Derivative Ginzburg-Landau Equation Research Articles

Engineering dissipative chirped solitons in the cardiac tissue under electromagnetic induction

This work deals with the engineering dissipative chirped solitary waves in the cardiac tissue under electromagnetic induction. Employing the a two-dimensional Ginzburg-Landau (GL) equation describing the spatiotemporal evolution of transmembrane potential in cardiac cells under magnetic flow effect, we use the phase imprint technique to reduce the model equation into a derivative GL equation (alias chirping model equation) that allow us to investigate the dynamics of dissipative chirp solitary wave propagating through a cardiac tissue. The baseband modulational instability (MI) of the chirping model is investigated and the analytical expression of the MI growth rate is derived. In the baseband MI regime, we use the approximative solitonlike solution of the chirping model to investigate the dynamics of transmembrane potential in cardiac cells. We show that the frequency chirp associated to the solitary wave is directly proportional to the wave amplitude and the nonlinear chirping parameter can be used to modulate the chirping amplitude. Also, our results show that the approximate solution of the chirping model can be used to generate bright (dissipative) solitons, dark solitons, breather solitons and first-order rogue wave in the cardiac cell under electromagnetic induction. Our theoretical results are confirmed by numerical simulations. The results of this work can motivate the investigations of qualitative new phenomena in the presence of large dissipation.

Large deviations for the 2-D derivative Ginzburg–Landau equation with multiplicative noise

This paper proves a Wentzell–Freidlin type large deviation principle for the 2-D derivative Ginzburg–Landau equation perturbed by a small multiplicative noise, based on the Laplace principle and weak convergence approach.

Recurrent solutions of the derivative Ginzburg–Landau equation with boundary forces

ABSTRACTIn this paper, we will establish the existence of the bounded solutions, periodic solutions, quasi-periodic solutions and almost periodic solutions for the derivative Ginzburg–Landau equation with time-dependent boundary external forces. A smoothing effect for the semigroup associated with linear Ginzburg–Landau operator is the crucial tool in establishing the main results.

On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation

We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation(1/2⩽α⩽1)in any spatial dimensionn⩾1with rough initial data. For1/2<α⩽1, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spacesMp,11-2α(1⩽p⩽∞). Forα=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data inB˙∞,10(ℝn)∩M∞,10(ℝn). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroupe-a+it-Δαto overcome the derivative in the nonlinear term.

Regularity of attractor for 3D derivative Ginzburg-Landau equation

Regularity of attractor for 3D derivative Ginzburg-Landau equation

Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation ut=(ν+i)Δu+λ→1⋅∇(|u|2u)+(λ→2⋅∇u)|u|2+α|u|2δu, where δ∈N, λ→1,λ→2 are complex constant vectors, ν∈[0,1], α∈C. For n≥3, we show that it is uniformly global well posed for all ν∈[0,1] if initial data u0 in modulation space M2,1s and Sobolev spaces Hs+n/2 (s>3) and ‖u0‖L2 is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in C(0,T;L2) if ν→0 and u0 in M2,1s or Hs+n/2 with s>4. For n=2, we obtain the local well-posedness results and inviscid limit with the Cauchy data in M1,1s (s>3) and ‖u0‖L1≪1.

Exponential attractor of the 3D derivative Ginzburg-Landau equation

In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.

Large deviations for the stochastic derivative Ginzburg–Landau equation with multiplicative noise

This paper first proves the existence of a unique mild solution to the stochastic derivative Ginzburg–Landau equation. The fixed point theorem for the corresponding truncated equation is used as the main tool. Since we restrict our study to the one-dimensional case, it is not necessary to introduce another Banach space and thus the estimates of the stochastic convolutions in the Banach space are avoided. Secondly, we also consider large deviations for the stochastic derivative Ginzburg–Landau equation perturbed by a small noise. Since the underlying space considered is Polish, using the weak convergence approach, we establish a large deviations principle by proving a Laplace principle.

Attractors of derivative complex Ginzburg-Landau equation in unbounded domains

The Ginzburg-Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics, and chemistry. In this paper, the derivative complex Ginzburg-Landau (DCGL) equation in an unbounded domain Ω ⊂ ℝ2 is studied. We extend the Gagliardo-Nirenberg inequality to the weighted Sobolev spaces introduced by S. V. Zelik. Applied this Gagliardo-Nirenberg inequality of the weighted Sobolev spaces and based on the technique for the semi-linear system of parabolic equations which has been developed by M. A. Efendiev and S. V. Zelik, the global attractor Open image in new window in the corresponding phase space is constructed, the upper bound of its Kolmogorov’s ɛ-entropy is obtained, and the spatial chaos of the attractor Open image in new window for DCGL equation in ℝ2 is detailed studied.

Fourier spectral approximation to long-time behaviour of the derivative three-dimensional Ginzburg–Landau equation

In this paper, we consider a derivative Ginzburg–Landau equation with periodic initial-value condition in three-dimensional space. A fully discrete Galerkin–Fourier spectral approximation scheme is constructed, and then the dynamical behaviour of the discrete system is analysed. Firstly, the existence of global attractors A N τ of the discrete system are proved by a priori estimate of the discrete solution. Next, the convergence of approximate attractors is proved by error estimates of the discrete solution. Furthermore, the long-time convergence as N → ∞ and τ → 0 simultaneously as well as the numerical long-time stability of the discrete scheme are obtained.

Global Existence of Solutions to the Derivative 2D Ginzburg–Landau Equation

In this paper we study a complex derivative Ginzburg–Landau equation with two spatial variables (2D). We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial boundary value problem of the derivative 2D Ginzburg–Landau equation and improve the known results.

Exponential Attractors for the Generalized Ginzburg-Landau Equation

In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation

Weak solutions to the two-dimensional derivative Ginzburg-Landau equation

In this paper, we deal with the generalized derivative Ginzburg-Landau equation in two spatial dimensions, and obtain the existence of global weak solutions for this equation subject to periodic boundary conditions.

Global existence theory for the two-dimensional derivative Ginzburg-Landau equation

The generalized derivative Ginzburg-Landau equation in two spatial dimensions is discussed. The existence and uniqueness of global solution arc obtained by Galerkin method and bya priori estimates on the solution inH 1-norm andH 2-norm.

Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions

In the present paper , we show the Gevrey class regularity of solutions for the generalized Ginzburg-Landau equation in two spatial dimensions. We also introduce an approximate inertial manifold for this system.

Finite dimensional behaviour for the derivative Ginzburg-Landau equation in two spatial dimensions

In this paper, we study the long time behaviour of the solutions for the generalized derivative Ginzburg-Landau equation in two spatial dimensions. We show the existence of the global attractor for this equation and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor.