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Register a new userEntire solutions of the magnetic Ginzburg-Landau equation in $mathbb{R}^4$
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We construct entire solutions of the magnetic Ginzburg-Landau equations in dimension 4 using Lyapunov-Schmidt reduction. The zero set of these solutions are close to the minimal submanifolds studied by Arezzo-Pacardcite{Arezzo}. We also show the existence of a saddle type solution to the equations, whose zero set consists of two vertical planes in $mathbb{R}^4$. These two types of solutions are believed to be energy minimizers of the corresponding energy functional and lie in the same connect component of the moduli space of entire solutions.
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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.
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.
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.
We consider half-harmonic maps from $mathbb{R}$ (or $mathbb{S}$) to $mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is $pm 1$, we prove that the deviation of any map $boldsymbol{u}:mathbb{R}to mathbb{S}$ from Mobius transformations can be controlled uniformly by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. This result resembles the quantitative rigidity estimate of degree $pm 1$ harmonic maps $mathbb{R}^2to mathbb{S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $boldsymbol{u}$ is already near to a Blaschke product, then the deviation of $boldsymbol{u}$ to Blaschke products can be controlled by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $boldsymbol{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${mathbb R}^2to {mathbb S}^2$.
In this chapter we review recent results concerning localized and extended dissipative solutions of the discrete complex Ginzburg-Landau equation. In particular, we discuss discrete diffraction effects arising both from linear and nonlinear properties, the existence of self-localized dissipative solitons in the presence of cubic-quintic terms and modulational instability induced by saturable nonlinearities. Dynamical stability properties of localized and extended dissipative discrete solitons are also discussed.
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