Landau Equation Research Articles

Explicit computation of jet functions in coordinate space

I review the main results leading to Factorization of QCD amplitudes in momentum space and, in view of the analogue results in coordinate space, the one-loop jet function in coordinate space is computed and Landau's equations for the Abelian radiative corrections to it are studied. Furthermore, two of these radiative corrections are reduced in quadrature.

Cutkosky’s theorem for massive one-loop Feynman integrals: part 1

We formulate and prove Cutkosky’s Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in Fotiadi et al. (Topology 4(2):159–191, 1965) . We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail.

Spectral Gap Formation to Kinetic Equations with Soft Potentials in Bounded Domain

It has been unknown in kinetic theory whether the linearized Boltzmann or Landau equation with soft potentials admits a spectral gap in the spatially inhomogeneous setting. Most of existing works indicate a negative answer because the spectrum of two linearized self-adjoint collision operators is accumulated to the origin in case of soft interactions. In the paper we rather prove it in an affirmative way when the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

Exact solutions to the fractional complex Ginzburg–Landau equation with time-dependent coefficients under quadratic–cubic and power law nonlinearities

Exact solutions to the fractional complex Ginzburg–Landau equation with time-dependent coefficients under quadratic–cubic and power law nonlinearities

Sn-cn, sn-dn, cn-dn Jacobi elliptic functions and solitons solutions in birefringent fibers for CGL equations with Hamiltonian perturbations and Kerr law nonlinearity

The current work is dedicated to the investigation of the propagation of progressive solitons in birefringent fibers for Complex Ginzburg–Landau equations with Hamiltonian perturbations and Kerr law nonlinearity. To achieve this goal, we use a modified F-expansion method which enables us to widen the class of solutions to these coupled equations and permit the propagation of different waves on the two lines. First of all, we constructed the solutions in terms of Jacobi elliptic functions in the forms of dn-sn, dn-cn, and cn-sn. After that, we obtained the progressive soliton solutions in the limiting approach, when the modulus of ellipticity approaches unity. These progressive soliton solutions include bright–dark, bright–bright, dark–dark, bright–front, dark–front. For these solutions to exist, the criteria are exhibited in form of parameter constraints. In addition, the visual representations of these solutions depicted in several figures surely play a key part in understanding the behavior and capturing some of the physical aspects of the model under consideration.

Chimera states under genuine local coupling

Chimera states, important forms of spatiotemporal self-organization in ensembles of identical oscillators, have been found in a wide variety of systems, provided that the coupling between the oscillators was nonlocal or global. Therefore, it is generally assumed that a locally coupled oscillatory medium, as described by the complex Ginzburg–Landau equation (CGLE), does not support chimera states. Here we show an alternative mechanism that does indeed lead to chimera states in a purely locally-coupled system, namely the interaction of an oscillatory medium, in the present case the CGLE, with a bistable internal degree of freedom.

Non-Hermitian control of optical turbulence in systems with fractional dispersion

We show an efficient mechanism to control optical turbulence in systems with different dispersion laws, including parabolic, sub-diffractive, hyper-diffractive or general fractional dispersion. The method is based on the modification of the energy cascade through spatial scales leading to turbulence: a non-Hermitian spatio-temporal periodic potential allows unidirectional coupling between modes in the excitation process. We prove a significant increase and reduction of the energy flow in turbulent states, by either condensing the excitation towards small wave-numbers or affecting the energy transfer towards large wave-number. The study is based on the complex Fractional Ginzburg–Landau equation, a universal model for pattern formation and turbulence in a wide range of systems. The enhancement or reduction of turbulence is indeed dependent on the imposed direction of the energy flow, controlled by the phase shift between the real and imaginary parts of the temporal oscillation of the non-Hermitian potential.

Collective effects in an incompressible electronic liquid.

Starting from Landau's kinetic equation, we show that an electronic liquid in d = 2, 3 spatial dimensions depicted by a Landau-type effective theory will become incompressible on condition that the Landau parameters satisfy either (i) [Formula: see text] or (ii) [Formula: see text]. Condition (i) is the Pomeranchuk instability in the current channel and suggests a quantum spin liquid (QSL) state with a spinon Fermi surface; while condition (ii) means that the strong repulsion in the charge channel leads to a conventional charge and thermal insulator. In the collisionless regime (ωτ ≫ 1) and the hydrodynamic regime (ωτ ≪ 1), the zero and first sound modes have been studied and classified by symmetries, including the longitudinal and transverse modes in d = 2, 3 and the higher angular momentum modes in d = 3. The sufficient (and/or necessary) conditions of these collective modes have been revealed. It has been demonstrated that some of these collective modes will behave in quite different manners under incompressibility condition (i) or (ii). Possible nematic QSL states and a hierarchy structure for gapless QSL states have been proposed in d = 3.

Modified stability analysis of double‐diffusive convection in viscoelastic fluid layer saturating porous media

AbstractThe present analysis mathematically investigates the thermohaline convection problem in viscoelastic fluid layer saturating porous media by utilizing the modified Boussinesq approximation. By performing linear stability analysis, the Darcy–Rayleigh numbers for stationary and oscillatory modes of convection are derived. The effects of different parameters describing the problem are studied numerically. In nonlinear stability analysis, the heat and mass transfer rates in the form of Nusselt and Sherwood numbers, respectively, are obtained for oscillatory convection using the derived Ginzburg–Landau equation. From the results, it is observed that overstability is the preferred mode of instability in linear stability. It is found that in linear double‐diffusive convection problems, the stress relaxation imparts a destabilizing effect whereas the strain retardation time, the coefficient of specific heat variation due to temperature, and the concentration gradient have a stabilizing effect on the system's stability. The numerical values of heat and mass transfer rates varied with the coefficient of specific heat showing that the heat transport decreases while the mass transport increases. Also, the stress relaxation time, the concentration gradient, and the gravity modulation's amplitude increase while the strain retardation time decreases the heat and mass transfer rates. The wavelength of oscillations remains unaltered with the variation of specific heat variation due to temperature. The modulation frequency does not affect the heat/mass transfer rate; though, the wavelength of oscillations decreases with increasing frequency.

Pattern-Selective Feedback Stabilization of Ginzburg–Landau Spiral Waves

The complex Ginzburg–Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg–Landau m-armed spiral waves have been investigated extensively. However, many multi-armed spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

New computational optical solitons for generalized complex Ginzburg–Landau equation by collective variables

The purpose of the paper is to implement the collective variable method to investigate the generalized complex Ginzburg–Landau equation, which characterizes the kinetics of solitons in respect of pulse parameters for fiber optics. The statistical simulations of the interacting system of ordinary differential equations that reflect all the collective variables included in the pulse ansatz have been successfully carried out using a well-known numerical methodology, the fourth-order Runge–Kutta technique. The collective variable method is employed to plot the pulse variation characteristics as a function of propagation distance. The amplitude, temporal position, width, chirp, frequency, and phase of the pulse are all depicted against the propagated coordinate, where the width, phase of soliton, amplitude, and chirp all show a strong periodicity. The numerical dynamics of solitons have also been exhibited against varying values of pulse parameters to highlight differences in collective variables. Other key bits of the current investigation are also determined.

Thermodynamics of ordering and mixing in plagioclase feldspars: atomistic modelling in favour of Landau theory

Plagioclase feldspars are key components of terrestrial planets and deriving their phase diagram and activity–composition relations has been the subject of many studies. Three main approaches have been used for thermodynamic modelling of the behaviour of plagioclase feldspars, and all three disagree in terms of the magnitude and relative importance of mixing energies and ordering energies. The present study uses Monte Carlo simulations based on classical atomistic modelling to provide independent constraints over ordering and mixing for anorthite and its solid solution with albite. Unsurprisingly, Al-avoidance plays a key role. The competition between first-neighbour and second-neighbour interactions is highlighted, providing simple explanations to the high degree of short-range order in macroscopically disordered plagioclase. The mixing properties obtained in this study support strong non-ideality, with negative mixing enthalpy and temperature-dependent mixing entropy. Ordering energies are consistent with studies using Landau theory. Unfortunately, the form of the Landau equations hindered parametrisation of a full thermodynamic model from albite to anorthite, even though the thermodynamic formalism remains simple. Reasons, perspectives, and implications for the phase diagram of ternary feldspars are discussed.

Interplay of Fracture and Martensite Transformation in Microstructures: A Coupled Problem.

We are witnessing a tremendous transition towards a society powered by net-zero carbon emission energy, with a corresponding escalating reliance on functional materials (FM). In recent years, the application of FM in multiphysics environments has brought new challenges to the mechanics and materials research communities. The underlying mechanism in FM, which governs several fundamental characteristics, is known as martensitic phase transformation (MPT). When it comes to the application of FM in the multiphysics context, a thorough understanding of the interplay between MPT and fracture plays a crucial role in FM design and application. In the present work, a coupled problem of crack nucleation and propagation and multivariant stress-induced MPT in elastic materials is presented using a finite element method based on Khachaturyan’s microelasticity theory. The problem is established based on a phase-field (PF) approach, which includes the Ginzburg–Landau equations with advanced thermodynamic potential and the variational formulation of Griffith’s theory. Therefore, the model consists of a coupled system of the Ginzburg–Landau equations and the static elasticity equation, and it characterizes evolution of distributions of austenite and two martensitic variants as well as crack growth in terms of corresponding order parameters. The numerical results show that crack growth does not begin until MPT has grown almost completely through the microstructure. Subsequent to the initial formation of the martensite variants, the initial crack propagates in such a way that its path mainly depends on the feature of martensite variant formations, the orientation and direction upon which the martensite plates are aligned, and the stress concentration between martensite plates. In addition, crack propagation behavior and martensite variant evaluations for different lattice orientation angles are presented and discussed in-detail.

Stability of Traveling Oscillating Fronts in Complex Ginzburg Landau Equations

Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star (x,t) = e^{-i \omega t} V_\star(x - ct)$ with a profile $V_{\star}$ which decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = \alpha U_{xx} + G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at $+ \infty$. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at $+ \infty$. Using slowly decaying exponential weights, the framework allows to derive appropriate resolvent estimates, semigroup techniques, and Gronwall estimates.

Dissipative solitons stabilized by nonlinear gradients in one spatial dimension: From deterministic to stochastic aspects, and a perspective

The purpose of this article is twofold. Firstly, to investigate the formation of localized spatiotemporal chaos in the complex cubic Ginzburg–Landau equation including nonlinear gradient terms. We found a transition to spatiotemporal disorder via quasiperiodicity accompanied by the fact that incommensurate satellite peaks arise around the fundamental frequency and its harmonics. Secondly, we review the influence of multiplicative noise on stationary pulses stabilized by nonlinear gradients. Numerical simulations show surprising results that are explained analytically. We found that multiplicative noise can induce a velocity change of propagating dissipative solitons. This completes previous communications on the two issues addressed in O. Descalzi et al., (2019, 2021, 2022).

Probing multi-particle unitarity with the Landau equations

We consider the 2\to 22→2 scattering amplitude of identical massive particles. We identify the Landau curves in the multi-particle region 16m^2 \leq s, t < 36m^216m2≤s,t<36m2. We systematically generate and select the relevant graphs and numerically solve the associated Landau equations for the leading singularity. We find an infinite sequence of Landau curves that accumulates at finite ss and tt on the physical sheet. We expect that such accumulations are generic for s,t > 16m^2s,t>16m2. Our analysis sheds new light on the complicated analytic structure of nonperturbative relativistic scattering amplitudes.

Optical Solitons with the Complex Ginzburg–Landau Equation with Kudryashov’s Law of Refractive Index

In this paper, the optical solitons for the complex Ginzburg–Landau equation with Kudryashov’s law of refractive index are established. An improved modified extended tanh–function technique is used to extract numerous solutions. Bright and dark solitons, as well as singular soliton solutions, are achieved. In addition, as the modulus of ellipticity approaches unity or zero, solutions are formulated in terms of Jacobi’s elliptic functions, which provide solitons and periodic wave solutions.

Unconditional error analysis of a linearized BDF2 virtual element method for nonlinear Ginzburg–Landau equation with variable time step

We consider a virtual element method in space for the nonlinear Ginzburg–Landau equation, while a linearized time-variable-step second order backward differentiation formula (BDF2) is adopted in time. The error splitting approach is used to prove the unconditional optimal error estimate of the derived scheme under the mild restriction on the ratio of adjacent time-steps ratios (similarly proposed in Liao and Zhang 2021, Zhang and Zhao 2021). By using the techniques of the discrete complementary convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels, we obtain the boundedness and error estimates of the solution of time-discrete system. Moreover, the optimal convergence in L2-norm for the fully discrete scheme is finally derived. Numerical examples on a set of polygonal meshes are given to validate our theoretical results.

Frequency modes of unstable spiral waves in two-dimensional Rosenzweig–MacArthur ecological networks

The existence of two frequency regimes in a two-dimensional (2D) Rosenzweig–MacArthur ecological network is debated. The semi-discrete approximation differentiates the two regimes, each described by a 2D complex Ginzburg–Landau equation. Using the standard theory of the linear stability analysis, a generalized expression for the modulational instability growth rate is derived for each frequency mode. The parametric study of the growth rate of modulational instability reveals its sensitivity to the changes in the recruitment rate of the resources. Moreover, direct numerical simulations are carried out to confirm our analytical results. Over the prolonged evolution of the perturbed plane wave solution, the high-frequency mode entertains spiral wave patterns. In contrast, the appearance of target waves manifests the low-frequency regime. In that context, we further explore the impact of the recruitment rate of resources and give the qualitative meaning of the obtained dynamical behaviors and their ecological implications. This work may additionally provide more insight into the mechanism leading to spiral and target waves in environmental systems.

Finite-amplitude instability of the buoyancy boundary layer in a thermally stratified medium

The finite-amplitude instability of the buoyancy-driven boundary layer is considered on a vertical plate immersed in a thermally stratified ambient medium, where the wall and surrounding fluid have different temperature gradients. Although the linear stability in this configuration has been investigated, the finite-amplitude solution arising from the critical instability has been studied only for specific parameter values. We extend this by using the amplitude expansion method. The primary bifurcations to the two-dimensional least unstable mode for different temperature gradient ratios ( $0 \leqslant \lambda \leqslant 10$ ) and Prandtl numbers ( $10^{-1} \leqslant Pr \leqslant 10^{4}$ ) are investigated. Only supercritical bifurcations are found to occur when $0 \leqslant \lambda < 2$ and $Pr \leqslant 2800$ , while subcritical bifurcations are also found for larger values of temperature gradient ratio and Prandtl number. Analysis of the contribution of the nonlinear terms in the Landau coefficient reveals that the interaction of the modification of the mean flow and second harmonic for velocity with the fundamental mode for temperature plays an important role in subcritical bifurcation. Based on the Landau equation, the threshold amplitude of the nonlinear equilibrium solution is discussed as well. These encouraging results should be helpful for understanding such a buoyancy-driven flow system.