Search Results (331)

Analytical solutions for the complex-valued nonlinear Gerdjikov–Ivanov (GI) equation have been studied extensively using integrability-based methods. In contrast, numerical and semi-analytical exploration remains relatively underdeveloped. Thus, the present study deploys both the traditional Adomian decomposition method (ADM) and its improved version (IADM) to explore the computational relevance of the GI equation to shock waves against a benchmark exact soliton solution. The findings indicate that both methods are effective in addressing the GI equation, with the improved method demonstrating an enhancement in the stability of the convergence under specific conditions. This work offers the first systematic semi-analytic and numerical evaluation of the GI equation, introducing practical implementation guidelines. 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

Traditional brain emulation approaches often rely on classical computational models that inadequately capture the stochastic, nonlinear, and potentially coherent features of biological neural systems. In this position paper, we introduce NeuroQ a quantum-inspired framework grounded in stochastic mechanics, particularly Nelson’s formulation. By reformulating the FitzHugh–Nagumo neuron model with structured noise, we derive a Schrödinger-like equation that encodes membrane dynamics in a quantum-like formalism. This formulation enables the use of quantum simulation strategies—including Hamiltonian encoding, variational eigensolvers, and continuous-variable models—for neural emulation. We outline a conceptual roadmap for implementing NeuroQ on near-term quantum platforms and discuss its broader implications for neuromorphic quantum hardware, artificial consciousness, and time-symmetric cognitive architectures. Rather than demonstrating a working prototype, this work aims to establish a coherent theoretical foundation for future research in quantum brain emulation. Full article

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 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

The chiral nonlinear Schrödinger equation (CNLSE) serves as a simplified model for characterizing edge states in the fractional quantum Hall effect. In this paper, we leverage the generalization and parameter inversion capabilities of physics-informed neural networks (PINNs) to investigate both forward and inverse problems of 1D and 2D CNLSEs. Specifically, a hybrid optimization strategy incorporating exponential learning rate decay is proposed to reconstruct data-driven solutions, including bright soliton for the 1D case and bright, dark soliton as well as periodic solutions for the 2D case. Moreover, we conduct a comprehensive discussion on varying parameter configurations derived from the equations and their corresponding solutions to evaluate the adaptability of the PINNs framework. The effects of residual points, network architectures, and weight settings are additionally examined. For the inverse problems, the coefficients of 1D and 2D CNLSEs are successfully identified using soliton solution data, and several factors that can impact the robustness of the proposed model, such as noise interference, time range, and observation moment are explored as well. Numerical experiments highlight the remarkable efficacy of PINNs in solution reconstruction and coefficient identification while revealing that observational noise exerts a more pronounced influence on accuracy compared to boundary perturbations. Our research offers new insights into simulating dynamics and discovering parameters of nonlinear chiral systems with deep learning. Full article

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, while showing nonexistence in the $L^2$-critical case. For $L^2$-supercritical exponents, we identify parameter regimes where ground states exist, characterized by a negative Lagrange multiplier. The analysis combines variational methods, scaling techniques, and the careful study of fibering maps to address challenges posed by competing nonlinearities and nonlocal interactions. Full article

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. 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

We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the set of all stable perturbations of the system, known as the ‘versal unfolding’. This construction yields a comprehensive classification of qualitatively distinct solutions and their metamorphoses into new topological forms, parametrized by the codimension of the bifurcation in each case. We illustrate these ideas through bifurcations in the simplest Friedmann models, the Oppenheimer-Snyder black hole, the evolution of causal geodesic congruences in cosmology and black hole spacetimes, crease flow on event horizons, and the Friedmann–Lemaître equations. Finally, we list open problems and briefly discuss emerging aspects such as partial differential equation stability of versal families, the general relativity landscape, and potential connections between gravitational versal unfoldings and those of the Maxwell, Dirac, and Schrödinger equations. Full article

In this paper we discuss some results on the existence of solutions for an elliptic system appearing in physical sciences. In particular the system appears when we look at standing wave solutions in the electrostatic situation for the Schrödinger equation coupled, with the minimal coupling rule, with the electromagnetic equations of Born–Infeld theory. Many difficulties appear, especially due to the fact we are in an unbounded domain (the whole space ${\mathbb{R}}^3$) and to the intrinsic nonlinear nature of the equations. We are able to prove the existence of a minimal energy solution by showing an approximating procedure that can be adapted depending on the value of the parameter p, which is in the nonlinearity. Full article

This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities. 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

External fields modify the confinement potential and electronic structure in a multiple quantum well system, affecting the light–matter interaction. Here, we present a theoretical study of the modulation of the nonlinear optical response simultaneously employing an intense non-resonant laser field and an electric field. Considering four occupied subbands, we focus on a GaAs/AlGaAs symmetric multiple quantum well system with five wells and six barriers. By solving the Schrödinger equation through the finite element method under the effective mass approximation, we determine the electronic structure and the nonlinear optical response using the density matrix formalism. The laser field dresses the confinement potential while the electric field breaks the inversion symmetry. The combined effect of both fields modifies the intersubband transition energies and the overlap of the wave functions. The results obtained demonstrate an active tunability of the nonlinear optical response, opening up the possibility of designing optoelectronic devices with tunable optical properties. 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