Search Results (41)

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering. Full article

This paper develops accurate three-dimensional trajectory tracking and anti-sway control strategies for offshore container cranes operating in an open-sea environment. A 5-DOF nonlinear dynamic model is developed that simultaneously accounts for the crane’s structural motion, trolley movement, spreader hoisting with variable rope length, and both lateral and longitudinal payload sway. The model further incorporates external disturbances induced by wave-excited ship motions. To ensure smooth, efficient, and accurate load transportation from the initial to the target position, an effective trajectory-planning scheme is proposed using a quintic polynomial trajectory refined by a ZVD shaper to suppress residual oscillations. A sliding mode control method is then designed to achieve accurate trajectory tracking and load-sway suppression under external disturbances. Numerical simulations demonstrate that the proposed trajectory planning method effectively reduces the residual oscillations and verifies the effectiveness and robustness of the proposed sliding mode control strategy. Full article

In this study, we investigate a nonlinear Schrödinger equation relevant to the evolution of optical beams in weakly nonlocal media. Utilizing the modified F-expansion method, we construct a variety of novel soliton solutions, including dark, bright, and wave solitons. These solutions are illustrated through comprehensive graphical simulations, including 2D contour plots and 3D surface profiles, to highlight their structural dynamics and propagation behavior. The effects of the temporal parameter on soliton formation and evolution are thoroughly analyzed, demonstrating its role in modulating soliton shape and stability. To further explore the system’s dynamics, chaos and sensitivity theories are employed, revealing the presence of complex chaotic behavior under perturbations. The outcomes underscore the versatility and richness of the present model in describing nonlinear wave phenomena. This work contributes to the theoretical understanding of soliton dynamics in weakly nonlocal nonlinear optical systems and supports advancements in photonic technologies. This study reports a novel soliton structure for the weak nonlocal cubic–quantic NLSE and also details the comprehensive chaotic and sensitivity analysis that represents the unexplored dynamical behavior of the model. This study further demonstrates how the underlying nonlinear structures, along with the novel solitons and chaotic dynamics, reflect key symmetry properties of the weakly nonlocal cubic–quintic Schrödinger model. These results enhanced the theoretical framework of the nonlocal nonlinear optics and offer potential implications in photonic waveguides, pulse shape, and optical communication systems. Full article

In a variety of applications, including signal processing, clock referencing, sensing, and others, microelectromechanical systems (MEMS) have been shown to be effective and broadly used. This study explores the dynamical response of a nonlinear MEMS resonator when subjected to a sudden mechanical shock under electrical excitation in the presence of quintic nonlinearity. The method of multiple scales (MMS) is utilized to construct the analytical formulas for analyzing the amplitude and phase response during primary resonance conditions. The analytical results are verified and compared with numerical simulations performed using the fourth-order Runge–Kutta method. Additionally, a parametric analysis is performed to examine the effect of different shock values on the resonator’s response and stability utilizing the Jacobian matrix. The agreement between analytical and numerical approaches proves MMS’s effectiveness in analyzing the shock impact on the MEMS resonator. The results provide valuable knowledge about the response and stability of MEMS resonators under mechanical shock, which is crucial for robust design in challenging conditions. Full article

The geometric content of chaos in nonlinear systems with multiple stabilities of high order is a challenge to computation. We introduce a single algorithmic framework to overcome this difficulty in the present study, where a parametrically forced oscillator with cubic–quintic nonlinearities is considered as an example. The framework starts with the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which is a self-learned algorithm that extracts an interpretable and correct model by simply analyzing time-series data. The resulting parsimonious model is well-validated, and besides being highly predictive, it also offers a solid base on which one can conduct further investigations. Based on this tested paradigm, we propose a unified diagnostic pathway that includes bifurcation analysis, computation of the Lyapunov exponent, power spectral analysis, and recurrence mapping to formally describe the dynamical features of the system. The main characteristic of the framework is an effective algorithm of computational basin analysis, which is able to display attractor basins and expose the fine scale riddled structures and fractal structures that are the indicators of extreme sensitivity to initial conditions. The primary contribution of this work is a comprehensive dynamical analysis of the DM-CQDO, revealing the intricate structure of its stability landscape and multi-stability. This integrated workflow identifies the period-doubling cascade as the primary route to chaos and quantifies the stabilizing effects of key system parameters. This study demonstrates a systematic methodology for applying a combination of data-driven discovery and classical analysis to investigate the complex dynamics of parametrically forced, high-order nonlinear systems. Full article

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, and self-focusing, arising from the balance between cubic and quintic nonlinearities. By employing the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method, we derive an extensive catalog of analytic solutions, multi-peakon structures, lump solitons, kinks, and bright and dark solitary waves, while periodic and singular solutions emerge as special cases. These outcomes are systematically constructed within a single framework and visualized through 2D, 3D, and contour plots under both anomalous and normal dispersion regimes. The analysis also addresses modulation instability (MI), interpreted as a sideband amplification of continuous-wave backgrounds that generates pulse trains and breather-type structures. Our results demonstrate that cubic–quintic contributions substantially affect MI gain spectrum, broadening instability bands and permitting MI beyond the anomalous-dispersion regime. These findings directly connect the obtained solution classes to experimentally observed routes for solitary wave shaping, pulse propagation, and instability and instability-driven waveform formation in optical communication devices, photonic platforms, and laser technologies. Full article

This study employs the improved modified extended tanh method (IMETM) to derive exact analytical solutions of a higher-order nonlinear Schrödinger (HNLS) model, incorporating $\beta$-fractional derivatives in both time and space. Unlike classical methods such as the inverse scattering transform or Hirota’s bilinear technique, which are typically limited to integrable systems and integer-order operators, the IMETM offers enhanced flexibility for handling fractional models and higher-order nonlinearities. It enables the systematic construction of diverse solution types—including Weierstrass elliptic, exponential, Jacobi elliptic, and bright solitons—within a unified algebraic framework. The inclusion of fractional derivatives introduces richer dynamical behavior, capturing nonlocal dispersion and temporal memory effects. Visual simulations illustrate how fractional parameters $\alpha$ (space) and $\beta$ (time) affect wave structures, revealing their impact on solution shape and stability. The proposed framework provides new insights into fractional NLS dynamics with potential applications in optical fiber communications, nonlinear optics, and related physical systems. Full article

This study focuses on the analysis of soliton solutions within the framework of the time-fractional cubic–quintic nonlinear Schrödinger equation (TFCQ-NLSE), a powerful model with broad applications in complex physical phenomena such as fiber optic communications, nonlinear optics, optical signal processing, and laser–tissue interactions in medical science. The nonlinear effects exhibited by the model—such as self-focusing, self-phase modulation, and wave mixing—are influenced by the combined impact of the cubic and quintic nonlinear terms. To explore the dynamics of this model, we apply a robust analytical technique known as the sub-ODE method, which reveals a diverse range of soliton structures and offers deep insight into laser pulse interactions. The investigation yields a rich set of explicit soliton solutions, including hyperbolic, rational, singular, bright, Jacobian elliptic, Weierstrass elliptic, and periodic solutions. These waveforms have significant real-world relevance: bright solitons are employed in fiber optic communications for distortion-free long-distance data transmission, while both bright and dark solitons are used in nonlinear optics to study light behavior in media with intensity-dependent refractive indices. Solitons also contribute to advancements in quantum technologies, precision measurement, and fiber laser systems, where hyperbolic and periodic solitons facilitate stable, high-intensity pulse generation. Additionally, in nonlinear acoustics, solitons describe wave propagation in media where amplitude influences wave speed. Overall, this work highlights the theoretical depth and practical utility of soliton dynamics in fractional nonlinear systems. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. 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

This study investigates a nonlinear Schrödinger equation that includes Kudryashov’s quintic power-law refractive index along with dual-form nonlocal nonlinearity. Employing dynamical systems theory, we analyze the model through a traveling-wave transformation, reducing it to a singular yet integrable traveling-wave system. The dynamical behavior of the corresponding regular system is examined, revealing phase trajectories bifurcations under varying parameter conditions. Furthermore, explicit solutions—including periodic, homoclinic, and heteroclinic solutions—are derived for distinct parameter regimes. Full article

This paper investigates the bifurcation dynamics and optical soliton solutions of the integrable quintic Kundu–Eckhaus (QKE) equation with an M-fractional derivative. By adding quintic nonlinearity and higher-order dispersion, this model expands on the nonlinear Schrödinger equation, which makes it especially applicable in explaining the propagation of high-power optical waves in fiber optics. To comprehend the behavior of the connected dynamical system, we categorize its equilibrium points, determine and analyze its Hamiltonian structure, and look at phase diagrams. Moreover, integrating along periodic trajectories yields soliton solutions. We achieve this by using the simplest equation approach and the modified extended Tanh method, which allow for a thorough investigation of soliton structures in the fractional QKE model. The model provides useful implications for reducing internet traffic congestion by including fractional temporal dynamics, which enables directed flow control to avoid bottlenecks. Periodic breather waves, bright and dark kinky periodic waves, periodic lump solitons, brilliant-dark double periodic waves, and multi-kink-shaped waves are among the several soliton solutions that are revealed by the analysis. The establishment of crucial parameter restrictions for soliton existence further demonstrates the usefulness of these solutions in optimizing optical communication systems. The theoretical results are confirmed by numerical simulations, highlighting their importance for practical uses. Full article

To achieve time–energy–impact multi-objective optimization in the trajectory control of underwater manipulators, this paper proposes a Fast Non-Dominated Sorting Tuna Swarm Optimization algorithm (FNS-TSO). The algorithm integrates a fast non-dominated sorting mechanism into the Tuna Swarm Optimization algorithm, improves initialization through Optimal Latin Hypercubic Sampling (OLHS) to enhance population distribution uniformity, and incorporates a nonlinear dynamic weight to refine the spiral foraging strategy, thereby strengthening algorithmic robustness. To verify FNS-TSO’s effectiveness, we conducted comparative evaluations using standard test functions against three established algorithms: Multi-Objective Particle Swarm Optimization (MOPSO), Multi-Objective Jellyfish Search Optimization (MOJSO), and the Non-Dominated-Sorting Genetic Algorithm (NSGA-II). Results demonstrate superior overall performance, particularly regarding convergence speed and solution diversity, with solution set distributions showing enhanced uniformity. In practical implementation, we applied FNS-TSO to the multi-objective optimization of an underwater manipulator using quintic spline curves for trajectory planning. Simulation outcomes reveal respective reductions of 11.03% in total operation time, 19.02% in energy consumption, and 24.69% in mechanical impacts, with the optimized manipulator achieving stable point-to-point motion transitions. Full article

The aim of this work is to model the nonlinear dynamics of conservative oscillators, with restoring force originating from even-order potentials. In particular, we extend our previous findings on inverting the time-integral equation that arises in the solution of such dynamical systems, a task that is almost always intractable in exact form. This is faced and solved by approximating the restoring force with its Chebyshev series truncated to order five; such a quintication approach yields a quinticate oscillator, whose associated time-integral can be inverted in closed form. Our solution procedure is based on the quinticate oscillator coefficients, upon which a second-order polynomial is constructed, which appears in the time-integrand of the quinticate problem, and whose roots determine the expression of the closed-form solution, as well as that of its period. The presented algorithm is implemented in the Mathematica software and validated on some conservative nonlinear oscillators taken from the relevant literature. Full article

The goal of this article is to reduce the vibration of a hybrid oscillator with a cubic–quintic nonlinear term under internal and external forces in the worst resonance case. To eliminate the harmful vibration in the system, the following strategies are suggested: nonlinear derivative feedback control (NDF), linear negative velocity feedback control (LNVC), nonlinear integral positive position feedback (NIPPF), integral resonant control (IRC), negative velocity with time delay (TD), and positive position feedback (PPF). It is discovered that the PPF control suppresses vibration more effectively than typical controllers, which reduces the vibration to 0.0001 with an effectiveness of 99.92%. Moreover, the main advantages of the PPF controller are its low cost and the fast response. The multiple time scale perturbation technique (MSPT) is used to apply the theoretical methodology and obtain a perturbed response. In order to close the concurrent primary and internal resonance case, frequency response (FR) equations are also used to study and analyze the system’s stability. The MATLAB software is used to complete and clarify all numerical topics. The FR curves are examined to determine the amplitude’s subsequent impact from variations in the parameters’ values. Lastly, a comparison of the numerical and analytical solutions is performed utilizing time history. Along with comparing the impact of our PPF damper on the hybrid oscillator, earlier research is also provided. Full article