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Register a new userBi-Lipschitz Mane projectors and finite-dimensional reduction for complex Ginzburg-Landau equation
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Anna Kostianko
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We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mane projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of 3D complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet-Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.
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The complex Ginzburg-Landau equation (CGLE) is a general model of spatially extended nonequilibrium systems. In this paper, an analytical method for a variable coefficient CGLE is presented to obtain exact solutions. Variable transformations for space and time variables with coefficient functions yield an imaginary time advection equation related to a complex valued characteristic curve. The variable coefficient CGLE is transformed into the nonlinear Schr{o}dinger equation (NLSE) on the complex valued characteristic curve. This result indicates that the analytical solutions of the NLSE generate that of the variable coefficient CGLE.
After a brief introduction to the complex Ginzburg-Landau equation, some of its important features in two space dimensions are reviewed. A comprehensive study of the various phases observed numerically in large systems over the whole parameter space is then presented. The nature of the transitions between these phases is investigated and some theoretical problems linked to the phase diagram are discussed.
For each given $ngeq 2$, we construct a family of entire solutions $u_varepsilon (z,t)$, $varepsilon>0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation begin{equation*} onumber Delta u+(1-|u|^2)u=0, quad (z,t) in mathbb{R}^2times mathbb{R} simeq mathbb{R}^3. end{equation*} These solutions are $2pi/varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $varepsilonto 0$ $$ u_varepsilon (z,t) approx prod_{j=1}^n Wleft( z- varepsilon^{-1} f_j(varepsilon t) right), $$ where $W(z) =w(r) e^{itheta} $, $z= re^{itheta},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ Delta W+(1-|W|^2)W=0 text{ in } mathbb{R}^2 $ and $$ f_j(t) = frac { sqrt{n-1} e^{it}e^{2 i (j-1)pi/ n }}{ sqrt{|logvarepsilon|}}, quad j=1,ldots, n. $$ Existence of these solutions was previously conjectured, being ${bf f}(t) = (f_1(t),ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ mathcal W_varepsilon ( {bf f} ) :=pi int_0^{2pi} Big ( , frac{|log varepsilon|} 2 sum_{k=1}^n|f_k(t)|^2-sum_{j eq k}log |f_j(t)-f_k(t)| , Big ) mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ lim_{|z| to +infty } |u_varepsilon (z,t)| = 1 quad hbox{uniformly in $t$} $$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.
We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically-stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically-stationary pulses in ultrafast fiber lasers.
The blow-up of solutions for the Cauchy problem of fractional Ginzburg-Landau equation with non-positive nonlinearity is shown by an ODE argument. Moreover, in one dimensional case, the optimal lifespan estimate for size of initial data is obtained.
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