Landau Equation Research Articles

Soliton-like nonlinear excitation in the FitzHugh–Nagumo cardiac model through the cubic–quintic complex Ginzburg–Landau equation

Soliton-like nonlinear excitation in the FitzHugh–Nagumo cardiac model through the cubic–quintic complex Ginzburg–Landau equation

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.

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.

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.

Binary robustness of random attractors for 2D-Ginzburg–Landau equations with Wong–Zakai noise

Consider a non-autonomous 2D-Ginzburg–Landau equation driven by Wong–Zakai noise or white noise, respectively, we first show the existence of pullback random attractors, which are random compact attracting sets indexed by two parameters: the size of Wong–Zakai noise and the current time. We then establish the robustness of the attractors when both parameters are simultaneously convergent. An essential difficulty arises from the possible loss of the convergence of solutions and only part convergence of solutions is available, which is a new phenomenon for 2D-GL equation distinguishing with the 1D case. So, by using part joint-convergence, regularity, eventual local-compactness and recurrence, we establish a binary robustness theorem of pullback random attractors and apply it to the weakly dissipative stochastic equation.

Ginzburg–Landau simulations of three-terminal operation of a superconducting nanowire cryotron

Abstract Superconducting nanowire cryotrons (nTrons) are expected to be used as interfaces for super-high-performance hybrid devices in which superconductor and semiconductor circuits are combined. However, nTrons are still under development, and diverse analyses of these devices are needed. Accordingly, we have developed a numerical technique to simulate the three-terminal operation of an nTron by using the finite element method to solve the time-dependent Ginzburg–Landau (TDGL) equation and the heat-diffusion equation. Simulations using this technique offer understanding of the dynamics of the order parameter, the thermal behavior, and the characteristics of three-terminal operation, and the TDGL model reproduces qualitatively the results of nTron experiments conducted at the Massachusetts Institute of Technology. In addition, we investigated how some geometric and physical parameters (the design elements) affect the operation characteristics. The TDGL model has far fewer free parameters compared with the lumped-element electrothermal model commonly used for simulating superconducting devices. Furthermore, the TDGL model provides time-dependent visual information about the superconducting state and the normal state, thereby offering insights into the relationship between nTron geometry and three-terminal operation. These simulation results offer a route to nTron optimization and the development of nTron applications.

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.

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.

Finite difference approximations of the spatially homogeneous Fokker–Planck–Landau equation

Finite difference methods are developed for approximating the spatially homogeneous Fokker–Planck–Landau equation for Coulomb collisions. The numerical methods apply the Fast Fourier Transform to improve time efficiency. Computational work is then done to compare the numerical approximations using FFT with the numerical method of Wollman (2017). In addition, some computations are done to verify the theoretical rate of convergence of the FPL equation proved in Strain and Guo (2008).

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.

Gevrey and Gelfand–Shilov smoothing effect for the spatially inhomogeneous Landau equation with hard potentials

In this work, we study the Cauchy problem of the spatially inhomogeneous Landau equation with hard potential in torus. It is showed that the smooth mild solution near Maxwellians to the Cauchy problem enjoys a Gelfand–Shilov regularizing effect in velocity variable and Gevrey smoothness property in space variable.

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

Multi-level storage in cleaved-gate ferroelectric FETs investigated by 3D phase-field-based quantum transport simulation

In this work, we investigate the feasibility of cleaved-gate ferroelectric FET (CG-FeFET) as multi-level cell (MLC) memory devices, by conducting 3-dimensional quantum transport simulations based on time-dependent-Ginzburg–Landau equation, and the non-equilibrium Green’s function method. Our results indicate that CG-FeFET can achieve multi-level operations by utilizing different thicknesses of the ferroelectric layer. We analyze the influence of electron–phonon interaction and also verify that CG-FeFET is robust to noise. Furthermore, we identify the critical role of the spacing between two ferroelectric layers in determining the memory window, considering the effects of polarization cancellation and electrostatic coupling. These findings provide valuable insights into designing stable and reliable nonvolatile memory technologies, which could offer potential solutions for high-density memory requirements.

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.

Bifurcations, chaotic behavior, and optical solutions for the complex Ginzburg–Landau equation

This research focuses on investigating an extended version of the Ginzburg–Landau (GL) equation that describes the motion of particles in a plasma. The other applications of this model can be found in optics, and other related fields. As part of our approach, we seek to discover some new wave solutions to the model, and then use the Galilean transformation in order to determine the model’s dynamical properties. After that, we use planar dynamical system theory to perform an in-depth bifurcation analysis, which gives us valuable insight into the system. By conducting thorough mathematical and bifurcation analyses, we identify the fundamental dynamics and critical points that regulate the system’s behavior. Moreover we explore the chaotic nature of the system, uncovering potential chaotic tendencies and their consequences. Further, we obtain several categories of optical solutions of the complex GL equation by utilizing new logarithmic transformations, offering valuable insight into its behavior and potential applications in optics. Our analytical technique yields closed-form solutions expressing elementary functions. In order to ensure the reliability of our findings, we rigorously validate the obtained solutions by substituting them back into the original model. Our research helps us better understand this equation’s characteristics and its relevance across a wide range of disciplines.

Global existence for an isotropic modification of the Boltzmann equation

Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau.Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the “very soft potentials” regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities.

Global solutions in the critical Sobolev space for the Landau equation

Global solutions in the critical Sobolev space for the Landau equation

The Effect of Throughflow and Gravitational Modulation on Weakly Nonlinear Bio-Thermal Convection in a Porous Medium Layer

We investigate the impact of periodically varying gravitational fields and a throughflow on the bio-thermal Darcy–Brinkman convection within a porous medium layer saturated with a Newtonian fluid containing gyrotactic microorganisms. The study includes an examination of two types of a throughflow: one directed against the gravity field and another one along it. We assume that the gravitational modulation has a small amplitude, quantified as a second-order smallness in the dimensionless parameter ϵ, which represents the supercritical parameter of the Rayleigh number. For weakly nonlinear convection, a Ginzburg–Landau (GL) equation with a periodic coefficient is derived in the third order in ϵ. To analyze the heat and mass transfer, we numerically solve the GL equation. The numerical results reveal that the vertical throughflow in the bio-thermal convection exhibits a dual nature, allowing for both augmentation and a reduction of the heat and mass transfers. We investigate the influence of variations in the Vadasz number, Peclet number, bioconvective Peclet number, frequency, and amplitude of modulation on the heat and mass transfer. The effects of these parameters are depicted graphically, illustrating that higher values of the Vadasz and Peclet numbers, as well as increased modulation amplitude, positively impact the heat and mass transfer. In addition, a comparative analysis of modulated and non-modulated systems shows a significant effect of the modulation on the stability of systems.