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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]

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]

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]

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 formation and propagation dynamics of the finite‐amplitude ion‐acoustic wave (IAW) structures (e.g., soliton, breather, rogue wave, etc.) is theoretically investigated in a plasma comprising of kappa distributed solar and cometary electrons of different temperatures, a hot H3O+$$ {\mathrm{H}}_3{\mathrm{O}}^{+} $$ drift ion component, and a pair of oppositely charged oxygen ion components. The modified‐KdV (mKdV) equation is derived in order to study the propagation dynamics of the ion‐acoustic solitary wave (IASW). It is then converted into the nonlinear Schrödinger equation (NLS) through appropriate algebraic manipulation in order to observe the amplitude modulation of the IAWs. Also, the appearance of envelope soliton and the possibility of breather structure formation have been studied from the NLS equation. The dependence of plasma parameters on the formation and propagation of IAW structures has been briefly discussed. The modified‐KdV equation is reduced in a dynamical system through the application of coordinate transformation. The existence of finite‐amplitude nonlinear and supernonlinear IAWs is demonstrated by phase plane analysis. Due to the fact that the results are primarily associated with cometary plasma, they possibly provide greater insight of the nonlinear characteristics of cometary plasma. [ABSTRACT FROM AUTHOR]

We discuss the integration of the Cauchy problem for the negative-order nonlinear Schrödinger equation in the class of rapidly decreasing functions via the inverse scattering problem method. In particular, we obtain the time dependence of scattering data of the Zakharov–Shabat system with the potential that is a solution of the considered problem. We give an explicit representation of the one-soliton solution of the negative-order nonlinear Schrödinger equation based on the obtained results. [ABSTRACT FROM AUTHOR]

This paper explore the new solitary wave solutions of the Biswas–Milovic equation and resonant nonlinear Schr o ¨ dinger equations with Kerr–Law nonlinearity. In mathematical physics, the Biswas–Milovic equation plays an important role. The dynamics of these solitons are well known to propagate over large distances in a few femto-seconds through these fibres. These dynamics are governed by the nonlinear Schrödinger's equation. The resonant nonlinear Schrödinger equation can be used to model the propagation of waves in fiber optics. The powerful analytical approach known as the extended simple equation method is employed to explore the solitary wave solutions of the Biswas–Milovic and resonant nonlinear Schr o ¨ dinger equations. These equations are primarily explored in the realm of solitons in nonlinear fiber optics. New dark, kink, anti-kink and singular periodic solitons are secured. Furthermore, 3D surface graphs, contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica. The constructed solutions will helpful to understand the dynamical framework of nonlinear Schr o ¨ dinger equations in the related physical phenomena. [ABSTRACT FROM AUTHOR]

Nonlinear partial evolution equations are mostly significant to illustrate critical phenomena in wave theory concerning real-world problems. The current study deals with the (2 + 1)-dimensional nonlinear Fokas model depicting the nonlinear pulse through the mono-mode optical fibers. Improved auxiliary equation and improved tanh schemes are executed on the considering governing system. Subsequently, a variety of optical soliton solutions with the nature of dynamic nonlinear waves are made accessible throughout the present exploration. Some of constructed solutions are figured out in 3D, 2D and contour sense for the visualization to the readers for making them understand of the characteristics of dynamic waves. The solitons are visible to be bright, dark, kink, anti-kink, singular kink, periodic, compacton, anti-compacton etc. in the current exploration. Involved free parameters are assigned with various numerical values and brought out the effects of nonlinear pulses in wave propagation along mono-mode optical fibers. The entire work might claim to be recorded in the literature as new aspects of research. [ABSTRACT FROM AUTHOR]

Pipe flows are commonly found in nature and industry as an effective mean of transporting fluids. They are primarily characterized by their resistance law, which relates the mean flow rate to the driving pressure gradient. Since Poiseuille and Hagen, various flow regimes and fluid rheologies have been investigated, but the behavior of shear-thickening suspensions, which jam above a critical shear stress, remains poorly understood despite important applications (e.g., concrete or food processing). In this study, we build on recent advances in the physics of shear-thickening suspensions to address their flow through pipes and establish their resistance law. We find that for discontinuously shear-thickening suspensions (large particule volume fractions), the flow rate saturates at high driving stress. Local pressure and velocity measurements reveal that this saturation stems from the emergence of a frictional soliton: a unique, localized, superdissipative, and backpropagating flow structure coexisting with the laminar frictionless flow phase observed at low driving stress. We characterize the remarkably steep effective rheology of the frictional soliton and show that it sets the resistance law at the whole pipe scale. These findings offer an unusual perspective on low-Reynolds suspension flows through pipes, intriguingly reminiscent of the transition to turbulence for simple fluids. They also provide a predictive law for the transport of such suspensions in pipe systems, with implications for a wide range of applications. [ABSTRACT FROM AUTHOR]

A prediction of rod wave type with great precision is extremely important in theoretical analysis and practical applications. Besides the well-known periodic motion and resonance, this paper studies the rod wave in a fractal space, and a fractal solitary wave is unlocked by the variational approach, the results reveal that the rod strain non-linearity and fractal dimensions affect greatly the wave travelling properties. This paper offers a new window for identifying a solitary wave from periodic motion easily and accurately. [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]

We consider k-equivariant wave maps from the exterior spatial domain R3∖B(0,1) into the target S3. This model has infinitely many topological solitons Qn,k​, which are indexed by their topological degree n∈Z. For each n∈Z and k≥1, we prove the existence and invariance of a Gibbs measure supported on the homotopy class of Qn,k​. As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives. [ABSTRACT FROM AUTHOR]

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma. [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]

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers. [ABSTRACT FROM AUTHOR]

Studies on rogue waves (RWs) over constant and periodic wave background are of contemporary interest in several areas of physics. These RWs arise not only on such backgrounds but also appear on double-periodic wave background. In this paper, we derive RW solutions over a double-periodic wave background of a fifth-order nonlinear Schrödinger equation. We create the double-periodic wave background with elliptic functions (in combinations of cn, sn and dn) as seed solutions in the first iteration of Darboux transformation. We then calculate the growth rate of instability for these double-periodic solutions under different values of elliptic modulus parameter. On the top of these background waves, we generate RWs with the help of second independent solution to the eigenvalue problem. Further, we analyze the differences that occur in the appearance of RWs with reference to the lower-order and higher-order dispersions terms. In addition to the above, we examine the derived solutions in detail for certain system and elliptic modulus parameter values and highlight some interesting features that we obtain from our studies. [ABSTRACT FROM AUTHOR]

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton). [ABSTRACT FROM AUTHOR]

In this study, we present a new analytical solution for the time-dependent Schrödinger equation for a free particle in one-dimensional case. The solution is derived by doing a non-linear transform to the linear Schrödinger equation and converting it into a Burger-like equation. We obtained an interesting non-stationary wave function where our soliton solution moves in time while maintaining its shape. The new solution is then analysed for three different cases: a periodic box, a box with hard wall boundary conditions and a periodic array of Dirac delta potentials. The resulting analytical solutions exhibit several interesting features including quantized soliton velocity and velocity bands. The analytical soliton solution that has been proposed, in our opinion, makes an important contribution to the study of quantum mechanics and we believe it will contribute significantly to our understanding of how particles behave in one-dimensional box potentials. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical investigation of the fractal modification of the (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation. A variational approach based on the two-scale fractal complex transformation and the variational principle is presented for solving this fractal equation. The fractal potential-YTSF equation can be transformed as the original potential-YTSF equation by means of the fractal complex transformation. Some fractal soliton-type solutions and fractal periodic wave solutions are provided by using the variational principle proposed by He, which are not touched in the existing literature. Some remarks about the variational formulation and the wave solutions for the original potential-YTSF equation by Manafian et al. [East Asian J. Appl. Math. 10(3) (2020) 549–565] are also given. Numerical results of the fractal wave solutions with different fractal dimensions and amplitudes are presented to show the propagation behavior. [ABSTRACT FROM AUTHOR]

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3 D representation and a 2 D representation, as well as a 2 D contour plot. [ABSTRACT FROM AUTHOR]

We propose a nonlinear quantum walk model inspired in a photonic implementation in which the polarization state of the light field plays the role of the coin-qubit. In particular, we take profit of the nonlinear polarization rotation occurring in optical media with Kerr nonlinearity, which allows to implement a nonlinear coin operator, one that depends on the state of the coin-qubit. We consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us to gain some insight into the existence of different solitonic structures, such as bright and dark solitons. We illustrate several properties of these solitons with numerical calculations, including the effect on them of an additional phase simulating an external electric field. [ABSTRACT FROM AUTHOR]

We have obtained the first-order solution of a three-coupled nonlinear Schrödinger equation based on the modified Darboux transformation. In addition, we have derived an expression for the distance between the rogue wave and breather. We have find that the amplitude of the rogue wave, the period of the breather, the distance between the rogue wave and breather, the transformation between the bright and dark rogue wave, and the transformation between the soliton and breather which are all affected by the values of the free parameters. [ABSTRACT FROM AUTHOR]

The present work strives to explore some qualitative analysis for the governing equation describing the dynamic response of a piezoelectro-magnetic circular rod. As a result of the integrability study of the governed equation, which furnishes valuable insights into its structure, solutions, and applications in various fields, we apply the well-known Ablowitz–Ramani–Segur (ARS) algorithm to prove the non-integrability of the governed equation in a Painlevé sense. The qualitative theory for planar integrable systems is applied to study the bifurcation of the solutions based on the values of rod material properties. Some new solutions for the governing equation are presented and they are categorized into solitary and double periodic functions. We display a 3D representation of the solutions in addition to investigating the influence of wave velocity on the obtained solution for the particular material of the rod. [ABSTRACT FROM AUTHOR]

In this work, we review and extend a version of the old attempt made by Louis de Broglie for interpreting quantum mechanics in realistic terms, namely, the double solution. In this theory, quantum particles are localized waves, i.e., solitons, that are solutions of relativistic nonlinear field equations. The theory that we present here is the natural extension of this old work and relies on a strong time-symmetry requiring the presence of advanced and retarded waves converging on particles. Using this method, we are able to justify wave–particle duality and to explain the violations of Bell's inequalities. Moreover, the theory recovers the predictions of the pilot-wave theory of de Broglie and Bohm, often known as Bohmian mechanics. As a direct consequence, we reinterpret the nonlocal action-at-a-distance in the pilot-wave theory. In the double solution developed here, there is fundamentally no action-at-a-distance but the theory requires a form of superdeterminism driven by time-symmetry. [ABSTRACT FROM AUTHOR]

We propose the Hirota bilinearization of the Fokas–Lenells derivative nonlinear Schrödinger equation (FLE) with a non-vanishing background. In the proposed method, we have introduced an auxiliary function to transform the equation into bilinear form. The use of an auxiliary function makes the method simpler than the ones reported earlier. Using the proposed method we have obtained the dark (and bright) one soliton solution. We have also discussed the properties of the obtained soliton and mentioned the criteria upon which the nature of the soliton being dark or bright depends upon. Then we have obtained the dark (and bright) two soliton solution and discussed the respective properties and also through asymptotic analysis showed how the phase between the two individual solitons changes before and after interaction. Eventually we have proposed the scheme for obtaining N soliton solutions. The proposed method can be extended to other nonlinear equations where straightforward bilinearization is not feasible. Later, we have introduced a gauge transformation which transforms the spectral problem of FLE into a spectral problem for the Landau–Lifshitz (LL) spin system. Soliton act as an information carrier and LL system exhibits a variety of nonlinear structures so the study is worth doing. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive a class of soliton solutions of the Schamel–KdV (SKdV) equation employing Darboux transformation with the help of the Lax pair. The SKdV equation is reduced to a suitable form by means of an effective transformation so that the Lax pair of the said equation can be derived using the Ablowitz–Kaup–Newell–Segur (AKNS) method. For the first time, we apply Darboux transformation through the Lax pair so as to obtain an effective solution to the SKdV equation, and this allows us to investigate new classes of soliton solutions of the SKdV equation. These solutions provide some new wave features, W-shaped soliton, breather-type soliton, etc. Effects of the nonlinear and dispersion coefficients are demonstrated numerically through two-dimensional graphs. Finally, for a better understanding of the dynamics of the model, some key three-dimensional plots of the wave solutions are presented. [ABSTRACT FROM AUTHOR]

We consider axially symmetric solutions of the U (1) gauged Skyrme model supplemented with a Callan–Witten (CW) anomaly density term. The main properties of the solutions are studied, several specific features introduced by the presence of the CW term being identified. We find that the solitons possess a nonzero angular momentum proportional to the electric charge, which in addition to the usual Coulomb part, acquires an extra (topological) contribution from the CW term. Specifically, it is shown that the slope of mass/energy M v s. electric charge Qe and angular momentum J can be both positive and negative. Furthermore, it is shown that the gauged Skyrmion persists even when the quartic (Skyrme) kinetic term disappears. [ABSTRACT FROM AUTHOR]

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion. [ABSTRACT FROM AUTHOR]

In this study, using the Kudryashov R function method and the generalized Kudryashov method, the (1 + 1) -dimensional resonant nonlinear Schrödinger equation with dual-power law nonlinearity has been effectively examined for finding optical soliton solutions. The Kudryashov R function technique has numerous advantages that make symbolic computing considerably simpler, particularly while dealing with strongly dispersive nonlinear equations. The generalised Kudryashov method is noteworthy due to its capacity to address a wide range of complex nonlinear ordinary differential equations (NLODEs) observed in diverse engineering, scientific and mathematical fields. Newly generated nonlinear Schrödinger equation is demonstrated by the resonant nonlinear Schrödinger equation used to describe nonlinear optical phenomena. In order to achieve the goal, the governing model was first transformed into a NLODE, and then the solution sets and solution functions were derived based on the definitions of the suggested approaches. Solitons are localized wave forms that retain their shape and stability as they propagate across optical fiber. Optical solitons are characterized by their ability to maintain their shape and amplitude during propagation, even when encountering other solitons. Using the suggested approaches, the singular, dark, kink-antikink and bright soliton solutions from the governing equation have been extracted. Under the appropriate selection of parameter values, 3D, 2D and contour graphs are shown to illustrate the physical characteristics of the obtained results. All generated solutions are demonstrated to be analytically stable by the analysis of modulation instability, which also reveals the stability and movement of the waves. These solutions have extensive implications in the field of telecommunications and nonlinear fiber optics and assist in understanding the physical phenomena underlying the equation. These methods are new and standardised, and they can be applied to solve a variety of mathematical and physical problems. [ABSTRACT FROM AUTHOR]

We propose and demonstrate a passively mode-locked all-fiber laser utilizing a multimode interference effect in graded-index multimode fiber (GIMF). The mode-locking is obtained due to the Kerr effect in GIMF, which induces the intensity modulation inside the ring cavity laser. We obtain stable soliton mode-locked pulses at a center wavelength of 1574.1 nm, with a fixed repetition rate of 22 MHz and a pulse duration of 650 fs, when the pump power is changed in the range from 66.2 to 140.0 mW. The maximum average output power and pulse energy are recorded at 9.82 mW and 448 pJ, respectively, at a pump power of 140 mW. This proves that a nonlinear optical response in GIMF can be used as a new modulation mechanism for obtaining ultra-short pulses based on all-fiber ring cavity. [ABSTRACT FROM AUTHOR]

In the present study, we focus on obtaining the optical soliton solutions of the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) equation using Kudryhashov methods. First, a wave transformation is applied to the DSSH equation, resulting in a nonlinear ordinary differential equation (NLODE). A balance number is deduced through the balancing of this NLODE. Subsequently, candidate solutions, inclusive of their respective derivatives and the wave transformation, are inserted into the DSSH equation. This results in an equation in polynomial form. Terms with equal power are grouped together in the new equation, and their coefficients are set to zero, leading to a system of algebraic equations. Solving this algebraic system facilitates the identification of undetermined parameters within the candidate solutions, thus yielding the solutions for the DSSH equation. Visualization of these solutions is executed through contour plots as well as two- and three-dimensional graphical representations. The contributions of this study are of substantial relevance for subsequent research in nonlinear science. [ABSTRACT FROM AUTHOR]

Chip‐scale optical frequency combs have attracted significant research interest and can be used in applications ranging from precision spectroscopy to telecom channel generators and lidar systems. In the time domain, microresonator based frequency combs correspond to self‐stabilized soliton pulses. In two distinct regimes, microresonators are shown to emit either bright solitons in the anomalous dispersion regime or dark solitons (a short time of darkness in a bright background signal) in the normal dispersion regime. Here, the dynamics of continuous‐wave‐laser‐driven soliton generation is investigated in the zero‐group‐velocity‐dispersion regime as well as the generation of solitons that are spectrally crossing different dispersion regimes. In the measurements, zero‐dispersion solitons with multipeak structures (soliton molecules) are observed with distinct and predictable spectral envelopes that are a result of fifth‐order dispersion of the resonators. Numerical simulations and the analysis of bifurcation structures agree well with the observed soliton states. This is the first observation of soliton generation that is governed by fifth‐order dispersion, which can have applications in ultrafast optics, telecom systems, and optical spectroscopy. [ABSTRACT FROM AUTHOR]

This study focuses on analyzing a coupled space-time fractional nonlinear Schrodinger equation, which has applications in describing non-relativistic quantum mechanical behavior. The investigation covers various aspects, including the examination of dynamical behaviors and the exploration of optical soliton solutions. The modified F-expansion method is employed to derive these soliton solutions. To visualize and interpret the physical characteristics of the solutions, they are plotted in 2D, 3D, and density plots with appropriate parameter settings. The dynamical behaviors of the equation are discussed by investigating bifurcations at equilibrium points, and the chaotic behavior of the perturbed dynamical system is demonstrated using chaos theory. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab. These findings provide a dynamic and powerful mathematical tool to address a range of nonlinear wave phenomena. Key discoveries include the identification of new solitary wave forms, as well as bifurcation and chaotic solutions. These unique and intriguing solutions have theoretical significance in understanding energy transfer and diffusion processes in mathematical models. [ABSTRACT FROM AUTHOR]

With a vanishing boundary condition, we consider a revised Riemann–Hilbert problem (RHP) for the derivative nonlinear Schrödinger equation (DNLS), where an integral factor is introduced such that the RHP satisfies the normalization condition. In the reflectionless situation, we construct the formulas for the th-order solutions of the DNLS equation, including the solitons and positons that respectively correspond to pairs of simple poles and one pair of th-order poles of the RHP. According to the Cauchy–Binet formula, we show the expressions for th-order solitons. Additionally, we give an explicit expression for the second-order positon and graphically describe evolutions of the third-order and fourth-order positons. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the integrable Lakshmanan–Porsezian–Daniel (LPD) equation, which was proposed as a model for the nonlinear spin excitations in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin. Our main purpose was to construct soliton and breather solutions on the nonconstant background for the integrable local and nonlocal LPD equations. Firstly, the Bäcklund transformations are constructed based on the pseudopotential of equations. Secondly, starting from the nonconstant initial solution sech and applying the obtained transformation, various nonlinear wave solutions of the local LPD equation are provided, including the time-periodic breather, W-shaped soliton, M-type soliton and two-soliton solutions, the elastic interactions between the two-soliton solutions are shown and the relationship between parameters and wave structures is discussed. Thirdly, beginning with the nonconstant initial solutions sech and tanh , the time-periodic breather, bell-shaped one-soliton and anti-bell-shaped one-soliton solutions of the nonlocal LPD equation are generated and these solutions possess no singularity. What is more, the time-periodic breather solutions exhibit the x-periodic background and double-periodic background, which is different from the previous results. The corresponding dynamics of these solutions related to the integrable local and nonlocal LPD equations are illustrated graphically. The results in this paper might be helpful for us to understand the nonlinear characteristics of magnetic materials. [ABSTRACT FROM AUTHOR]

The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction. [ABSTRACT FROM AUTHOR]

In this paper, a five-order Korteweg–de Vries (KdV) equation is studied, which is used to describe the nonlinear phenomena in the fluids, especially those of the surface gravity waves and internal waves in the stratified fluids. (a) Via the symbolic calculation, this KdV equation cannot pass the Painlevé test without any constraint conditions. By virtue of the ansatz method, bell-shape and kink soliton solutions of this KdV equation are attained. (b) Via the bilinear method, multisoliton solutions of this KdV equation are obtained under some constraint conditions. Propagation and interaction of the multisoliton are discussed. Soliton interaction is elastic, that is to say, they have no effect on each other's amplitude and speed except for phase shift. We hope that our results will be useful for experimental studies of surface gravity waves and internal waves since the coefficients of this KdV equation are all expressed in terms of physical constants, depths, and densities of the fluid. [ABSTRACT FROM AUTHOR]

We report the study on pulse compression using air-core photonic band-gap fibre filled with air. The distortion of the output pulse is minimised using a distortion compensating fibre (DCF) for different input and output pulse energies at different values of atomic gas pressure. In the present work, 110 m long DCF was used and the pulses from a picosecond / femtosecond pulse were linearly stretched and then recompressed in 12 m of the air-core fibre. With the band-pass filter, the output pulses were as short as 1.2 ps. The peak power obtained was 100 W with the use of the filter. The output peak power was generated using a filter by simulation and the dispersion characterisation shows that with increasing order of the input pulse, dispersion also changes in the output pulse. The simulation is done by finite difference time domain method in a soliton pulse. [ABSTRACT FROM AUTHOR]

We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space H 1 × L 2. This model has solitary waves with speeds − 1 < c < 1. When | c | approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space. [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 C0 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 C0, and for asymmetric soliton solutions, the propagation constant k decreases nonlinearly with C0. The minimum point of k(C0) is just the demarkation between the symmetric soliton solutions and the asymmetric soliton solutions. If C0 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 results of a theoretical study on Benjamin–Ono (BO) soliton evolution are presented in a simple model of a two-layer ocean with a shear flow and viscosity. The upper layer is assumed to move with a constant speed relative to the lower layer with a tangential discontinuity in the flow profile. It is shown that in the long-wave approximation, such a model can be appropriate. If the flow is supercritical, i.e., its speed (U) exceeds the speed of long linear waves ( c 1 ), then BO solitons experience "explosive-type" enhancement due to viscosity, such that their amplitudes increase to infinity in a finite time. In the subcritical regime, when U < c 1 , BO solitons experience very slow decay due to viscosity. Soliton amplitude decays with time as A ∼ t − 1 / 2 or A ∼ t − 1 / 3 , depending on whether both layers are weakly viscous (the former case) or only the lower layer is viscous (the latter case). Estimates of "explosion time" are presented for real oceanic parameters. [ABSTRACT FROM AUTHOR]

The kinetic Alfvén solitons in a plasma with regularized kappa distributed electron are studied under the small‐amplitude assumption; the analytical solution of the kinetic Alfvén soliton is obtained using the reductive perturbation method. The numerical calculation reveals that the width and amplitude of the soliton decrease with the increase of the cut‐off parameter α$$ \alpha $$ for fixed spectral index κ$$ \kappa $$. However, the width and amplitude of the soliton increase for small spectral index and decrease for a larger spectral index in the case of the fixed cut‐off parameter. The properties of the soliton can explain the observed electrostatic structure observed in the Earth's inner magnetosphere. [ABSTRACT FROM AUTHOR]

In this work we develop a time-symmetric soliton theory for quantum particles inspired from works by de Broglie and Bohm. We consider explicitly a non-linear Klein–Gordon theory leading to monopolar oscillating solitons. We show that the theory is able to reproduce the main results of the pilot-wave interpretation for non interacting particles in a external electromagnetic field. In this regime, using the time symmetry of the theory, we are also able to explain quantum entanglement between several solitons and we reproduce the famous pilot-wave nonlocality associated with the de Broglie-Bohm theory. [ABSTRACT FROM AUTHOR]

An analytical work on intrinsic localized modes in a two-dimensional Heisenberg ferromagnet on the checkerboard lattice is presented. Taking advantage of an asymptotic method, the governing lattice dynamical equations are reduced to one (2 + 1)-dimensional nonlinear Schrödinger equation. In our work, we obtain two types of nonlinear localized mode solutions, namely, Brillouin zone center modes and Brillouin zone corner modes. The occurrence conditions for these intrinsic localized modes are given in detail. Especially, we find that the competition between the Dzialozinskii-Moriya interaction and the next-nearest neighbor interaction of the checkerboard ferromagnet has an effect on the local structure of the Brillouin zone corner acoustic mode. [ABSTRACT FROM AUTHOR]

Electrostatic nonlinear random Langmuir structures have been propagated in stochastic magnetospheres, clouds and solar wind. A theoretical description of Langmuir waves can be modeled by Schrödinger and Zakharov models with stochastic terms. It was explained that the stochastic parameter affects the forcing, collapsing in strongly density turbulence and density crystalline structures. The unified method has been implemented to provide new stochastic solutions for a Zakharov system in subsonic limit with noises via the Itô sense. This unified approach provides a variety of advantages, such as avoiding difficult calculations and explicitly providing pivotal solutions. It is easy to use, efficient, and precise. The induced generated energy during the collapsing of solar Langmuir wave bursts and clouds is determined by the solitonic formations. In addition, the collapsing strong turbulence or forcing density crystalline structures depend mainly on stochastic processes. Furthermore, electrostatic waves in clouds that may collapse are represented sometimes as dissipative shapes. So, the results of this investigation could be applicable to observations of energy seeding and collapsing in clouds. This energy is based on the electrostatic field and its related densities' perturbation in subsonic limits. Finally, it has been explored how noise parameters in the Itô sense affect the solar wind Langmuir waves' properties. So, the findings of this discussion may be applicable to real observations of energy collapsing and seeding in clouds. [ABSTRACT FROM AUTHOR]

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution's quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated. [ABSTRACT FROM AUTHOR]

The aim of this paper is to elucidate the existence of patterns for Keller–Segel-type models that are solutions of the traveling pulse form. The idea is to search for transport mechanisms that describe this type of waves with compact support, which we find in the so-called nonlinear diffusion through saturated flux mechanisms for the movement cell. At the same time, we analyze various transport operators for the chemoattractant. The techniques used combine the analysis of the phase diagram in dynamic systems together with its counterpart in the system of partial differential equations through the concept of entropic solution and the admissible jump conditions of the Rankine–Hugoniot type. We found traveling pulse waves of two types that correspond to those found experimentally. [ABSTRACT FROM AUTHOR]

A meshless finite point method (FPM) is developed in this paper for the numerical solution of the nonlinear improved Boussinesq equation. A time discrete technique is used to approximate time derivatives, and then a linearized procedure is presented to deal with the nonlinearity. To achieve stable convergence numerical results in space, the stabilized moving least squares approximation is used to obtain the shape function, and then the FPM is adopted to establish the linear system of discrete algebraic equations. To enhance the accuracy and convergence order in time, the Richardson extrapolation is finally incorporated into the FPM. Numerical results show that the FPM is fourth‐order accuracy in both space and time and can obtain highly accurate results in simulating the propagation of a single solitary wave, the interaction of two solitary waves, the solitary wave break‐up and the solution blow‐up phenomena. [ABSTRACT FROM AUTHOR]