Complex Ginzburg Landau Type Equation Research Articles

Existence and decay estimates of solutions to complex Ginzburg–Landau type equations

This paper deals with the initial-boundary value problem (denoted by (CGL)) for the complex Ginzburg–Landau type equation ∂u∂t−(λ+iα)Δu+(κ+iβ)|u|q−1u−γu=0 with initial data u0∈Lp(Ω) in the case 1<q<1+2p/N, where Ω is bounded or unbounded in RN, λ>0, α,β,γ,κ∈R. There are a lot of studies on local and global existence of solutions to (CGL) including the physically relevant case q=3 and κ>0. This paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view. The physically relevant case can be considered as a special case of the results. Moreover, in the case κ<0, local and global existence of solutions with the decay estimate ‖u(t)‖Lp(Ω)≤c1e−c2t (c1,c2 are positive constants) is obtained under some conditions. The key to the local existence is to construct a semigroup {et[(λ+iα)Δ]} and its Lp-Lq estimate. On the other hand, the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with Re〈|v|p−2v,−(λ+iα)Δv〉.

Approximate solutions for half-dark solitons in spinor non-equilibrium Polariton condensates

In this work I generalize and apply an analytical approximation to analyze 1D states of non-equilibrium spinor polariton Bose–Einstein condensates (BEC). Solutions for the condensate wave functions carrying black solitons and half-dark solitons are presented. The derivation is based on the non-conservative Lagrangian formalism for complex Ginzburg–Landau type equations (cGLE), which provides ordinary differential equations for the parameters of the dark soliton solutions in their dynamic environment. Explicit expressions for the stationary dark soliton solution are stated. Subsequently the method is extended to spin sensitive polariton condensates, which yields ordinary differential equations for the parameters of half-dark solitons. Finally a stationary case with explicit expressions for half-dark solitons is presented.

Pullback attractors for the non-autonomous complex Ginzburg–Landau type equation with p-Laplacian

In this paper, we are concerned with the long-time behavior of the non-autonomous complex Ginzburg–Landau type equation with p-Laplacian. We first prove the existence of pullback absorbing sets in L2(Ω)∩W01,p(Ω)∩Lq(Ω) for the process {U(t,τ)}t⩾τ corresponding to the non-autonomous complex Ginzburg–Landau type equation with p-Laplacian. Next, the existence of a pullback attractor in L2(Ω) is established by the Sobolev compactness embedding theorem. Finally, we prove the existence of a pullback attractor in W01,p(Ω) for the process {U(t,τ)}t⩾τ by asymptotic a priori estimates.

Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data

This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $$ \dfrac {\partial u}{\partial t} -(\lambda +{\rm i} \alpha )\Delta u +(\kappa +{\rm i} \beta )|u|^{q-1}u-\gamma u=0 $$ in $\mathbb {R}^{N}\times (0,\infty )$ with $L^{p}$-initial data $u_{0}$ in the subcritical case ($1\leq q 0$, $p>1$, ${\rm i} =\sqrt {-1}$ and $N\in \mathbb {N}$. The proof is based on the $L^{p}$-$L^{q}$ estimates of the linear semigroup $\{\exp (t(\lambda +{\rm i} \alpha )\Delta )\}$ and usual fixed-point argument.

Dissipative ion-acoustic solitary and shock waves in a plasma with superthermal electrons

The excitation of a nonlinear ion-acoustic (IA) wave packet in a viscous plasma consisting of superthermal electrons and cold ions is investigated. It is assumed that the superthermal components obey the Kappa distribution. The standard reductive perturbation technique is employed to obtain a cubic complex Ginzburg–Landau type equation, which governs the dynamics of the amplitude of IA wave packets in a viscous plasma. In a weakly dissipative regime, by means of a perturbation method adopted from non-integrable dynamical systems theory, explicit soliton solutions are obtained of the dissipative dynamical system, determined as fixed points of a two-dimensional system of evolution equations describing the soliton amplitude and velocity. The stability of bright solitons is addressed using adiabatic perturbation theory in a weakly dissipative case. In a general dissipation regime, a novel type of travelling solitary (dissipative solitons) and shock-type solutions are derived for an envelope equation. Moreover, the dependence of the solitary and shock wave characteristics on relevant physical parameters of the problem is studied. It is shown that dissipative solitons do exist, sustained by a mutual balance between loss and gain terms, in addition to the interplay between dispersion and nonlinearity. The dependence of solitary and shock wave velocities on the parameters of the system are investigated. To feature distinctive characteristics of a dissipative soliton, it is compared with the bright soliton solution of the nonlinear Schrödinger equation.

Monotonicity method applied to complex Ginzburg–Landau type equations

Global existence and smoothing effect are established for the complex Ginzburg–Landau type equation in which the real and imaginary parts of the complex coefficient are multiplied by nonlinear terms with different powers. The proof is based on monotonicity methods described by subdifferential operators. The key lies in two modified inequalities.

Exponential attractor for the 3D Ginzburg–Landau type equation

In this paper, we consider a complex Ginzburg–Landau type equation with periodic initial value condition in three spatial dimensions. Sufficient conditions for existence and uniqueness of global solutions are obtained by uniform a priori estimates of solutions. Furthermore, the existence of a global attractor with finite Hausdorff and fractal dimensions is proved. Finally, the existence of the exponential attractor is proved.

Theory of Magnetodynamics Induced by Spin Torque in Perpendicularly Magnetized Thin Films

A nonlinear model of spin-wave excitation using a point contact in a thin ferromagnetic film is introduced. Large-amplitude magnetic solitary waves are computed, which help explain recent spin-torque experiments. Numerical simulations of the fully nonlinear model predict excitation frequencies in excess of 0.2 THz for contact diameters smaller than 6 nm. Simulations also predict a saturation and redshift of the frequency at currents large enough to invert the magnetization under the point contact. The theory is approximated by a cubic complex Ginzburg-Landau type equation. The mode's nonlinear frequency shift is found by use of perturbation techniques, whose results agree with those of direct numerical simulations.

Life-span of smooth solutions to the complex Ginzburg–Landau type equation on a torus

An upper bound of the life-span of smooth solutions to the complex Ginzburg–Landau equation with periodic boundary condition in one space dimension is given explicitly in terms of an integral mean of the Cauchy data in the case where the interaction is focusing.

Pattern formation of a reaction-diffusion system with self-consistent flow in the amoeboid organismPhysarumplasmodium

The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on the basis of spatio-temporal pattern formation by local contraction-oscillators. This biological system can be regarded as a reaction-diffusion system which has spatial interaction by active flow of protoplasmic sol in the cell. Paying attention to the physiological evidence that the flow is determined by contraction pattern in the plasmodium, a reaction-diffusion system having self-determined flow arises. Such a coupling of reaction-diffusion-advection is a characteristic of the biological system, and is expected to relate with control mechanism of amoeboid behaviours. Hence, we have studied effects of the self-determined flow on pattern formation of simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial solution, the envelope dynamics follows the complex Ginzburg-Landau type equation just after bifurcation occurs at finite wave number. The flow term affects the nonlinear term of the equation through the critical wave number squared. Contrary to this, wave number isn't explicitly effective with lack of flow or constant flow. Thus, spatial size of pattern is especially important for regulating pattern formation in the plasmodium. On the other hand, the flow term is negligible in the vicinity of bifurcation at infinitely small wave number, and therefore the pattern formation by simple reaction-diffusion will also hold. A physiological role of pattern formation as above is discussed.

On the relationship of periodic wavetrains and solitary waves of complex Ginzburg-Landau type equations

The relationship between periodic wavetrains and solitary waves in complex Ginzburg-Landau type equations, such as those that model optical fiber amplifiers, is studied in the nonlinear Schrödinger limit with a Melnikov method. An important example, the cubic complex Ginzburg-Landau equation, is studied in detail. For this equation it is found that in the NLS limit particular families of periodic wavetrains all deform asymptotically to a single, persisting, stationary, nonlinear Schrödinger soliton as their periods tend to infinity.

Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion

We consider the evolution of a gasless combustion front. We derive coupled complex Ginzburg-Landau type equations for the amplitudes of waves along the front as functions of slow temporal and spatial variables. The equations are written in characteristic variables and involve averaged terms which reflect the fact that in the slowest time scale, the effect on one wave, of a second wave traveling with the group velocity in the opposite direction on the intermediate time scale, enters only through its average. Solutions of the amplitude equations in the form of traveling, standing, and quasiperiodic waves are found, and regions of stability for these solutions are determined. In particular we find that the traveling and quasiperiodic (including standing) waves are not stable simultaneously. Finally, we observe that the stability analysis for coupled complex Ginzburg-Landau equations with averaged terms differs from that for coupled complex Ginzburg-Landau equations with the averages removed.