Landau Equation Research Articles

Ginzburg–Landau equations involving different effects and their solitary waves

Ginzburg–Landau equations describe a wide range of phenomena involving superconductivity, superfluidity, etc. In the present paper, Ginzburg–Landau equations involving distinct laws are considered, and as a consequence, their solitary waves in the presence of perturbation terms are formally derived using the Kudryashov method. The effect of Kerr and parabolic laws on the dynamics of solitary waves is examined in detail. The outcomes of the current paper present suitable ways to control the width and amplitude of solitary waves. The authors believe that the results reported in the current study will contribute significantly to studies related to Ginzburg–Landau equations with distinct laws.

Effects of final state interactions on Landau singularities

In certain kinematic and particle mass configurations, triangle singularities may lead to line-shapes which mimic the effects of resonances. This well-known effect is scrutinized here in the presence of final-state rescattering. The goal is achieved first by utilizing general arguments provided by Landau equations, and second by applying a modern scattering formalism with explicit two- and three-body unitarity.

A Particle Method for the Multispecies Landau Equation.

The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method (Carrillo et al. in J. Comput. Phys. 7:100066, 2020) has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies Bobylev-Krook-Wu (BKW) solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.

New physics opportunities in triangle singularity

In this paper, we show that loop-induced processes involving new physics particles can readily satisfy Landau Equation and trigger triangular singularities at high-energy colliders, leading to fully visible Standard Model final states. In addition, a t-channel triangular singularity can also occur during particle scattering processes that may extend to low momentum exchange. We discuss several typical scenarios in supersymmetric and extended Higgs models, and three types of final-state kinematic features at the collider. Four-particle vertices in new physics can play a significant role in high energy diagrams. We identify an “everything on shell” triangular singularity diagram only involving bosonic couplings, which has the potential to completely avoid large virtuality suppression. We further show that such a virtuality-free diagram is missing in the Standard Model at electroweak scale, and it becomes available in new physics models.

Analysis of a Crank–Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg–Landau equation

A fast element-free Galerkin (EFG) method is proposed in this paper for solving the nonlinear complex Ginzburg–Landau equation. A second-order accurate time semi-discrete system is presented by using the Crank–Nicolson scheme for the temporal discretization, and then a meshless fully discrete system is established by using the EFG method for the spatial discretization. In the proposed EFG method, Nitsche’s technique is used to impose the essential boundary conditions in a weak sense, and the reproducing kernel gradient smoothing integration is used to accelerate the calculation. Theoretical errors for the time semi-discrete system and the fully discrete EFG system are analyzed in detail. Optimal error estimates of the fully discrete Crank–Nicolson EFG method are obtained in both L2 and H1 norms. Numerical results validate the theoretical results and the effectiveness of the method.

Optical solitons of the (1+1)-dimensional perturbed complex Ginzburg–Landau equation having the Kerr law in the absence of the chromatic dispersion

Optical solitons of the (1+1)-dimensional perturbed complex Ginzburg–Landau equation having the Kerr law in the absence of the chromatic dispersion

Impact of magnetic field variation on thermally annealed tantalum-filled vertical superconducting interconnects for scalable quantum computing systems

Efforts to scale down advanced quantum processors with more numbers of qubits offer challenges since the integration of qubits is a significant obstacle in the path. Superconducting vertical interconnects in 3D IC integration can provide a crucial solution for wiring complexities, as shorter interconnections reduce energy loss when working in cryogenic conditions with better signal fidelity. Using the time-dependent Ginzburg–Landau equation, this simulation study analyzes vertical superconducting interconnects filled with materials like tantalum, niobium, and thermally annealed tantalum. It also investigates how London penetration depth (temperature-dependent) and magnetic fields affect Cooper pair density and, consequently, current density.

Effect of grain size and grain boundary on the energy storage performance of polycrystalline ferroelectrics

Dielectric capacitors based on polycrystalline ferroelectrics have attracted much attention due to their significant power density and fast charge–discharge speed. The energy storage performance of polycrystalline ferroelectrics is highly dependent on the grain size and grain boundary. Here, the effect of grain size and grain boundary on the domain structures and polarization–electric field (P–E) hysteresis loops of polycrystalline ferroelectrics are investigated by using a phase-field model based on the time dependent Ginzburg–Landau (TDGL) equation. It is found that the depolarization field in the grain boundary induces the vortex domain when the grain size is reduced or the grain boundary thickness increases in certain extent, resulting in slender P–E loops, which contributes to an improvement in the energy storage efficiency and density. However, as the grain size further decreases or the grain boundary thickness further increases, the energy storage density decreases, which is attributed to the concurrent reduction in both the remnant and saturation polarizations. This study provides a considerable insight for optimizing the energy storage performance by carefully adjusting the grain size and grain boundary thickness in polycrystalline ferroelectrics.

Correction: Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Correction: Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Dynamics of N-elastically longitudinal coupled rotating pendulums with smooth and discontinuous nonlinearities: Generation of chaotic bursting with many orbits as a solution

The dynamics of a mechanical network, consisting of a certain number (N) of cells, in which each unit cell consists of the irrational coupling of a simple pendulum and inclined mass-spring oscillator, is investigated. The single cell was recently introduced by Ning Han and Qingjie Cao, and it has strong irrational nonlinearity, exhibiting smooth and discontinuous characteristics due to the geometry configuration. The obtained network has a large degree of freedom and consequently has richer dynamics as compared to a single one. The set of discrete equations of this network is found, while the first equation is driven by an external sinusoidal force; the variables here are the angles between the vertical direction and the directions of mass in each unit cell. By using the non-driven version of this set of equations, the equilibrium points are found as well as their stability using the Jacobian matrix. Then, by using the multiple time scale method, the solutions of the set of the network equations are found for weak amplitude oscillations of the system, while the large amplitude solutions are found numerically, showing the ability of the network to transform the shape of some solutions at the input to other forms as the signals evolve in the network. One notices the generation of simple chaotic signals as well as the train of impulse signals. What is important here is the generation of the chaotic bursting with many orbits by the system, which are the simultaneous oscillations around several fixed points. For the case of numerous cells, the system can be used as a waveguide; this is why the set of the equation of the network is reduced to the extended complex Ginzburg Landau (ECGL) equation, with complex nonlinear dispersion and nonlocality terms, using the Tanuiti's reduction perturbation methods. The dissipative dark compacton and peakon as solutions for the ECGL equation are then found. The input-output matching of the network is next investigated via the transfer function of the system near the equilibrium points. The inverse Fourier transform of the transfer function multiplied by the Fourier transform of the input sinusoidal force gives the solution of the network equation for low amplitude oscillations. Finally, the supratransmission condition is studied, leading to the fact that the input sinusoidal signal with frequency nearly inside the forbidden gap zone gradually transforms into the train of chaotic bursting.

An investigation of rapid surface melting in nanowires

The investigation into virtual melting phenomena in nanowires holds significant relevance owing to its profound impact on material durability under extreme loading conditions. Thus, the exploration of this pivotal plastic deformation mechanism is undertaken utilizing the phase-field methodology. Employing a monolithic solver, we solve the coupled highly nonlinear time-dependent Ginzburg–Landau equation and dynamic elasticity equation. Our analysis encompasses the consideration of surface tension stress in conjunction with a coherent solid–liquid interface subjected to uniaxial transformation strain, thereby unveiling intriguing facets of melting phenomena. The investigation delves into the influence of transformation strain, kinetic coefficient, and temperature on the thickness of the solid–liquid interface and its corresponding velocity. This analysis is conducted through meticulous comparison with existing experimental data and analytical solutions. Moreover, employing the phase-field method yields precise descriptions of the system kinetics, capturing virtual melting phenomena in both pristine and flawed nanowire configurations.

Non-Hermitian mode management in periodically modulated waveguide amplifiers

We propose an efficient mode management scheme in active nonlinear multimode fibers based on non-Hermitian potentials in the longitudinal direction with an antisymmetric transverse profile. The proposal takes advantage of the nonlinear saturation toward particular mode configurations, which can be tuned by the non-Hermitian potential. We demonstrate flexible control of the beam profile within the parameter space of the applied potential with various possibilities, such as improving the beam quality by condensing photons to the lowest order Hermite mode, exciting higher order modes, or engineering a desired mode profile as a combination of modes. The effect is also analytically predicted using a mode-expansion approach, showing good agreement with the full model calculations based on the complex Ginzburg–Landau equation. This study was performed for 1D planar guiding structures, yet the results can be extended to 2D fibers and could be very useful in applications of fiber amplifiers and lasers.

New Coupled Optical Solitons to Birefringent Fibers for Complex Ginzburg–Landau Equations with Hamiltonian Perturbations and Kerr Law Nonlinearity

In this study, we use an analytical method tailored for the in-depth exploration of coupled nonlinear partial differential equations (NLPDEs), with a primary focus on the dynamics of solitons. Traditional methods are quite effective for solving individual nonlinear partial differential equations (NLPDEs). However, their performance diminishes notably when addressing systems of coupled NLPDEs. This decline in effectiveness is mainly due to the complex interaction terms that arise in these coupled systems. Commonly, researchers have attempted to simplify coupled NLPDEs into single equations by imposing proportional relationships between various solutions. Unfortunately, this simplification often leads to a significant deviation from the true physical phenomena that these equations aim to describe. Our approach is distinctively advantageous in its straightforwardness and precision, offering a clearer and more insightful analytical perspective for examining coupled NLPDEs. It is capable of concurrently facilitating the propagation of different soliton types in two distinct systems through a single process. It also supports the spontaneous emergence of similar solitons in both systems with minimal restrictions. It has been extensively used to investigate a wide array of new coupled progressive solitons in birefringent fibers, specifically for complex Ginzburg–Landau Equations (CGLEs) involving Hamiltonian perturbations and Kerr law nonlinearity. The resulting solitons, with comprehensive 2D and 3D visualizations, showcase a variety of coupled soliton configurations, including several that are unprecedented in the field. This innovative approach not only addresses a significant gap in existing methodologies but also broadens the horizons for future research in optical communications and related disciplines.

Local regularity for the space-homogenous Landau equation with very soft potentials

Local regularity for the space-homogenous Landau equation with very soft potentials

Towards the Construction of an Analog Solver for the Schrödinger and Ginzburg–Landau Equations Based on a Transmission Line

Towards the Construction of an Analog Solver for the Schrödinger and Ginzburg–Landau Equations Based on a Transmission Line

Isolated Spiral Solutions of the Ginzburg–Landau Equation

Using finite element simulation, we study the main features of rotating wave solutions of the cubic complex Ginzburg–Landau equation. To focus on the characteristics of the waves themselves, we have used a circular domain to avoid the effects of irregular boundaries; we have inhibited the formation of defects using an Archimedean wave centered at the origin as the initial condition, and we have chosen a domain size long enough to contain several spiral arms but not so long as to promote long-wave instabilities. This allows us to focus on the geometric features of the solutions often overlooked in traditional works with random initial conditions in large domains. We show with our simulations that the convective and absolute stabilities differ from those predicted for plane waves. Likewise, we show that the appearance of spirals and anti-spirals can occur in any of the quadrants of the parameter space and depends fundamentally on the initial conditions. Finally, regarding the stability of spiral waves against sideband disturbances, we show that the persistence of two-dimensional spiral waves, if fulfilled, does not correspond to the Eckhaus criterion. Our simulations allow us to corroborate the idea that two-dimensional spirals depend on the size of the domain and that they require new analytical results whose trends are identified in this work.

Instability, bifurcation and nonlinear dynamics of Poiseuille flow in fluid overlying an anisotropic and inhomogeneous porous domain

The present study focuses on the finite amplitude analysis of Poiseuille flow in an anisotropic and inhomogeneous porous domain that underlies a fluid domain. The nonlinear interactions are studied by imposing finite amplitude disturbances to the Poiseuille flow. The former interactions in terms of modal amplitudes dictate the fundamental mode, the distorted mean flow, the second harmonic and the distorted fundamental mode. The harmonics are solved progressively in increasing order of the least stable mode obtained from the linear theory to ascertain the cubic Landau equation, which in turn helps to determine the bifurcation phenomena. The presented weakly nonlinear theory predicts the existence of subcritical transition to turbulence of Poiseuille flow in such superposed systems. In general, on moving away from the bifurcation point, it is found that a decrease in the value of inhomogeneity (in terms of Ai), Darcy number (δ) and an increase in the value of depth ratio (dˆ; the ratio of fluid domain thickness to that of porous domain) favours subcritical bifurcation. For the considered variation of parameters, the bifurcation, either subcritical or supercritical, remains the same irrespective of the value of media anisotropy (ξ) in the vicinity of the bifurcation point except for dˆ=0.2,Ai=1. In such a situation, subcritical (supercritical) bifurcation is witnessed for ξ=0.001,0.01,0.1 (1,3). Furthermore, in contrast to isotropic and homogeneous porous media, both subcritical and supercritical bifurcations are observed when moving away from the bifurcation point. A correspondence between the type of mode via linear theory and the type of bifurcation via nonlinear theory is witnessed, which is further affirmed by the secondary flow patterns. Finally, the presented theoretical results reveal an early onset of subcritical transition to turbulence in comparison with isotropic and homogeneous porous media.

Efficient simulation of complex Ginzburg–Landau equations using high-order exponential-type methods

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies.

Global Existence for Long Wave Hopf Unstable Spatially Extended Systems with a Conservation Law

AbstractWe are interested in reaction–diffusion systems, with a conservation law, exhibiting a Hopf bifurcation at the spatial wave number $$ k = 0 $$ k = 0 . With the help of a multiple scaling perturbation ansatz a Ginzburg–Landau equation coupled to a scalar conservation law can be derived as an amplitude system for the approximate description of the dynamics of the original reaction–diffusion system near the first instability. We use the amplitude system to show the global existence of all solutions starting in a small neighborhood of the weakly unstable ground state for original systems posed on a large spatial interval with periodic boundary conditions.

Dynamical large deviations for long-range interacting inhomogeneous systems without collective effects.

We consider the long-term evolution of a spatially inhomogeneous long-range interacting N-body system. Placing ourselves in the dynamically hot limit, i.e., assuming that the system only weakly amplifies perturbations, we derive a large deviation principle for the system's empirical angle-averaged distribution function. This result extends the classical ensemble-averaged kinetic theory given by the so-called inhomogeneous Landau equation, as it specifies the probability of typical and large dynamical fluctuations. We detail the main properties of the associated large deviation Hamiltonian, particularly focusing on how it complies with the system's conservation laws and possesses a gradient structure.