Landau Equation Research Articles

Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

Abstract In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.

Mega riverbed-patterns: linear and weakly nonlinear perspectives.

In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre–manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart–Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.

Odd-viscosity-induced instability of a thin film with variable density

The stability of a two-dimensional gravity-driven thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a uniformly heated inclined plane is discussed. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow. A theoretical model is implemented, which captures the dependence of the surface tension on temperature, and the model also allows for variation in the density of the liquid with a thermal difference. The coupled effect of odd viscosity, variable density, and surface tension has been investigated both analytically and numerically. A nonlinear evolution equation of the free surface is derived by the method of systematic asymptotic expansion. A linear stability analysis is carried out, which yields the critical conditions for the onset of instability in long-wave perturbations. New interesting results illustrating how the critical Reynolds number depends on the odd viscosity as well as other various dimensionless parameters have been obtained. In addition, a weakly nonlinear stability analysis is performed based on the method of multiple scales from which a complex Ginzburg–Landau equation is obtained. It is observed that the film not only has supercritical stable and subcritical unstable zones, but also unconditional stable and explosive zones. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. Employing the Crank–Nicolson method in a periodic domain, the spatiotemporal evolution of the model has been analyzed numerically for different values of the odd viscosity as well as other flow parameters. Nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The results are conducive to the further development of related experiments.

Dynamics of modulated waves and localized energy in a Burridge and Knopoff model of earthquake with velocity-dependant and hydrodynamics friction forces

We investigate both analytically and numerically the dynamics of modulated waves in a modified Burridge and Knopoff mechanical model of earthquake. We take into account the velocity-dependent and hydrodynamics friction forces between the rock blocks of the system. The theoretical framework for the analysis is presented using the multiple scale expansion method. We show that the system can be governed by the complex Ginzburg–Landau equation, which admits soliton solutions. The modulational instability of the system is performed, showing that the bandwidth of instability increases with the friction forces parameter. The planar wave solution used as the initial condition leads to more unstable waves in the system with chaotic-like behaviour, allowing energy localization. It is also found that the wave parameters, namely, the frequency, magnitude and velocity (as well as the energy of the system) strongly depend on η. In the non-friction limit, the obtained waves propagate with a coherent behaviour for a very short time, before disappearing. Our results suggest that the damages caused by an earthquake can be important as the friction forces increase and may excite the possibility of relative predictions for the dynamics of waves propagating in the earth in the field of nonlinear science.

The Study of the Numerical Treatment of the Real Ginzburg–Landau Equation Using the Galerkin Method

The real Ginzburg–Landau equation is studied in this article. We proceed and use the Galerkin method in combination with the compactness theorem to show that the solution of this problem exists and is unique in appropriate Sobolev spaces. This is proceeded by the designing of a reliable nonlinear numerical scheme from the afore mentioned problem and further show that this scheme is stable. Furthermore, the optimal rate of convergence of the scheme is determined in some appropriate spaces with emphasizes on the fact that the numerical solution from this scheme preserves all the qualitative properties of the exact solution and the numerical experiments are conducted with the help of an example to justify the theory.

Sensitive behavior and optical solitons of complex fractional Ginzburg–Landau equation: A comparative paradigm

This article obtains the optical solitons of the complex fractional Ginzburg–Landau equation by the hypothesis of traveling wave and generalized projective Riccati equation scheme. There are four conditions, Kerr law, parabolic law, power law and dual power law of nonlinearity associated with the model. The constraint conditions for the existence of these solutions have also been discussed. Moreover, the physical significance of the constructed solutions has been provided using graphical representation. A comparative study is made by using two distinct definitions of fractional derivatives namely as Beta and M-truncated. Furthermore, a quantitative overview is also included, which involves solutions to the model under discussion. The complex Ginzburg–Landau equation is subjected to a comprehensive sensitivity analysis. Finally, the modulation instability (MI) analysis of proposed model is also carried out on the basis of linear stability analysis. A dispersion relation is obtained between the wave number and frequency.

New optical solitons for complex Ginzburg–Landau equation with beta derivatives via two integration algorithms

New optical solitons for complex Ginzburg–Landau equation with beta derivatives via two integration algorithms

Novel vortex structures in a three-dimensional superconductor under a helical magnetic field from a chiral helimagnet

We have investigated vortex structures in three-dimensional superconductors under a helical magnetic field from a chiral helimagnet numerically. In order to obtain vortex structures, we solve the three-dimensional Ginzburg–Landau equations with the finite element method. A spatial distribution of the helical magnetic field corresponds to a spatial distribution of magnetic moments in the chiral helimagnet. Then, the magnetic field is parallel to a yz-plane and helical rotation along a helical axis (x-axis). An arrangement of the vortex is different from the spatial oscillation in each component of the magnetic field. In this calculation, two-types of vortex structures appear. One is an appearance of vortices perpendicular to a xy-plane and a zx-plane of the superconductor. The other arrangement is that vortices are tilted toward the helical axis, although the magnetic field along the helical axis is zero. The arrangement of the tilted vortex depends on a direction of a rotation in the helical magnetic field.

Nonlinear stability of fast invading fronts in a Ginzburg–Landau equation with an additional conservation law

We consider traveling front solutions connecting an invading state to an unstable ground state in a Ginzburg–Landau equation with an additional conservation law. This system appears as the generic amplitude equation for Turing pattern forming systems admitting a conservation law structure such as the Bénard–Marangoni problem. We prove the nonlinear stability of sufficiently fast fronts with respect to perturbations which are exponentially localized ahead of the front. The proof is based on the use of exponential weights ahead of the front to stabilize the ground state. The main challenges are the lack of a comparison principle and the fact that the invading state is only diffusively stable, i.e. perturbations of the invading state decay polynomially in time.

Asymptotic behaviour of non-autonomous discrete complex Ginzburg–Landau equations driven by nonlinear noise

In this paper, we consider dynamical behaviour for stochastic discrete complex Ginzburg–Landau equations driven by locally Lipschitz nonlinear noise. We prove the existence and uniqueness of solutions. Thus the solution operators generate mean random dynamical systems. Finally, the existence and uniqueness for weak pullback random attractors in are established.

Exploration of anisotropy on nonlinear stability of thermohaline viscoelastic porous convection

The nonlinear stability of thermohaline convection in a horizontal layer of an anisotropic porous medium saturated with an Oldroyd-B fluid is studied using a modified perturbation method. Anisotropy in permeability, thermal and solute diffusivities is modelled as second-rank tensors. The cubic Landau equations for steady and oscillatory bifurcating solutions are derived. It is apparent that subcritical instability is possible depending on the choices of governing parameters indicating the linear stability theory is not sufficient to capture the onset of convection. The effect of strain retardation parameter, solute Darcy-Rayleigh number, thermal anisotropy parameter as well as solute anisotropy parameter is to delay, while the effect of stress relaxation parameter, Lewis number and the mechanical anisotropy parameter is to hasten the onset of oscillatory convection. It is found that the stationary bifurcating solution is subcritical while that of oscillatory is supercritical at higher values of solute Darcy-Rayleigh number. The results of Maxwell fluid are obtained as a limiting case. Heat and mass transports are estimated using Nusselt number and Sherwood number, respectively and by tuning the anisotropy parameters it is possible to control the same.

On trigonometric cosine, square, sawtooth, and triangular wave‐type rotational modulations on triple‐diffusive convection in salted water

AbstractThe effect of sinusoidal (trigonometric cosine) and nonsinusoidal (square, sawtooth, and triangular) rotational modulations on triple‐diffusive convection in a Newtonian fluid is studied in this paper. The derived Ginzburg–Landau equation for the stationary mode of convection is used to quantify heat and mass transport. The enhancement of heat and mass transport in water due to the addition of different salts is explained in terms of the thermophysical properties of the fluid and the aqueous solutes. The flow equations are reduced to a Lorenz‐type scheme of nonlinear evolution equations using a truncated Fourier series. The equations describing the growth of convection amplitudes are solved using the multidomain spectral collocation method. With the aid of thermophysical properties of the fluid, numerical computations are performed for different values of the revised Rayleigh number (), Taylor number (), the amplitude of modulation , and modulation frequency . A comparison of heat and mass transport in different combinations of aqueous solutions is made. The study reveals that all modes of rotational modulation can control the rate of heat and mass transport. It is noted that amongst all other modulations, the square wave modulation is the most destabilizing one.

Preconditioned method for the nonlinear complex Ginzburg–Landau equations

Preconditioned method for the nonlinear complex Ginzburg–Landau equations

Asymptotic behavior of random coupled Ginzburg\u2013Landau equation driven by colored noise on unbounded domains

In this paper, a random coupled Ginzburg–Landau equation driven by colored noise on unbounded domains is considered, in which the nonlinear term satisfies a local Lipschitz condition. It is shown that the random attractor of such a coupled Ginzburg–Landau equation is a singleton set, and the components of solutions are very close when the coupling parameter becomes large enough.

Interfacial phase change effect on a viscous falling film having odd viscosity down an inclined plane

We investigate the phase change effect at the thin film interface with broken time-reversal-symmetry falling down an inclined flat wall. We consider the impact of evaporation and as well as condensation at the fluid-vapor interface separately. The non-zero odd part of the Cauchy stress tensor with the odd viscosity coefficient, which arises due to broken-time-reversal-symmetry, adds an attractive characteristic in the flow by stabilizing the effect of evaporation/condensation. We study the long-wave instabilities of the ununiform film by deriving a non-linear evolution equation in the classical long wave expansion method framework. The one-equation model can track the free surface evolution and accounts for viscosity, gravity, surface tension, and phase change (evaporation/condensation) effects. Linear stability analysis confirms that the odd viscosity has no significant impact on the instability threshold of an evaporating film as a contrast of condensate film, where the critical Reynolds number increases with the odd-viscosity parameter μ. While studying the weakly non-linear waves, we use the method of multiple scales to obtain the Complex Ginzburg Landau equation that shows both supercritical and subcritical solutions are possible for evaporating and condensate film. Interestingly, while one subcritical region is visible for an evaporating film, two subcritical unstable regions are found for condensate film. The numerical solution of the free-surface equation illustrates the finite-amplitude behavior that tends to dry out for evaporating film and rapidly increase film thickness for condensate film. The presence of odd viscosity slows down both the film thinning process for evaporation and the film thickening process for condensation. Furthermore, odd-viscosity cannot prevent film rupture in case of evaporation.

High-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian: Discretization and simulations

In this paper, we propose a three-level linearized finite difference scheme for the high-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian. The Crank–Nicolson scheme is used for time discretization, and the fractional Laplacian is discretized by the fractional centered difference scheme. The proposed difference scheme (i.e. a linear system) can be solved efficiently by fast Fourier transform (FFT) and complex conjugate gradient method, since the coefficient matrix is a multi-level block Toeplitz matrix. Furthermore, we analyze the unique solvability and boundedness of solution of the difference scheme by the discrete energy method. It is also proved that the difference scheme is unconditionally stable and second-order accurate in time and space with respect to l∞-norm. Finally, several numerical examples are provided to validate the theoretical results.

Stability of Vacuum for the Boltzmann Equation with Moderately Soft Potentials

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter $s\in (0,1)$, i.e. with $\gamma+2s\in(0,2]$ on the whole space $\mathbb{R}^3$. We prove that if the initial data $f_{\text{in}}$ are close to the vacuum solution $f_{\text{vac}}=0$ in an appropriate weighted norm then the solution $f$ remains regular globally in time and approaches a solution to a linear transport equation. Our proof uses $L^2$ estimates and we prove a multitude of new estimates involving the Boltzmann kernel without angular cut-off. Moreover, we rely on various previous works including those of Gressman--Strain, Henderson--Snelson--Tarfulea and Silverstre. From the point of view of the long time behavior we treat the Boltzmann collisional operator perturbatively. Thus an important challenge of this problem is to exploit the dispersive properties of the transport operator to prove integrable time decay of the collisional operator. This requires the most care and to successfully overcome this difficulty we draw inspiration from Luk's work [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11] and that of Smulevici [Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2(2):Art. 11, 55, 2016]. In particular, to get at least integrable time decay we need to consolidate the decay coming from the space-time weights and the decay coming from commuting vector fields.

New optical soliton solutions via generalized Kudryashov’s scheme for Ginzburg–Landau equation in fractal order

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

Landau equation for self-gravitating classical and quantum particles: application to dark matter

We develop the kinetic theory of classical and quantum particles (fermions and bosons) in gravitational interaction. The kinetic theory of quantum particles may have applications in the context of dark matter. For simplicity, we consider an infinite and spatially homogeneous system (or make a local approximation) and neglect collective effects. This leads to the quantum Landau equation derived heuristically in (Physica A 332: 89, 2004). We establish its main properties: conservation laws, H-theorem, equilibrium state, relaxation time, quantum diffusion and friction coefficients, quantum Rosenbluth potentials, self-consistent evolution, (thermal) bath approximation, quantum Fokker–Planck equation, quantum King model $$\ldots $$ For bosonic particles, the Landau equation can describe the process of Bose–Einstein condensation. We discuss the relation of our study with the works of Levkov et al. (Phys Rev Lett 121: 151301, 2018) and Bar-Or et al. (Astrophys J 871: 28, 2019) on fuzzy dark matter halos and the formation of Bose stars and solitons.

Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

We consider a homogeneous plasma composed of $N$ particles of the same electric charge which interact through a Coulomb potential. In the large plasma parameter limit, classical kinetic theories justify that the empirical density is the solution of the Balescu-Guernsey-Lenard equation, at leading order. This is a law of large numbers. The Balescu-Guernsey-Lenard equation is approximated by the Landau equation for scales much smaller than the Debye length. In order to describe typical and rare fluctuations, we compute for the first time a large deviation principle for dynamical paths of the empirical density, within the Landau approximation. We obtain a large deviation Hamiltonian that describes fluctuations and rare excursions of the empirical density, in the large plasma parameter limit. We obtain this large deviation Hamiltonian either from the Boltzmann large deviation Hamiltonian in the grazing collision limit, or directly from the dynamics, extending the classical kinetic theory for plasmas within the Landau approximation. We also derive the large deviation Hamiltonian for the empirical density of $N$ particles, each of which is governed by a Markov process, and coupled in a mean field way. We explain that the plasma large deviation Hamiltonian is not the one of $N$ particles coupled in a mean-field way.