On L 2 -decay of weak solutions to a class of fractional complex Ginzburg–Landau equations

The recent findings confirm that bacteria communicate each other through chemical and electrical signals. Bacteria use chemical signaling molecules which are called as quorum sensing molecules( QSMs) or autoinducers. Moreover, the ion channels in bacteria conduct a long-range electrical signaling within biofilm communities through propagated waves of potassium ions and biofilms attracts other bacterial species too. Both communication process are used by bacteria to make their own survival strategies. In this article, we model this bacterial communication mechanism by complex Ginzburg- Landau equation and discuss the formation of patterns depending on kinematic viscosity associated with internal noise. Again, the potassium wave propagation is described by the non-linear Schrodinger equation in a dissipative environment. By adding perturbation to non-linear Schrodinger equation one arrives at Complex Ginzburg-Landau equation. In this paper we emphasize that at the cellular level(bacteria) we use Complex Ginzburg - Landau equation as a perturbed Non- linear Schrodinger equation to understand the bacterial communication as well as pattern formation in Biofilms for certain range of kinematic viscosity which can be tested in laboratory experiment. Here, the perturbation is due to the existence of non thermal fluctuations associated to the finite size of the bacteria. It sheds new light on the relevance of quantum formalism in understanding the cell to cell communication.

We present analytical methods whereby weak and strong turbulence are predicted in the D dimensional complex Ginzburg Landau (CGL) equation \(A_{t}=RA+(1+i\nu)\Delta A-(1+i\mu)A\vert A\vert^{2q}\) on a periodic domain [0,1]. Strong (hard) turbulence is characterised by large fluctuations away from space & time averages while no such fluctuations occur in weak (soft) turbulence. In the Δ-ν plane, there are different areas where weak & strong behaviour can occur. In the strong case (Δ, ν → ±∞, ‡∞), the corresponding areas go out to the inviscid limit where the CGL equation becomes the NLS equation in which a finite time singularity occurs when qD ≥ 2. A new infinite set of differential inequalities for the “lattice” of functionals \(F_{n,m}=\int\left[{\vert\nabla^{n-1}A\vert}^{2m}+\alpha_{n,m}\vert A\vert^{2m\left[q(n-1)+1\right]}\right]d\underline{x}\) enables us to construct large time upper bounds on the F n,m . The occurrence of strong spiky turbulence is predicted for qD = 2 by showing that exponents of R in the upper bounds of F n,m & ‖A‖∞ in the strong regions are dependent on the quantity |ν| which gets large in the inviscid limit. The critical value qD = 2 plays an important role: when qD > 2 the CGL equation has some similarities with the 3D Navier equations. A comparison is made between the two & the possibility of having a 1D system which mimics some limited features of the Navier Stokes equations is discussed.

Lower and upper bounds of temporal decay for solutions to [formula omitted]-dimensional hyperviscous Navier–Stokes equations

This paper examines the bifurcation and stability of the solutionsof the complex Ginzburg--Landau equation(CGLE). The structure of thebifurcated solutions shall be explored as well. We investigate twodifferent modes of the CGLE. The first mode of the CGLE containsonly an unstable cubic term and the second mode contains not only acubic term but a quintic term. The solutions of the cubic CGLEbifurcate from the trivial solution to an attractor supercriticallyin some parameter range. However, for the cubic-quintic CGLE, asubcritical bifurcation is obtained. Due to the global attractor, weobtain a saddle node bifurcation point $\lambda_c$. By thoroughlyinvestigating the structure and transition of the solutions of theCGLE, we confirm that the bifurcated solutions are homeomorphic to$S^1$ and contain steady state solutions.

We study the one-dimensional supercritical real and complex subcritical Ginzburg- Landau equations. In the real case, with periodic boundary conditions in a finite box, we construct analytically nucleation solutions allowing transitions between stable plane waves. Moreover we construct vortex solutions and study the Eckhaus bifurcation in a finite box. The Lyapounov functional for the above stationary solutions is also discussed. In the complex case we study stationary localized solutions in the region where there exist coexistence of homogeneous attractors. Using a matching approach we construct analytically pulses and show that their appearance is related to a saddle-node bifurcation. Our approximation scheme presented here is valid through the whole intermediate range of parameters between the variational and conservative limit. We perform numerical simulations which are in good agreement with our theoretical predictions. This review is based on the original works [42–44]. KeywordsPlane WaveContinuous LineBifurcation CurveLandau EquationPhase GradientThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows

We show that solutions of the initial boundary value problem for the coupled system of Kuramoto-Sivashinsky and Ginzburg-Landau (KS–GL) equations continuously depend on parameters of the system, and under some restrictions on parameters all solutions of initial boundary value problem for KS–GL equations tend to zero as with an exponential rate.

In this paper, complex Ginzburg-Landau (CGL) equations governed by p-Laplacian are studied. We discuss the global existence of solutions for the initial-boundary value problem of the equation in general domains. The global solvability of the initialboundary value problem for the case when p = 2 is already examined by several authors provided that parameters appearing in CGL equations satisfy a suitable condition. Our approach to CGL equations is based on the theory of parabolic equations with nonmonotone perturbations. By using this method together with some approximate procedure and a diagonal argument, the global solvability is shown without assuming any growth conditions on the nonlinear terms.

On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

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.

Dynamics of defects in the vector complex Ginzburg–Landau equation

It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|σ−2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 < σ < 2 and exponentially decay in time if σ = 2. In the special case of a suitable force field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided 1 < σ < 2. We also prove that for non-zero body forces decaying at a power-time rate, the solutions decay at analogous power-time rates if σ > 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.

Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms

Solitary waves of the complex Ginzburg-Landau equation with anti-cubic nonlinearity

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg–Landau equation (CGL), [Formula: see text] with [Formula: see text], are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when [Formula: see text] and [Formula: see text] large enough.