysis, we find that the above system has Painlevé-integrable property. A set of the auto-Bäcklund transformations and some solutions are derived by the virtue of the truncated Painlevé method. We obtain certain bilinear forms via some seed solutions. According to the mentioned bilinear form, we derive the multiple-soliton solutions on some nonzero backgrounds. Based on the soliton solutions and conjugation transformations, the higher-order breather solutions on certain nonzero backgrounds have been obtained. Via some conjugation transformations, hybrid solutions formed from the breathers and solitons on certain nonzero backgrounds have been derived. We also graphically show the interactions between those solitons and breathers. [ABSTRACT FROM AUTHOR]

The geometry of solutions to curvature flows is highly dependent on properties of the speed function that drives the flow. Here, we study a specific kind of solution, namely the analogue of the bowl soliton [studied in Clutterbuck et al. (Calc Var Partial Differ Equ 29(3):281–293, 2007)] to a very large class of geometric flows, and present a comprehensive classification of these speed functions based on their asymptotic geometry. Our results here are among the very few that relate the geometry of solutions to such very general nonlinear geometric flows to algebraic properties of the speed function. In addition to existence and uniqueness, we provide precise criteria for the speed function which determine whether the corresponding translator is an entire graph or asymptotically cylindrical. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we also describe the asymptotics of the translator at infinity. It is remarkable that we do not require any concavity or high regularity on the speed function. [ABSTRACT FROM AUTHOR]

The application of limited polarization and gain control to the recently demonstrated mode-locked ultralong ultrafast, EDFA-based and Raman-assisted pulsed ring fibre lasers is shown to reduce output pulse duration and noise, as well as to increase pulse peak power in cavities of 10 km with repetition rates as low as 38 kHz and pulse durations below 200 fs FWHM, which remain effectively depolarized at the output in the optimized case, accompanied by improved mode-locking stability. The impact of polarization state optimization prior to the doped-fibre amplification section on these devices is studied for different lengths of fibre, harmonic modes, and Raman amplification values. Peak power is nearly doubled thanks to temporal compression and spectral broadening, with a wide choice of regimes of operation dynamically available for a given cavity length, allowing the modification of laser output towards targeted pulse characteristics.

In this study, let's find the soliton solutions of the perturbed nonlinear Chen–Lee–Liu equation via the new fractional derivative operator in following form iℵhyp,tαƛ(x,t)+aℵhyp,x2αƛ(x,t)+ib|ƛ(x,t)|ℵhyp,xαƛ=i[λℵhyp,xαƛ(x,t)+θℵhyp,xα|ƛ(x,t)|2mƛ(x,t)+σƛ(x,t)ℵhyp,xα(|ƛ(x,t)|2m),by using the extended hyperbolic method. This equation is one of the most widely used models in mathematics and physics, which requires the study of this equation with different and practical methods. One of these methods is the extended hyperbolic approach, which is discussed and analyzed in this article. Since this equation has a very wide application in particle physics, how to study it is very important. Therefore, it is very important to use methods that include a wide range of answers. This method can also be very useful because it has a variety of answers, which we can see in the obtained answers. The solutions obtained in this article are new and more accurate than the studies done so far.

This study investigates soliton solutions to the (2 + 1)-dimensional space fractional Heisenberg ferromagnetic spin chain equation incorporating beta fractional derivatives that explain the nonlinear propagation of spin wave in the ferromagnetic material with magnetic interactions including the both classical and semi-classical frameworks. This model has applications in spintronics, plasma physics, magnet theory, and condensed matter physics. Employing the two-potential extended Kudryashov and (G′/G, 1/G)-expansion methods, we derive some original and generic solutions composed of trigonometric functions, hyperbolic functions, and their rational forms. For particular values of the parameters, these solutions offer V-shaped, bell-shaped, kink, periodic, flat kink, and singular periodic solitons. The physical behavior of these solitons is demonstrated through three-dimensional, contour, and two-dimensional plots. The results are important in the study of phase transitions in ferromagnetic materials, lattice vibrations in condensed matter, and the propagation of shock waves in plasma. This study also demonstrates the potential and reliability of the employed strategies.