Search Results (502)

Optical solitons have emerged as a highly active research domain in nonlinear fiber optics, driving significant advancements and enabling a wide range of practical applications. This study investigates the dynamics of dark solitons in systems governed by the resonant nonlinear Schrödinger equation (RNLSE). For the RNLSE with third-order (3OD) and fourth-order (4OD) dispersions, the dark soliton solution of the equation in the (1+1)-dimensional case is derived using the F-expansion method, and the analytical study is extended to the (2+1)-dimensional case via the self-similar method. Subsequently, the nonlinear equation incorporating perturbation terms is further studied, with particular attention given to the dark soliton solutions in both one and two dimensions. The soliton dynamics are illustrated through graphical representations to elucidate their propagation characteristics. Finally, modulation instability analysis is conducted to evaluate the stability of the nonlinear system. These theoretical findings provide a solid foundation for experimental investigations of dark solitons within the systems governed by the RNLSE model. Full article

This study focuses on investigating the asymptotic and oscillatory behavior of a new class of fourth-order nonlinear neutral differential equations. This research aims to achieve a qualitative advancement in the analysis and understanding of the relationships between the corresponding function and its derivatives. By utilizing various techniques, innovative criteria have been developed to ensure the oscillation of all solutions of the studied equations without resorting to additional constraints. Effective analytical tools are provided, contributing to a deeper theoretical understanding and expanding their application scope. The paper concludes by presenting examples that illustrate the practical impact of the results, highlighting the theoretical value of the research in the field of functional differential equations. Full article

In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation $(a$$\varphi \left(u\right)$$u^{‴})$+q$(u\circ h)=0$, which is of a functional type with a delay. The oscillation behavior of solutions of fourth-order delay equations has been studied using many techniques, but previous results did not take into account the existence of the function $\varphi$ except in second-order studies. The existence of $\varphi$ increases the difficulty of obtaining monotonic and asymptotic properties of the solutions and also increases the possibility of applying the results to a larger area of special cases. We present two criteria to ensure the oscillation of the solutions of the studied equation for two different cases of $\varphi$. Our approach is based on using the comparison principle with equations of the first or second order to benefit from recent developments in studying the oscillation of these orders. We also provide several examples and compare our results with previous ones to illustrate the novelty and effectiveness. Full article

This paper investigates the initial-boundary value problem for a fourth-order pseudo-parabolic equation with a nonlocal source: $u_t+{\Delta }^{2}u-\Delta u_t={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$. By employing the Galerkin method, the potential well method, and the construction of an energy functional, we establish threshold conditions for both the global existence and finite-time blow-up of solutions. Additionally, under the assumption of low initial energy $J\left(u_0\right)<d$, an upper bound for the blow-up time is derived. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This paper presents some of the most significant findings in the design of a hydraulic servomechanism for flight controls, which were primarily achieved by the first author during his activity in an aviation institute. These results are grouped into four main topics. The first one outlines a classical theory, from the 1950s–1970s, of the analysis of nonlinear automatic systems and namely the issue of absolute stability. The uninformed public may be misled by the adjective “absolute”. This is not a “maximalist” solution of stability but rather highlights in the system of equations a nonlinear function that describes, for the case of hydraulic servomechanisms, the flow-control dependence in the distributor spool. This function is odd, and it is therefore located in quadrants 1 and 3. The decision regarding stability is made within the so-called Lurie problem and is materialized by a matrix inequality, called the Lefschetz condition, which must be satisfied by the parameters of the electrohydraulic servomechanism and also by the components of the control feedback vector. Another approach starts from a classical theorem of V. M. Popov, extended in a stochastic framework by T. Morozan and I. Ursu, which ends with the description of the local and global spool valve flow-control characteristics that ensure stability in the large with respect to bounded perturbations for the mechano-hydraulic servomechanism. We add that a conjecture regarding the more pronounced flexibility of mathematical models in relation to mathematical instruments (theories) was used. Furthermore, the second topic concerns, the importance of the impedance characteristic of the mechano-hydraulic servomechanism in preventing flutter of the flight controls is emphasized. Impedance, also called dynamic stiffness, is defined as the ratio, in a dynamic regime, between the output exerted force (at the actuator rod of the servomechanism) and the displacement induced by this force under the assumption of a blocked input. It is demonstrated in the paper that there are two forms of the impedance function: one that favors the appearance of flutter and another that allows for flutter damping. It is interesting to note that these theoretical considerations were established in the institute’s reports some time before their introduction in the Aviation Regulation AvP.970. However, it was precisely the absence of the impedance criterion in the regulation at the appropriate time that ultimately led, by chance or not, to a disaster: the crash of a prototype due to tailplane flutter. A third topic shows how an important problem in the theory of automatic systems of the 1970s–1980s, namely the robust synthesis of the servomechanism, is formulated, applied and solved in the case of an electrohydraulic servomechanism. In general, the solution of a robust servomechanism problem consists of two distinct components: a servo-compensator, in fact an internal model of the exogenous dynamics, and a stabilizing compensator. These components are adapted in the case of an electrohydraulic servomechanism. In addition to the classical case mentioned above, a synthesis problem of an anti-windup (anti-saturation) compensator is formulated and solved. The fourth topic, and the last one presented in detail, is the synthesis of a fuzzy supervised neurocontrol (FSNC) for the position tracking of an electrohydraulic servomechanism, with experimental validation, in the laboratory, of this control law. The neurocontrol module is designed using a single-layered perceptron architecture. Neurocontrol is in principle optimal, but it is not free from saturation. To this end, in order to counteract saturation, a Mamdani-type fuzzy logic was developed, which takes control when neurocontrol has saturated. It returns to neurocontrol when it returns to normal, respectively, when saturation is eliminated. What distinguishes this FSNC law is its simplicity and efficiency and especially the fact that against quite a few opponents in the field, it still works very well on quite complicated physical systems. Finally, a brief section reviews some recent works by the authors, in which current approaches to hydraulic servomechanisms are presented: the backstepping control synthesis technique, input delay treated with Lyapunov–Krasovskii functionals, and critical stability treated with Lyapunov–Malkin theory. Full article

Icing on transmission lines in cold regions can cause asymmetry in the conductor cross-section. This asymmetry can lead to low-frequency, large-amplitude oscillations, posing a serious threat to the stability and safety of power transmission systems. In this study, the aerodynamic characteristics of crescent-shaped and sector-shaped iced eight-bundled conductors were systematically investigated over an angle of attack range from 0° to 180°. A combined approach involving wind tunnel tests and high-precision computational fluid dynamics (CFD) simulations was adopted. In the wind tunnel tests, static aerodynamic coefficients and dynamic time series data were obtained using a high-precision aerodynamic balance and a turbulence grid. In the CFD simulations, transient flow structures and vortex shedding mechanisms were analyzed based on the Reynolds-averaged Navier–Stokes (RANS) equations with the SST k-ω turbulence model. A comprehensive comparison between the two ice accretion geometries was conducted. The results revealed distinct aerodynamic instability mechanisms and frequency-domain characteristics. The analysis was supported by Fourier’s fourth-order harmonic decomposition and energy spectrum analysis. It was found that crescent-shaped ice, due to its streamlined leading edge, induced a dominant single vortex shedding. In this case, the first-order harmonic accounted for 67.7% of the total energy. In contrast, the prismatic shape of sector-shaped ice caused migration of the separation point and introduced broadband energy input. Stability thresholds were determined using the Den Hartog criterion. Sector-shaped iced conductors exhibited significant negative aerodynamic damping under ten distinct operating conditions. Compared to the crescent-shaped case, the instability risk range increased by 60%. The strong agreement between simulation and experimental results validated the reliability of the numerical approach. This study establishes a multiscale analytical framework for understanding galloping mechanisms of iced conductors. It also identifies early warning indicators in the frequency domain and provides essential guidance for the design of more effective anti-galloping control strategies in resilient power transmission systems. Full article

In this paper a new generalized fractal equation for studying the behaviour of self-similar beams using the Timoshenko beam theory is introduced. This equation is established in fractal dimensions by applying the concept of fractal continuum calculus $F^{\alpha }$-CC introduced recently by Balankin and Elizarraraz in order to study engineering phenomena in complex bodies. Ultimately, the achieved formulation is a fourth-order fractal single equation generated by superposing a shear deformation on an Euler–Bernoulli beam. A mapping of the Timoshenko principle onto self-similar beams in the integer space into a corresponding principle for fractal continuum space is formulated employing local fractional differential operators. Consequently, the single equation that describes the stress/strain of a fractal Timoshenko beam is solved, which is simple, exact, and algorithmic as an alternative description of the fractal bending of beams. Therefore, the elastic curve function and rotation function can be described. Illustrative examples of classical beams are presented and show both the benefits and the efficiency of the suggested model. Full article

This paper is devoted to a class of general nonlocal fourth-order elliptic equation with concave–convex nonlinearities. First, using the $Z_2$-mountain pass theorem in critical point theory, we obtain the existence of infinitely many large energy solutions. Then, using the dual fountain theorem, we prove that the equation has infinitely many negative energy solutions, whose energy converges at 0. Our results extend and complement existing findings in the literature. Full article

In this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserves both mass and energy through provable conservation properties. Under suitable assumptions on the exact solution, we establish upper and lower bounds for the numerical solution in the infinity norm, and further prove that the errors are fourth-order accurate in space and second-order in time in both the 2-norm and infinity norm. A detailed description of the nonlinear system solver at each time step is provided. We validate the proposed method through numerical experiments that demonstrate its efficiency, including fourth-order convergence (when sufficiently small time steps are used) and machine-level accuracy in the relative errors of mass and energy. Full article

In this work, using the weight function technique, we introduce a new family of fourth-order iterative methods optimal in the sense of Kung and Traub for scalar equations, generalizing Jarratt’s method. Through Taylor series expansions, we confirm that all members of this family achieve fourth-order convergence when derivatives up to the fourth order are bounded. Additionally, a stability analysis is performed on quadratic polynomials using complex discrete dynamics, enabling differentiation among the methods based on their stability. To demonstrate practical applicability, a numerical example illustrates the effectiveness of the proposed family. Extending our findings to Banach spaces, we conduct local convergence analyses on a specific subfamily containing Jarratt’s method, requiring only boundedness of the first derivative. This significantly broadens the method’s applicability to more general spaces and reduces constraints on higher-order derivatives. Finally, additional examples validate the existence and uniqueness of approximate solutions in Banach spaces, provided the initial estimate lies within the locally determined convergence radius obtained using majorizing functions. Full article

Implementing a computationally efficient numerical model for a single streamer discharge is essential to understand the complex processes such as lightning initiation and electrical discharges in high voltage systems. In this paper, we present a streamer discharge simulation in air, by solving one-dimensional (1D) drift diffusion reaction (DDR) equations for charged species with the disc approximation for electric field. A recently developed fourth-order space and time-centered implicit finite difference method (FDM) with a flux-corrected transport (FCT) method is applied to solve the DDR equations, followed by a comparative simulation using the well-established explicit FDM with FCT. The results demonstrate good agreement between implicit and explicit FDMs, verifying their reliability for streamer modeling. The total electrons, total charge, streamer position, and hence the streamer bridging time obtained using the FDMs with FCT agree with the same streamer computed in the literature using different numerical methods and dimensions. The electric field is obtained with good accuracy due to the inclusion of image charges representing the electrodes in the disc method. This accuracy can be further improved by introducing more image charges. Both implicit and explicit FDMs effectively capture the key streamer behavior, including the variations in charged particle densities and electric field. However, the implicit FDM is computationally more efficient. Full article

In this article, fourth-order systems of ordinary differential equations are studied. These systems are of a special form, which is used in modeling gene regulatory networks. The nonlinear part depends on the regulatory matrix W, which describes the interrelation between network elements. The behavior of solutions heavily depends on this matrix and other parameters. We research the evolution of trajectories. Two approaches are employed for this. The first approach combines a fourth-order system of two two-dimensional systems and then introduces specific perturbations. This results in a system with periodic attractors that may exhibit sensitive dependence on initial conditions. The second approach involves extending a previously identified system with chaotic solution behavior to a fourth-order system. By skillfully scanning multiple parameters, this method can produce four-dimensional chaotic systems. Full article

An analytical study of the nonlinear response of imperfect stiffened doubly curved shells made of functionally graded material (FGM) is presented. The formulation of the problem is based on the first-order shear deformation shell theory in conjunction with the von Kármán geometrical nonlinear strain–displacement relationships. The nonlinear equations of the motion of stiffened double-curved shells based on the extended Sanders’s theory were derived using Galerkin’s method. The material properties vary in the direction of thickness according to the linear rule of mixture. The effect of both longitudinal and transverse stiffeners was considered using Lekhnitsky’s technique. The fundamental frequencies of the stiffened shell are compared with the FE solutions obtained by using the ABAQUS 6.14 software. A stepwise approximation technique is applied to model the functionally graded shell. The resulting nonlinear ordinary differential equations were solved numerically by using the fourth-order Runge–Kutta method. Closed-form solutions for nonlinear frequency–amplitude responses were obtained using He’s energy method. The effect of power index, functionally graded stiffeners, geometrical parameters, and initial imperfection on the nonlinear response of the stiffened shell are considered and discussed. Full article

In this study, we discuss the semilocal convergence analysis of a fourth-order iterative method in Banach spaces. We assume the Fréchet derivative satisfies the Lipschitz continuity condition, obtains suitable recurrence relations, and determines the domain of convergence under appropriate initial estimates. In addition, the uniqueness domain for the solution and the error bounds are obtained. Next, several numerical examples, one which includes a Chandrasekhar integral equation, are carried out to apply the theoretical findings for semilocal convergence. Then, a final overview is provided. Full article