Ginzburg-Landau Equation Research Articles

Critical Casimir effect in a disordered O(2)-symmetric model.

The critical Casimir effect appears when critical fluctuations of an order parameter interact with classical boundaries. We investigate this effect in the setting of a Landau-Ginzburg model with continuous symmetry in the presence of quenched disorder. The quenched free energy is written as an asymptotic series of moments of the model's partition function. Our main result is that, in the presence of a strong disorder, Goldstone modes of the system contribute either with either an attractive or a repulsive force. This result was obtained using the distributional zeta-function method without relying on any particular ansatz in the functional space of the moments of the partition function.

Partially “invisible” pulsation of asymmetric soliton molecules

Pulsation is a universal phenomenon in all disciplines of nonlinear science, which generally exhibit various nonlinear soliton dynamic modes in nonlinear optics. With the emergence of real-time spectral measurement technology, a new phenomenon of optical soliton pulsation, “invisible” soliton pulsation, has gradually attracted researchers’ attention. In this paper, by solving the coupled Ginzburg–Landau equation, we have discovered the partially “invisible” pulsation phenomenon of asymmetric soliton molecules (SMs) composed of two unequal-intensity pulses in a normal-dispersion Mamyshev oscillator for the first time. It is indicated that the fluctuation periods of each internal pulse in asymmetric SMs in terms of peak power, energy, and relative phase difference are all four roundtrips. However, the oscillation period of asymmetric SMs energy is two roundtrips, only half of the above period. Further research shows that the oscillation of the relative phase difference between two internal pulses in SMs is related to the variation of their intensity difference, which can also affect the position change of the spectral modulation peak. This work will enrich the research on “invisible” soliton pulsation and is of great significance for promoting the development of nonlinear dissipative systems.

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Vortex waves in fluid–structure interaction with high Froude number and a damped structure

In this paper, we study the non-linear dynamic response generated as a result of a fluid–structure interaction between a flexible structure and a flowing fluid, when the structure is subjected to non-linear excitations. In first place, the use of semi-discrete approximations allowed us to show that the motion of a flexible structure coupled with a surrounding fluid flowing could be modelled and analysed via a coupled Complex Cubic Ginzburg–Landau equations (CCGLEs). Through the obtained CCGLEs, we were able to show that modulational instability (MI) is the main mechanism responsible for the generation of vortex shedding. Moreover, we showed that the stability of continuous wave depends on the coupling parameters between the fluid and the structure. Secondly, using a mathematical method, namely the G’/G expansion method, we found that vortex wave trains could be generated as cylindrical waves. These results are highly significant from a theoretical point of view and could be a plus to explain the process of generation of Kármán Vortex as a consequence of unstable coupling between two continuous wave in the fluid–structure system. Moreover, considering the industrial interest, such as floating wind turbines, this work aims to provide an additional understanding of the interactions between a flexible body and a surrounding flow.

Calabi-Yau four-, five-, sixfolds as Pwn hypersurfaces: Machine learning, approximation, and generation

Calabi-Yau fourfolds may be constructed as hypersurfaces in weighted projective spaces of complex dimension five defined via weight systems of six weights. In this work, neural networks were implemented to learn the Calabi-Yau Hodge numbers from the weight systems, where gradient saliency and symbolic regression then inspired a truncation of the Landau-Ginzburg model formula for the Hodge numbers of any dimensional Calabi-Yau constructed in this way. The approximation always provides a tight lower bound, is shown to be dramatically quicker to compute (with computation times reduced by up to 4 orders of magnitude), and gives remarkably accurate results for systems with large weights. Additionally, complementary datasets of weight systems satisfying the necessary but insufficient conditions for transversality were constructed, including considerations of the interior point, reflexivity, and intradivisibility properties, overall producing a classification of this weight system landscape, further confirmed with machine learning methods. Using the knowledge of this classification and the properties of the presented approximation, a novel dataset of transverse weight systems consisting of seven weights was generated for a sum of weights ≤200, producing a new database of Calabi-Yau fivefolds, with their respective topological properties computed. Furthermore, an equivalent database of candidate Calabi-Yau sixfolds was generated with approximated Hodge numbers. Published by the American Physical Society 2024

Stability of viscoelastic film on a slippery inclined plane

The linear and weakly nonlinear stability of viscoelastic film flowing down a slippery inclined plane is investigated analytically. Under the assumption of the long wave approximation, the first-order Benny equation of Oldroyd-B fluid thin film with slip condition is obtained. Through the normal mode analysis, the neutral stability curve and the temporal growth rates are calculated to explore the linear stability of the film. Linear results show that the critical Reynolds number decreases with the increase in slip length and viscoelastic parameter and that the liquid film may exhibit pure elastic instability. For the nonlinear stability analysis, both hydrodynamic instability and elastic instability are discussed. The primary bifurcations in the phase plane are identified by calculating the Landau coefficient, i.e., the unconditional stable region, the supercritical region, the subcritical region, and the explosive region. The dependence of primary bifurcation regions upon the slip length and Deborah number are studied, and the results indicate that the slip boundary and viscoelasticity destabilizes the flow. According to the Ginzburg–Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.

Dynamics of femtosecond lasers and induced plasma in non-Kerr nonlinear transparent materials: Competing effects of electron–hole radiative recombinations and single-electron diffusions

Modern laser micromachining utilizes ultrashort optical fields, such as femtosecond lasers, to perform high-precision processings on solid materials, including cutting, drilling, ablation, polishing, and scripturing. Femtosecond laser-based material processings can induce a plasma of free electrons whose density depends on physical phenomena such as single-electron diffusions, multiphoton ionization, and electron–hole radiative recombinations. In this work, we examine the dynamics of femtosecond lasers in transparent materials with non-Kerr nonlinearity, taking into account the generation of an electron plasma. In these specific materials, a balance between the nonlinearity and the group-velocity dispersion of the optical medium can favor the formation of optical filaments propagating with a permanent shape by virtue of their “solitonic” features. We are interested in the effects of the competition between electron–hole radiative recombination and single-electron diffusion processes on the spatiotemporal profiles of the propagating optical field and of the plasma density. The model features a complex Ginzburg–Landau equation with an optical nonlinearity of a general saturable form and a Kth-order nonlinearity term accounting for K-photon ionization processes, coupled to a rate equation for the electron plasma density where the present terms are representing avalanche ionizations, single-electron diffusion, and electron–hole radiative recombination processes. The modulational-instability analysis suggests that the continuous-wave regime will be stabilized by strong electron–hole radiative recombination processes for a fixed value of the single-electron diffusion coefficient, a stability enhanced by an increase in K. In the nonlinear regime, numerical simulations of the model equations for different combinations of the nonlinearity-saturation exponents and different values of the photon number K unveil soliton train structures forming from the laser field propagation and the time evolution of the plasma density. These structures turn out to be either dissipative soliton trains in the absence of electron–hole radiative recombinations or soliton crystals when electron–hole radiative recombination processes are taken into consideration to balance the damping effect caused by single-electron diffusions.

Optical soliton solutions of complex Ginzburg–Landau equation with triple power law and modulation instability

This paper examines the complex Ginzburg Landau equation, which describes pulse propagation inside a fiber with the triple power law of self-phase modulation. Since the effect of parameter selection has become very important in relevant model studies recently, self-phase modulation has been added to the complex Ginzburg Landau equation, which has been studied in the literature, and it is aimed at investigating the analytical solutions of the presented equation. Adding the triple power law of the self-phase modulation parameter to the model, in addition to existing studies in the literature, emphasizes the innovative aspect and importance of the study. The first aim is to reveal bright and singular solitons using the new Kudryashov method. The new Kudryashov method is a technique that is frequently used in the literature, is effective for generating analytical solutions, provides ease of operation, and can be applied to a wide class of nonlinear partial differential equations. The second goal is to show that the obtained solutions have modulation stability. By using modulation instability analysis, the gain spectrum is formed for different parameter values. Graphic presentations support the findings. Moreover, bright and singular soliton portraits are demonstrated with 3D and 2D graphs. The novelty of the study lies in the fact that the relevant model has not been studied before with an effective method such as the new Kudryashov method, and the modulation instability has been studied for the first time.

Numerical Study on Monopole Production and Deconfinement Transition in Two-Condensate Charged Systems

The condensed matter Bose system may contain effective monopole quasiparticles in its excitation spectrum. In this paper, we first accomplish the mapping of the two-band Ginzburg–Landau theory to the extended CP1 model, and then perform the Monte Carlo simulations on the 50×50×50 cubic lattice with periodic boundary conditions. With the numerical data of monopole density and magnetic susceptibility, we indicate that there exists a monopole–antimonopole deconfinement transition for the two-band superconducting system with the critical temperature above 70 K. We also suggest the possible detection of this new monopole plasma phase in high-Tc iron-based superconductors.

Effects of throughflow and gravity modulation on triple-diffusive convection in a sparsely packed porous medium

This article investigates the impact of four different gravity modulations on triple-diffusive convection in a porous layer with throughflow. The study conducts a weakly nonlinear stability analysis. Additionally, cubic Ginzburg–Landau equation is derived to observe the effects of throughflow, gravity modulation, and concentration. The solution of Ginzburg–Landau equation is used to calculate heat transfer and mass transport in terms of the Nusselt and Sherwood numbers, respectively. Furthermore, the mean Nusselt and Sherwood numbers are calculated. The study investigates the influence of Lewis numbers, solutal Rayleigh numbers, and Peclet numbers on the stability and instability of the system graphically for the four types of gravity modulations. The results show that the Sawtooth modulation destabilizes the system, while the square waveform modulation stabilizes it. The mean Nusselt number decreases with both the first Lewis number Le 1 and the second Lewis number Le 2, as well as the mean Sherwood number, which decreases with the first solutal Rayleigh number Rs 1 and the second solutal Rayleigh number Rs 2. This indicates system stability. Furthermore, it is noted that the heat and mass transport for triangular modulation fall between those of sawtooth modulation and square modulation.

Strain tuning on Van der Waals negative capacitance transistors

Negative differential capacitance induced by the energy barrier formed by the nonlinear ferroelectric insulating layer during the phase transition in negative capacitance field effect transistors (NCFETs) can break through the fundamental thermionic limit of the metal-oxide-semiconductor field effect transistors (MOSFETs), which is set at a subthreshold swing (SS) of 60 mV/dec at room temperature for low-power switching. Strain engineering has been shown effective for modulating both intrinsic and coupled properties of ferroelectric materials. However, while the modulation of negative capacitance (NC) by strain primarily remains within theoretical predictions, there is a lack of experimental studies to explore its effect on NCFETs. This study introduces strain to NCFETs by utilizing the magnetostriction of Terfenol-D upon the applied magnetic field. Results reveal a decrease in SSmin from 43.5 mV/dec to 15.2 mV/dec with a compressive strain increase from 0 to −0.005%. Additionally, theoretical results by using time-evolving Ginzburg-Landau equations and Kirchhoff's laws demonstrate that compressive strain enhances both the amplitude and duration of transient NC in CuInP2S6, while vice versa under tensile strain. This work offers valuable insights for the design of NCFET devices.

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain.

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equationwith periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

Chirped optical solitons for the complex Ginzburg–Landau equation with Hamiltonian perturbations and Kerr law nonlinearity

Abstract What the motivation of this paper is to provide chirped optical solitons for the complex Ginzburg–Landau equation with Hamiltonian perturbations and Kerr law nonlinearity. We get 19 exact chirped solutions by utilizing trial equation method and the complete discriminant system for polynomial method, which are richer than the solutions acquired in existing papers. We draw the two-dimensional graphs of amplitudes and corresponding chirps in order to verify the existence of the solutions and discuss the dynamical properties of the solutions. To our knowledge, this is the first time that comprehensive set of exact chirped solutions of the governing equation in the paper are obtained. The model and the results obtained in this paper may help explain some nonlinear problems.

Phase field simulations on the rate‐ and grain‐size‐dependent crack propagation of polycrystalline NiTi shape memory alloy

AbstractA three‐dimensional non‐isothermal phase field model of polycrystalline NiTi shape memory alloy (SMA) was established based on the Ginzburg–Landau's theory, which considered the interaction between crack growth and martensite transformation (MT). Then, the crack propagation of polycrystalline NiTi SMA and its dependence on the system's grain size and loading rate were studied by the phase field simulations. The results show that the proposed phase field model can characterize the MT behavior, temperature field changes, and stress‐displacement response of NiTi SMA during the crack growth with different grain sizes and loading rates. That is the higher the loading rate, the higher the fracture toughness is; the larger the grain size, the higher the fracture toughness is and the more significant the influence of grain boundary on the crack growth path becomes. Meanwhile, as the grain size increases, the fracture mode gradually changes from the transgranular fracture (for the fine‐grained alloy with a grain size ≈ 15 nm) to the intergranular one (for the coarse‐grained alloy with a grain size ≈ 100 nm).

Exact solutions to the fractional complex Ginzburg-Landau equation with cubic-quintic and Kerr law nonlinearities

The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved (G′/G2) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.

Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski [27], predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit this conjecture through the lens of mirror symmetry for a Fano pair (X,D), where X is a smooth Fano variety and D is a simple normal crossing divisor. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of D. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev [32]. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it.

Intermittency analysis of charged particles generated in Xe-Xe~collisions at $\sqrt{s_{\rm{NN}}}$ = 5.44 TeV using the AMPT model

The multiplicity fluctuations are sensitive to QCD phase transition and to the presence of critical point in QCD phase diagram. At critical point a system undergoing phase transition is characterized by large fluctuations in the observables which is an important tool to understand the dynamics of particle production in heavy-ion interactions and phase changes. Multiplicity fluctuations of produced particles is an important observable to characterize the evolving system. Using scaling exponent obtained from the normalized factorial moments of the number of charged hadrons in the two dimensional (\eta,\phiη,ϕ) phase space, one can learn about the dynamics of system created in these collisions. Events generated using Xe-Xe collisions at $ = 5.44 $ TeV with string-melting (SM) version of the AMPT model are analyzed and the scaling exponent (\nu)(ν) for various p_TpT intervals is determined. It is observed that the calculated value of \nuν is larger than the universal value 1.304, as is obtained from Ginzburg-Landau theory for second order phase transition. Here we will also present the results of the dependence of the scaling exponent on the transverse momentum bin width.

Exact periodic solution family of the complex cubic-quintic Ginzburg–Landau equation with intrapulse Raman scattering

Exact periodic solution family of the complex cubic-quintic Ginzburg–Landau equation with intrapulse Raman scattering

Revisitation of “implicit quiescent optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion": generalized temporal evolution

AbstractThis paper is a revisitation to the study on the retrieval of quiescent optical solitons to the complex Ginzburg–Landau equation that is considered with generalized temporal evolution and nonlinear chromatic dispersion. The results are recovered with the application of Lie symmetry. Apart from a couple of self-phase modulation structures where the results are in quadratures, the integrals are with a range of special functions. Two new forms of self-phase modulation structures are addressed in the paper.

Influence of Two-Frequency Rotational Modulation on the Dynamics of the Rayleigh–Bénard Convection in Water-Based Nanoliquids with Either AA7072 or AA7075 Nanoparticles

The effect of time-periodic two-frequency rotation modulation on Rayleigh–Bénard convection in water with either AA7072 or AA7075 nanoparticles is investigated. The single-phase description of the Khanafer–Vafai–Lightstone model is used for modeling the nanoliquids. An asymptotic expansion procedure is adopted in the case of the linear stability to obtain the correction (due to modulation) to the Rayleigh number at marginal stability of unmodulated convection. A nonlinear regime of convection is considered with a nonautonomous generalized Lorenz model as the governing equation. The method of multiscales is then employed to obtain the coupled nonautonomous Ginzburg–Landau equations with cubic nonlinearity from the Lorenz model. These equations are presented in the phase-amplitude form and the amplitude is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and moderate frequencies of modulation are considered. We found that there is a threshold frequency beyond which the system behavior reverses. At frequencies below the threshold, the mean Nusselt number increases with an increase in the amplitude of modulation while an opposite influence is seen for values above the threshold. Such a behavior is a consequence of what is analogously seen in the case of the critical Rayleigh number. The influence of two-frequency modulation is more pronounced on the results of the linear and nonlinear regimes compared to that of the single-frequency one. The heat transport is enhanced due to the presence of dilute concentration of suspended nanoparticles (either AA7072 or AA7075 nanoalloys) in water. The influence of nanoparticles is to modify the threshold values generating chaos but it does not qualitatively alter the dynamical behavior of the system. The plots of Lyapunov exponents reveal that there is no possibility of hyper-chaos in the generalized Lorenz model when there is a rotational modulation.