Search Results (375)

This paper studies a nonlinear Schrödinger equation with a variable-order time-fractional derivative. Because classical L1 and L1-2 schemes are not directly applicable to variable-order fractional operators, an improved L1-2 discretization with dynamically updated convolution weights is developed based on the Coimbra-type definition, in which the fractional order is evaluated at the current time. By combining the proposed temporal approximation with the Galerkin finite element method for spatial discretization and a linearized extrapolation technique for the nonlinear terms, a fully discrete numerical scheme is constructed. The unconditional stability of the scheme is rigorously proven, and optimal error estimates are established under a mild time step restriction. Numerical experiments are presented to confirm the theoretical results and to demonstrate the effectiveness of the method in capturing the influence of time-dependent memory effects on wave propagation. A key numerical observation is that stronger memory effects may suppress wave packet evolution, which is qualitatively reminiscent of a Zeno-like inhibition phenomenon. Full article

We report on the generation and spectral shaping of ultrabroadband terahertz-to-infrared radiation (>119 THz) from air plasma excited by a conventional tightly focused femtosecond Ti:Sa laser pulse with a duration of 35 fs assisted by its second harmonic (SH). A controllable and large frequency detuning between the SH and blueshifted component of the fundamental spectrum was achieved by utilizing spectral broadening of the fundamental pulse under filamentation and adjusting the longitudinal separation of the two cascaded filaments. For convenience, the resulting ultrabroadband emission is divided into a low-frequency part (<30 THz), an intermediate-frequency part (~50 THz), and a high-frequency part (~100 THz) that can be optimized with the filaments’ longitudinal separation. We attribute such ultrabroadband THz radiation generation to the excitation of photocurrent from the nonlinear interaction of SH with both the field at the fundamental frequency and its blueshifted component acquired during filamentation. Theoretical calculations based on time-dependent Schrödinger equation, as well as the Maxwell–Schrödinger equation for spectral broadening dynamics, reproduced the spectral features as well as the distinct dependence of the low- and high-frequency THz components. Full article

This paper introduces the Quantum-Inspired Impulsive Continuous Hopfield Network (Q-ICHN), a novel hybrid control framework designed to handle non-smooth, high-energy perturbations in nonlinear dynamical systems. Standard Continuous Hopfield Networks (CHNs) rely on sigmoidal activation functions that are prone to gradient saturation, which leads to an insufficient corrective response when the system undergoes large deviations from equilibrium. To overcome this shortcoming, the proposed Q-ICHN adopts a wave-packet-based activation function grounded in the stationary Schrödinger equation, yielding a non-monotonic and oscillatory activation profile that sustains effective compensatory dynamics across a broad range of states. Furthermore, the proposed framework incorporates Madelung’s quantum potential into the control architecture, thereby enabling a fundamental reshaping of the system’s energy landscape. Specifically, this induces a tunneling-like mechanism that allows the system to circumvent local minima and rapidly recover from impulsive disturbances, manifested as a sharpened attractor structure in the phase-space domain. Together, these properties yield enhanced convergence behavior and improved robustness over traditional neural control approaches. To rigorously assess its merits, the performance of the Q-ICHN is evaluated through a large-scale benchmark involving 20 established control methods, including Sliding Mode Control (SMC), Model Predictive Control (MPC), and Backstepping. The experimental results obtained across 20 heterogeneous scenarios demonstrate that the proposed model achieves a 48% reduction in Mean Squared Error (MSE) relative to the classical ICHN. In addition, the Q-ICHN exhibits improved smoothness, reflected in a 30% reduction in jerk with respect to high-gain robust controllers, and enhanced reliability, validated by superior spectral purity and a 34% reduction in integrated variance under stochastic perturbations. Collectively, these results underscore the potential of quantum-inspired activation mechanisms to favorably balance control responsiveness and harmonic stability, providing a robust framework for handling both continuous dynamics and impulsive effects. Full article

This study presents a systematic investigation of nonlinear wave interactions in a (2+1)-dimensional nonlinear Schrödinger equation with a space–time-symmetric potential. We focus on the interaction dynamics of high-order line-soliton solutions and on the anomalous scattering phenomena exhibited by high-order lump solutions, which correspond to fully localized spatiotemporal optical wave packets. Using the generalized Darboux transformation, we obtain, for the first time, explicit high-order line-soliton solutions for this model. A rigorous asymptotic analysis framework is developed to characterize the behavior of these solutions on both long and short time scales. Furthermore, high-order lump solutions are constructed, and their decomposition and anomalous scattering properties are elucidated. This work provides new insights into complex wave dynamics in higher-dimensional integrable systems and their implications for multidimensional beam propagation in nonlinear optical media. Full article

This article proposes a novel time-space two-grid high-order compact difference scheme for the one-dimensional nonlinear Schrödinger equation subject to Dirichlet boundary conditions. In comparison with the fully nonlinear compact difference scheme, the proposed methodology combines a small-scale nonlinear fourth-order compact difference algorithm on a time-space coarse grid and a large-scale linearized correction compact difference algorithm on a fine grid. In contrast to the time two-grid compact difference method, the proposed scheme applies the two-grid technique in both the spatial and temporal domains, thereby further improving computational efficiency. Solutions from the coarse grid are projected onto the fine grid via a temporally linear and spatially cubic Lagrange interpolation operator. Unconditional stability and optimal convergence rates, which are fourth-order in space and second-order in time, are proven in both the discrete $L^2$ and $L^{\infty }$ norms, without any constraints on the grid ratio. In addition to the standard techniques of the energy method, a discrete Sobolev inequality and an a priori error estimate are employed to demonstrate stability and high-order convergence. Finally, the theoretical results are validated through numerical experiments, which confirm the robustness and reliability of the proposed approach. A single-soliton experiment demonstrates that, compared with the fully nonlinear compact difference scheme, the proposed method achieves a significant reduction in CPU time while maintaining a comparable level of accuracy. Additional experiments further illustrate the algorithm’s effectiveness in simulating two-soliton interactions and soliton birth. These findings establish the proposed scheme as a highly efficient alternative to conventional nonlinear approaches. Full article

Scholars are widely concerned about the research of nonlinear Rossby waves due to their essential importance in understanding the geophysical fluid dynamics. The effects of different topographies on the propagation of barotropic Rossby waves are discussed in this paper. Starting from the classical shallow water equation of uniformly rotating fluid with bottom topography, a new Schrödinger model equation of nonlinear Rossby wave amplitude is obtained by multi-scale spatial-temporal transformations and perturbation expansion method, which has an advantage in characterizing the propagation of the blocking for atmospheres. The evolutionary dynamics of dipole blocking are discussed analytically and are simulated numerically via changing terrain parameters for sinusoidal topography, slope topography, and roughed topography, respectively. The results show that the amplitude increase for sinusoidal bottom topography makes the dipole blocking move faster and enhances the intensity significantly. For sloped topography, the intensity of dipole blocking slowly decreases with increasing topographic slope. At the same time, the effect of the frequency for roughed topography agrees with the slope effect on the dynamics of nonlinear envelope solitary Rossby waves. This theoretical attempt gives a new explanation of the topographic Rossby waves. Full article

In this paper, we study the existence of normalized solutions for anisotropic nonlinear Schrödinger equation with potentials which are bounded and converge to positive constants at infinity. In the mass subcritical case, we show the energy functional is bounded below on the $L^2$-sphere and prove the existence of a global minimizer. In the critical case, we establish a similar result under a condition on the mass. For the supercritical case, we introduce a Pohozaev-Nehari manifold and prove the existence of a positive-energy solution via the minimax methods. The compactness is recovered through detailed analysis involving the associated limit problem and strict monotonicity conditions on the potentials. To the best of our knowledge, this is the first study on the existence of normalized solutions for anisotropic nonlinear Schrödinger equation, and our approach provides a unified variational framework for handling anisotropic fractional operators with competing nonlinearities. Full article

In this paper, we investigate the multiplicity and concentration of normalized solutions to a fractional Kirchhoff–Schrödinger problem with logarithmic nonlinearity. By combining the Pohozaev identity, the penalization technique, and the concentration–compactness principle, we overcome the twofold difficulties caused by the Kirchhoff term and the logarithmic nonlinearity and establish the validity of the (PS) condition. On this basis, we employ the Ljusternik–Schnirelmann category theory to prove the multiplicity of solutions, linking the number of solutions to the topological category of the set M in which the potential function $V\left(x\right)$ attains its minimum. Finally, we analyze the concentration behavior and algebraic decay properties of these normalized solutions as $\epsilon \to 0$. Full article

This work presents the first systematic development of the inverse scattering transform for matrix nonlinear Schrödinger equations in the case where the discrete spectrum has triple poles, under nonzero boundary conditions at infinity. These systems arise physically as reductions modeling spinor Bose-Einstein condensates with hyperfine spin $F=1$ and find applications in nonlinear optics. A uniformization variable is employed to map the underlying Riemann surface to the complex plane, enabling a complete characterization of the analyticity, symmetries, and asymptotic behaviors of the Jost functions and scattering data. Extending the established framework for simple and double poles, we show that $\mathrm{rank} P(x,t,z_n)=3$ requires a third-order zero of $\det a\left(z\right)$ at $z=z_n$, while $\mathrm{rank} P(x,t,z_n)=2$ necessitates a fourth-order zero—a nontrivial feature absent in lower-order cases. The discrete spectrum for both rank configurations is fully characterized, and the full singular behavior near a triple pole is derived, respecting the quartet symmetry $z_n$, $z_n^{*}$, $v{k}_0^2/z_n$, $v{k}_0^2/z_n^{*}$ imposed by the nonzero boundary conditions. Solving the resulting matrix Riemann-Hilbert problem with triple poles yields the potential reconstruction formula and, in the reflectionless case, explicit expressions for general N-triple-pole soliton solutions, with a detailed example for $N=1$ presented to illustrate the construction. Full article

We investigate a stochastic coupled nonlinear Schrödinger (Manakov-type) system for option price and volatility wave fields within the Ivancevic adaptive-wave option-pricing paradigm, and derive exact wave families together with statistical diagnostics of the resulting dynamics. This system combines behavioral market effects with classical efficient-market dynamics and incorporates a controlled stochastic volatility component. Randomness in both the option price and volatility is incorporated via white noise, and a system of stochastic partial differential equations (PDEs) is developed that governs the joint evolution of option prices and stock price volatility. We derive advanced solutions of the proposed system using a newly created methodology. The obtained solutions are expressions of cnoidal, snoidal, dnoidal, hyperbolic, trigonometric, and exponential functions. The stochastic dynamical investigation, together with the statistical measures are presented. The autocorrelation function (ACF) of squared returns for the obtained analytical solutions is demonstrated to show distinct differences in second-order temporal dependence, while asymmetries in the temporal evolution of the fluctuations are depicted via leverage correlation (LC). The probability distribution function (PDF) dynamics of the soliton solutions illustrate prominent temporal variability and non-stationary statistical dynamics. Differences in dynamical coupling between the two components of the considered system are presented via phase velocity cross-correlation analysis and are supported by phase difference dynamics visualizations. The strength and structure of coupling between components are displayed via the amplitude cross-correlation function. Mean amplitude dynamics and variance as a function of noise intensity $\sigma$, provide a systematic influence of stochastic forcing on their energy and a quantitative measure of stochastic dispersion of soliton solutions. All the results are displayed in 3D and 2D graphs of the stochastics and statistical dynamics of the obtained solutions. Full article

The full symmetry group is found for a system of nonlinear schrödinger equations describing the propagation of optical pulses in an isotropic media. It is shown, in particular, that the six-dimensional symmetry group found is composed of a scaling transformation and a rotation of the four-dimensional space, thereby proving that the symmetry group preserves the shape of solutions. A symmetry classification of one-dimensional subalgebras of the Lie algebra is performed and yields, in particular, the symmetry reduction to the most general system of equations satisfied by the solitary waves of the equation. Explicit soliton solutions of the equation are found by largely autonomous technics. The found solitons are used to recursively generate two new ones by means of two iterations using the symmetry group. Other properties of the system are also highlighted, as well as the possible connections between the theories of symmetry groups and Darboux transformations inspired by this study. Full article

This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of metric-affine gravity (MAG) and the stochastic variational method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, and discuss the geometrical significance of torsion within this context. Subsequently, we summarize SVM, a powerful technique that facilitates quantization while effectively incorporating geometrical effects. By integrating these frameworks, we evaluate how the geometrical structures originating from torsion affect quantum fluctuations, demonstrating that they induce non-linearity in quantum mechanics. Notably, torsion, traditionally believed to influence only spin degrees of freedom, can also affect spinless degrees of freedom via quantum fluctuations. Furthermore, extending beyond the results of previous work [Koide and van de Venn, Phys. Rev. A112, 052217 (2025)], we investigate the competitive interplay between the Levi-Civita curvature and torsion within the non-linearity of the Schrödinger equation. Finally, we discuss the structural parallelism between SVM and information geometry, highlighting that the splitting of time derivatives in stochastic processes corresponds to the dual connections in statistical manifolds. These insights pave the way for future extensions to gravity theories involving non-metricity and are expected to deepen our understanding of unresolved cosmological problems. Full article

Rogue waves are sudden, extreme events that pose a threat to offshore structures’ safety. Accurately replicating nonlinear rogue waves in laboratory settings is challenging but crucial for evaluating extreme loads. Recently, the time-reversal (TR) method based on the time-reversal feature of nonlinear water wave equations, such as the cubic Schrödinger equation, has shown breakthroughs in experimental rogue wave generation. However, when generating rogue waves of large steepness and strong nonlinearity (especially high-order rogue waves), this method encounters issues such as significantly insufficient wave height and weakened nonlinear characteristics. In this article, a modified time-reversal (MTR) method is proposed based on the dynamic transfer function between the rogue wave surface history and the motion of the wave-generator paddle. MTR adopts a two-round (just like TR) but seven-step procedure for high-order rogue wave generation. Using MTR, high-order rogue waves with respect to 1st–5th-order Peregrine breathers are successfully generated in a physical wave flume. Analysis of shape indices and the energy spectrum shows that MTR greatly improves the quality of high-order rogue wave generation over the TR method. It does this by increasing the focused wave height, improving wave profile accuracy, and better preserving the highly nonlinear features of rogue waves. Using the proposed MTR method, a fifth-order rogue wave was generated with a maximum steepness of 0.03. This exceeds previous studies, where the maximum wave steepness was typically around 0.01. Consequently, this work nearly triples the wave steepness compared to earlier results, yielding the steepest fifth-order rogue wave observed in water wave research. Full article

The purpose of this study is to construct diverse forms of exact soliton solutions and conduct a comprehensive qualitative analysis. For this aim, we use the Gross–Pitaevskii system, which belongs to the family of nonlinear Schrödinger equations. This model is considered to be iconic and significant because it has potential applications in applied sciences, such as in physics, where it is used to exemplify quantum systems like Bose–Einstein condensates and illustrate the propagation of waves in optical fibers. Employing analytical techniques, the modified sine–cosine/sinh–cosh and extended rational sinh–Gordon expansion methods, we extract several waves from solutions in the shape of trigonometric, hyperbolic, and rational forms. To further deepen our insights related to the system’s behavior, we execute a detailed dynamical analysis, including sensitivity, bifurcation, and chaos, using the corresponding Hamiltonian structure. We also derive the instability modulation using linear stability theory. Using Mathematica, we systematically simulate and verify all constructed results and present some solutions for appropriate parameter values using 2D, 3D, and contour plots. The outcomes provide fruitful insights relevant to multiple scientific domains, including optical fiber technology, plasma, and condensed matter physics. This work contributes to the ongoing study of nonlinear models by applying novel solution techniques and offering a broader perspective on the complex behavior of such systems. The novelty of this study lies in the fact that the proposed model has not been previously explored using the aforementioned advanced methods and comprehensive dynamical analyses. Full article

The paper retrieves quiescent dispersive solitons in dispersion-flattened optical fibers having nonlinear chromatic dispersion and the Kerr law of self-phase modulation. The platform model is the Schrödinger–Hirota equation. The enhanced direct algebraic method has made this retrieval possible. The intermediary functions are Jacobi’s elliptic function and Weierstrass’ elliptic function. The final results appear with parameter constraints for the existence of such solitons. Full article