U of M MATH 5076 - Symmetry-Breaking Solutions of the Ginzburg–Landau Equation

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1063-7761/04/9905- $26.00 © 2004 MAIK “Nauka/Interperiodica”1090 Journal of Experimental and Theoretical Physics, Vol. 99, No. 5, 2004, pp. 1090–1107.From Zhurnal Éksperimental’no œ i Teoretichesko œ Fiziki, Vol. 126, No. 5, 2004, pp. 1249–1266.Original English Text Copyright © 2004 by Ovchinnikov, Sigal. 1. INTRODUCTIONThe Ginzburg–Landau equation describes, amongother things, macroscopic stationary states of superflu-ids, Bose–Einstein condensation, and solitary waves inplasmas. In recent years, it has become a subject ofactive mathematical research (see monographs [1–3]and reviews [4–7] for some of the recent references).This equation is simple to write,(1.1)where (in the case of the entire plane ! 2 ) ψ : ! 2 " ,with the boundary condition(1.2)but not easy to analyze. In fact, so far only radially sym-metric solutions, i.e., solutions of the form ψ n ( x ) = f n ( r ) e in θ , where r and θ are polar coordinates for x ∈ ! 2 ,are known for (1.1) and (1.2) (see [8–17]). Solutions ψ n are called n vortices. We note that n = deg ψ n , wheredeg ψ , the degree (or vorticity) of ψ (satisfying (1.2)) isthe total index (winding number) at ∞ of ψ consideredas a vector field on ! 2 , i.e.,for sufficiently large R .The existence and properties of the vortex solutionswere established only recently. The known facts are asfollows.Δψ– ψ21–( )ψ+ 0,=ψ 1 as x ∞,degψ := 12π------argψ( )dx R=∫ (i) Existence and uniqueness (modulo symmetrytransformations and in a class of radially symmetricfunctions) [10–13].(ii) Stability for | n | ≤ 1 and instability for | n | > 1([13], earlier results on stability for the disc are dueto [15–17]).(iii) Uniqueness of ψ ± 1 (again, modulo symmetrytransformation) in a class of functions ψ with deg ψ = ± 1 and < ∞ [16].Therefore, the next question is: Are there nonradi-ally symmetric solutions?In this paper, we present results indicating that suchsolutions exist. There are two key ingredients in ouranalysis. First, we characterize nonradially symmetricsolutions as critical points of the intervortex energyfunction described below (see also [18]). Second, weseek solutions having certain point symmetries. Thelatter fact reduces the number of free parametersdescribing such solutions to one (the size of the corre-sponding polygon of vortices).Solutions breaking the rotational symmetry werefound to exist in the case of the Ginzburg–Landau equa-tion in the ball B R = { x ∈ ! 2 | | x | ≤ R } with the bound-ary condition = e in θ and | n | ≥ 2 (see [1, 2], The-orem IX.1). However, in the case of the ball, there is anexternal mechanism leading to the symmetry breaking:the boundary condition. It repels vortices, forcing theirconfinement. On the other hand, the energy is loweredby breaking up multiple vortices into (+1)- (or ( − 1)-)vortices and merging vortices of opposite signs. Thus,for R that are not very small, the lowest energy isreached by a configuration of | n | vortices of vorticitiesψ21–( )2∫ψ∂BR Symmetry-Breaking Solutionsof the Ginzburg–Landau Equation ¶ Yu. N. Ovchinnikov a,b and I. M. Sigal c,d a Landau Institute, Moscow, Russia b Max-Planck-Institute for Physics of Complex Systems, Dresden, Germany c University of Toronto, Canada d University of Notre Dame, USAe-mail: [email protected] Received March 1, 2004 Abstract —We consider the question of the existence of nonradial solutions of the Ginzburg–Landau equation.We present results indicating that such solutions exist. We seek such solutions as saddle points of the renormal-ized Ginzburg–Landau free-energy functional. There are two main points in our analysis: searching for solu-tions that have certain point symmetries and characterizing saddle-point solutions in terms of critical pointsof certain intervortex energy function. The latter critical points correspond to forceless vortex configurations. © 2004 MAIK “Nauka/Interperiodica”. SOLIDSElectronic Properties ¶ This article was submitted by authors in English.JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 99 No. 5 2004 SYMMETRY-BREAKING SOLUTIONS OF THE GINZBURG–LANDAU EQUATION 1091 ± 1 depending on the sign of n , which, obviously, is notrotationally symmetric.This paper is organized as follows. In Sections 2and 3, we review some material in [13]: the variationalformulation of the problem and some specific proper-ties of vortex solutions. In Section 4, we define the int-ervortex energy and discuss its properties. In particular,we discuss the correlation term in (the upper bound on)the expansion of the intervortex energy for large inter-vortex separations and a definition of G -symmetric vor-tex energies, where G is a subgroup of the symmetrygroup of (1.1).In Section 5, we consider point symmetries ( C N v ),present one of our main results, Theorem 5.1, on theexistence of critical points for C N v -symmetric intervor-tex energies, and derive some general relations forthose energies. In Section 6, we prove Theorem 5.1 anddiscuss some other cases.Finally, in our five appendices all the hard analyticand numerical work is concentrated. In these appendi-ces, we compute various asymptotic expansionsbeyond the leading order. We feel that these appendicesare of interest on their own because they address rathersubtle computational issues.2. RENORMALIZED GINZBURG–LANDAU ENERGYIt is a straightforward observation that Eq. (1.1) isthe equation for critical points of the functional(2.1)Indeed, if we define the variational derivative ∂ ψ ! ( ψ )of ! by(2.2)for any path ψ λ such that ψ 0 = ψ and = ξ ,then the left-hand side of Eq. (1.1) is equal to = for ! ( ψ ) given by (2.1).Equation (2.1) is the celebrated Ginzburg–Landau(free) energy. However, there is a problem with it in ourcontext. It is shown in [13] that, if ψ is an arbitraryC1-vector field on !2 such that |ψ| 1 as |x| ∞uniformly in = x/|x| and degψ ≠ 0, then !(ψ) = ∞.We renormalize the Ginzburg–Landau energy func-tional as follows (see [13]). Let χ(x) be a smooth posi-! ψ( )12---∇ψ212---ψ21–( )2+  .∫=Re ξ∂ψ! ψ( )∫λ∂∂! ψλ( )λ 0==λ∂∂ψλλ 0=∂ψ! ψ( ) ∂ψ! ψ( )xˆtive


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U of M MATH 5076 - Symmetry-Breaking Solutions of the Ginzburg–Landau Equation

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