Search Results (125)

In this article, we investigate the regularization and qualitative properties of parabolic Ginzburg–Landau equations in variable exponent Herz spaces. These spaces capture both local and global behavior, providing a natural framework for our analysis. We establish boundedness results for fractional Bessel–Riesz operators, construct examples highlighting their advantage over classical Riesz potentials, and recover several known theorems as special cases. As an application, we analyze a parabolic Ginzburg–Landau operator with VMO coefficients, showing that our estimates ensure the boundedness and continuity of solutions. Full article

It is well known that nonlinear partial differential equations (NLPDEs) can only be solved numerically and that fourth-order NLPDEs in their derivatives require unconventional methods. This paper explains spectral numerical methods for obtaining a numerical solution by Fast Fourier Transform (FFT), implemented under Python in tis version 3.1 and their libraries (NumPy, Tkinter). Examples of NLPDEs typical of Condensed Matter Physics to be solved numerically are the conserved Cahn–Hilliard, Swift–Hohenberg and conserved Swift–Hohenberg equations. The last two equations are solved by the first- and second-order exponential integrator method, while the first of these equations is solved by the conventional FFT method. The Cahn–Hilliard equation, a phase-field model with an extended Ginzburg–Landau-like functional, is solved in two-dimensional (2D) to reproduce the evolution of the microstructure of an amorphous alloy Ce₇₅Al_{25 − x}Gax, which is compared with the experimental micrography of the literature. Finally, three-dimensional (3D) simulations were performed using numerical solutions by FFT. The second-order exponential integrator method algorithm for the Swift–Hohenberg equation implementation is successfully obtained under Python by FFT to simulate different 3D patterns that cannot be obtained with the conventional FFT method. All these 2D/3D simulations have applications in Materials Science and Engineering. Full article

This paper provides highly dispersive optical soliton solutions to the perturbed complex Ginzburg–Landau equation. The self-phase modulation structures are maintained in three forms, which are derived from the power law of nonlinearity with arbitrary intensity. The paper employs the semi-inverse variational principle as its integration scheme, as conventional methods are incapable for it. The amplitude–width relation of the solitons is reconstructed by employing Cardano’s method to solve a cubic polynomial equation. Also presented are the necessary parameter constraints that naturally arise from the scheme. These findings enhance our understanding of soliton dynamics and pave the way for further research into more complex nonlinear systems. Future studies may explore the implications of these results in various physical contexts, potentially leading to novel applications in fields such as fiber optics and quantum fluid dynamics. Full article

This paper investigates quiescent solitons in optical fibers and crystals, modeled by the complicated Ginzburg–Landau equation incorporating nonlinear chromatic dispersion and six self-phase modulation structures introduced by Kudryashov. The model is formulated with linear temporal evolution and analyzed using Lie symmetry methods. The study also identified parameter constraints under which solutions exist. Full article

To improve the sampling efficiency of Markov Chain Monte Carlo in complex parameter spaces, this paper proposes an adaptive sampling method that integrates a swarm intelligence mechanism called the BeeSwarm-MH algorithm. The method combines global exploration by scout bees with local exploitation by worker bees. It employs multi-stage perturbation intensities and adaptive step-size tuning to enable efficient posterior sampling. Focusing on Bayesian inference for parameter estimation in the soliton solutions of the two-dimensional complex Ginzburg–Landau equation, we design a dedicated inference framework to systematically compare the performance of BeeSwarm-MH with the classical Metropolis–Hastings algorithm. Experimental results demonstrate that BeeSwarm-MH achieves comparable estimation accuracy while significantly reducing the required number of iterations and total computation time for convergence. Moreover, it exhibits superior global search capabilities and adaptive features, offering a practical approach for efficient Bayesian inference in complex physical models. Full article

Ferroelectric materials, characterized by spontaneous electric polarization, exhibit remarkable parallels with fluid dynamics, where polarization flux behaves similarly to fluid flow. Understanding polarization distribution in confined geometries at the nanoscale is crucial for both fundamental physics and technological applications. Here, we show that the classical Bernoulli principle, which describes the conservation of the energy flux along velocity streamlines in a moving fluid, can be extended to the conservation of polarization flux in ferroelectric nanorods with varying cross-sectional areas. Geometric constrictions lead to an increase in polarization, resembling fluid acceleration in a narrowing pipe, while expansions cause a decrease. Beyond a critical expansion, phase separation occurs, giving rise to topological polarization structures such as polarization bubbles, curls and Hopfions. This effect extends to soft ferroelectrics, including ferroelectric nematic liquid crystals, where polarization flux conservation governs the formation of complex mesoscale states. Full article

In this paper, we prove the local well-posedness of classical solutions $(\psi ,A,\varphi )$ to the $nD(n\ge 3)$ time-dependent Ginzburg–Landau model in superconductivity with the choice of Coulomb gauge and the main assumptions ${\psi }_0,A_0\in H^s\left({\mathbb{R}}^n\right)$ with $\mathrm{div}{A}_0=0$ in ${\mathbb{R}}^n$ and $s>{\displaystyle \frac{n}{2}}$. This result can be used in the proof of regularity criterion and global-in-time well-posedness of the strong solution. Full article

The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. Full article

We investigate the time evolution of the quark condensate toward a chiral symmetry broken phase in hot and dense quark matter using a field-theoretic quark model with nonlocal chiral-invariant four-fermion coupling. By purposely selecting a parameter set in which inhomogeneous phases are energetically disfavored, we nonetheless observe the emergence of metastable patterned configurations that appear to persist for remarkably long timescales. These findings suggest that even when not fully stable, inhomogeneous phases may play a significant role in the dynamics of chiral symmetry breaking and restoration. To gain deeper insight into these phenomena, we also analyze the impact of the dimensionality of coordinate space on both the formation and stability of inhomogeneous chiral condensates. Full article

This paper is devoted to studying the existence and uniqueness of mild solutions for semilinear fractional evolution equations with the Hilfer–Katugampola fractional derivative and under the nonlocal multi-point condition. The analysis is based on analytic semigroup theory, the Krasnoselskii fixed-point theorem, and the Banach fixed-point theorem. An application to a time-fractional real Ginzburg–Landau equation is also given to illustrate the applicability of our results. Furthermore, we determine some conditions to make the control (Bifurcation) parameter in the Ginzburg–Landau equation sufficiently small. Full article

This manuscript studies the M-fractional Landau–Ginzburg–Higgs (M-fLGH) equation in comprehending superconductivity and drift cyclotron waves in radially inhomogeneous plasmas, especially for coherent ion cyclotron wave propagation, aiming to explore the soliton solutions, the parameter’s effect, and modulation instability. Here, we propose a novel approach, namely a newly improved Kudryashov’s method that integrates the combination of the unified method with the generalized Kudryashov’s method. By employing the modified F-expansion and the newly improved Kudryashov’s method, we investigate the soliton wave solutions for the M-fLGH model. The solutions are in trigonometric, rational, exponential, and hyperbolic forms. We present the effect of system parameters and fractional parameters. For special values of free parameters, we derive some novel phenomena such as kink wave, anti-kink wave, periodic lump wave with soliton, interaction of kink and periodic lump wave, interaction of anti-kink and periodic wave, periodic wave, solitonic wave, multi-lump wave in periodic form, and so on. The modulation instability criterion assesses the conditions that dictate the stability or instability of soliton solutions, highlighting the interplay between fractional order and system parameters. This study advances the theoretical understanding of fractional LGH models and provides valuable insights into practical applications in plasma physics, optical communication, and fluid dynamics. Full article

Based on two-band time-dependent Ginzburg–Landau theory, we study the electromagnetic properties of mesoscopic type-1.5 superconductors with different defect configurations. We perform numerical simulations with the finite element method, and give direct evidence for the existence of a vortex cluster phase in the presence of nonmagnetic impurity. In addition, we also investigate the depinning critical current of the magnetic vortex cluster induced by the isotropic or anisotropic defect structure under the external current. Our theoretical results thus indicate that the diversity of impurity deposition has a significant influence on the semi-Meissner state in type-1.5 superconductors. Full article

The Bogomolny approach to the Ginzburg–Landau equations in the context of strong and semi-strong necessary conditions is formulated for various superconducting structures in a quasi-one-dimensional description, considering both flat and curved geometries. This formulation is justified by a perturbative approach to the Ginzburg–Landau theory applied to a superconducting structure that is polarized by an electric charge moving across two neighboring quantum dots. The situation considered involves an interface between a Josephson junction and a semiconductor quantum dot system in a one-dimensional setting. Full article

The impact of dynamic processes on equilibrium properties is a fundamental issue in condensed matter physics. This study investigates the intrinsic ferromagnetism generated by memory effects in the low-dimensional continuous symmetry Landau–Ginzburg model, demonstrating how memory effects can suppress fluctuations and stabilize long-range magnetic order. Our results provide compelling evidence that tuning dynamical processes can significantly alter the behavior of systems in equilibrium. We quantitatively evaluate how the emergence of the ferromagnetic phase depends on memory effects and confirm the presence of ferromagnetism through simulations of hysteresis loops, spontaneous magnetization, and magnetic domain structures in the 1D continuous symmetry Landau–Ginzburg model. This research offers both theoretical and numerical insights for identifying new phases of matter by dynamically modifying equilibrium properties. Full article

The Inverse Faraday Effect (IFE) is a phenomenon that enables non-thermal magnetization in various types of materials through the interaction with circularly polarized light. This study investigates the impact of single defects on the ability of circularly polarized radiation to switch between distinct superconducting current states, when the magnetic flux through a superconducting ring equals half the quantum flux, ${\mathsf{\Phi }}_0/2$. Using both analytical methods within the standard Ginzburg–Landau theory and numerical simulations based on the stochastic time-dependent Ginzburg–Landau approach, we demonstrate that while circularly polarized light can effectively switch between current-carrying superconducting states, the presence of a single defect significantly affects this switching mechanism. We establish critical temperature conditions above which the switching effect completely disappears, offering insights into the limitations imposed by a single defect on the dynamics of light-induced IFE-based magnetization in superconductors. Full article