- PDF Free Download

Transcription

1 Rev. Mat. Iberoam, 17 (1), 49{419 Dynamical instability of symmetric vortices Lus Almeida and Yan Guo Abstract. Using the Maxwell-Higgs model, we prove that linearly unstable symmetric vortices in the Ginzburg-Landau theory are dynamically unstable in the H 1 norm (which is the natural norm for the problem). In this work we study the dynamic instability of the radial solutions of the Ginzburg-Landau equations in R, (1) 8 >< > curl A + i (D; D)= ;D + (jj ; 1) = where R ;! C is the Higgs eld, or condensed wave function (jj is proportional to the local density of Cooper pairs), and A is the gauge potential 1-form (it can also be seen as the vector potential of the magnetic eld). The covariant derivative is D = r ; ia, with i = p ;1. The electric eld is absent in the stationary model, and H = curla is the magnetic eld. The dimensionless coupling constant is positive, < 1 corresponding to superconductors of type I and >1 to those of type II. Solutions of (1) are critical points of the Helmholtz free energy associated to the Ginzburg-Landau model, which we maywriteas () E = ZR 1 jcurl Aj + 1 jdj + 8 (jj ; 1) 49

2 41 L. Almeida and Y. Guo There is a vortex number (charge) associated with every nite energy solution of (1). It can be dened as n = 1 Z R H = 1 Z lim Adx N!1 jxj=n This number, which is always an integer, has a topological meaning { it is the winding number of the Higgs eld (see, for instance, [7]). In the early seventies, Nielson and Olesen ([8]) interpreted the nite energy solutions of (1) as string-like eld congurations and, soon after, a family of topologically non-trivial solutions (one for every integer value of the topological degree n and positive real value of the parameter ) was constructed mathematically by Berger and Chen ([]) and Plohr ([9]). In polar coordinates (r ) R these radial solutions (a ) are of the form (3) a(r )=a 1 dx 1 + a dx = ns(r) d (r )=R(r) e in where n Z and > are arbitrary. Here r = p (x 1 ) +(x ) and =tan ;1 (x =x 1 ). For studying the dynamics, one should consider the action on the Minkowski space-time R 1+ (we add the time coordinate, t, which will also be denoted by x ), with a metric g, = 1 with signature (; + +). The action is then given by (see [7, 1.9.a) and b)]) A = 1 (4) = 1 ZR1+ jf 1 j ;jd j + X j=1 (jd j j ;jf j j )+ 4 (jj ; 1) ZR1+ g ij g kl F ik F jl + g ij D i D j + 4 (jj ; 1) where, in the last expression we used the Einstein convention for summing over repeated indices (we will continue to do so below). Here, A is now the electro-magnetic potential { it has also an electric potential component A. We denote partial derivatives x, for = 1. Then, since F A A for, the electric eld is given by F j, j =1, and ;F 1 is the magnetic eld. We denoted covariant derivatives with respect to space or time variables by D ; ia. As usual, we will raise and lower indices by using the metric g. For instance, D i = g ij D j, and thus D = ;D, while D 1 = D 1.

3 Dynamical instability of symmetric vortices 411 Dynamical stability is then investigated using the Maxwell-Higgs system, which can be written as (5) 8 F = ;j D D + (jj ; 1) = The charge and current densities are given by j =Im( D )=; i ( D ; D ) for = 1, and the conserved energy is 1 ZR jf j + jd j + 4 (jj ; 1) dx 1 dx The Maxwell-Higgs model is invariant under a gauge transformation ( A ;! A e i We will work under the temporal gauge condition A, and thus we just need to consider the variations of the spatial components of A (we will be back to working with a -dimensional A, with real components A 1 and A, and a C ' R valued Higgs eld ). Let v =(W ) T R 4 be a perturbation of the radially symmetric vortex (a ), where W = A ; a and = ;. The full nonlinear Maxwell-Higgs system, in terms of v, can be written as (6) 8 >< t (@ 1 W 1 W ) ; t( ; )= i t t ) d v dt + E (a ) v = N(v) Here, E (a ) denotes the second order variation of () around the vortices (a ), and the nonlinear term N(v) isequal to i k k ) ; W k j j ; W k ( + ) ; a k j j ;iw j ; i@ j (W j ) ; Wj ; a j W j C A ;Wj ; (j j ( +)+ ) 1

4 41 L. Almeida and Y. Guo where 1 k, and we have an implicit sum over 1 j. Ever since the construction of these stationary radially symmetric vortices, their stability against initial perturbations with the same charge has been an interesting problem for both mathematicians and physicists. In a classical paper of [3], the question of stability was addressed by numerical and formal analysis. The study of the linear operator E (a ) indicates that for 1 and all charges n the vortices are linearly stable. On the other hand, for >1 and jnj, the vortices are linearly unstable. Unlike in the nite dimensional dynamical system, the passage form linear growing modes to a genuine nonlinear instability in an innite dimensional partial dierential equation is quite delicate. This is due to the possible presence of the continuous spectrum for the linearized operator and to severe high order perturbations arising from the nonlinearity. In previous works ([4] and [1]), the dynamical instabilityofvortices with large coupling constant was proven in the norm kfk X = kfk H 1 (R ) + kfk 1 The kk 1 was needed to control the H growth estimate. In this work, we improve the passage from linear instability to nonlinear dynamical instability in the more natural H 1 norm by a rened bootstrap argument. Let the initial perturbation be of the order. Within a time-interval of the order of j ln j we can estimate the H norm of the perturbation only by its H 1 norm (without any extra assumptions on its L 1 norm). A similar argument has also been used in [5]. In fact, in Theorem 1, we show that if the linear operator E (a ) has a negative direction, then the vortex is dynamically unstable in H 1 norm. For given positive constants and ", and for any small parameter >, we dene the associated escape time T by (7) e T = " For a xed appropriately chosen ",as ;! the dynamical instability will occur within t T. Lemma 1. Let v(t) be a solution of the full Maxwell-Higgs system (6). Assume (8) (9) kv()k H + kv t ()k H 1 C kv(t)k H 1 + kv t (t)k C e t

5 Dynamical instability of symmetric vortices 413 for t T, where > and C is independent of t. Then, there exist C 1 " > such that if t min ft T g then (1) kv(t)k H + kv t (t)k H 1 C 1 e t C 1 " where T is dened in (7). Proof. We shall estimate kvk H in terms of kvk H 1 by energytype estimates. Taking one spatial l x l through both equations in (6) we obtain (11) 8 >< > Here L 1 (v) t (@ l W l W ) ; t(@ l l ) = t(@ l l )+ t t ) d dt (@ lv) ; L(@ l v)=l 1 (v)+@ l N(v) i (@ l@ k kl k kl ) ;@ l a k ( + ) ; a k (@ l l ) ; W l jj ; i@ l a j ; iw jl ; i@ j W l l (jaj + jj ) ; W l (a j ) l( ) 1 C A Dene y(t) =kv(t)k H + kv t (t)k H 1. Using estimate (3.18) in [4] with suciently small ", we have that (1) y(t) C (kvk X + kvk X) C (kvk H + 1 kv tk ) d y() d + C kv(t)k H 1 + C (kv()k H + kv t()k H 1) We notice that due to typographical errors, the square was omitted in (3.18) for both kv(t)k H 1 and the initial data kv()k H + kv t ()k H 1.

6 414 L. Almeida and Y. Guo Since kvk X C kvk H, (13) H y(t) C (kvk H + kvk ) C (kvk H + 1 kv tk ) d y() d + C kv(t)k H 1 + C (kv()k H + kv t()k H 1) We now dene (14) T =sup n t for all s [ t] kv(s)k H + kv t (s)k H 1 min For t min ft T g, since kv(t)k H 1, we have Moreover, from (9), C (kvk H + kvk H ) 4 C (kvk H + 1 kv tk ) d + C kv(t)k H 1 n 8 C 1 oo C (C e ) d + C e t C e t Therefore, using (8), we obtain from (13) (15) y(t) y() d + C e t Now, proceeding as in the proof of the Gronwall inequality, we deduce that e ;(=)t y() d C e t;(1=)t = C e (3=)t Integrating over t, we obtain e ;(=)t y() d C e (3=)s ds = C e (3=)t

7 Therefore, Dynamical instability of symmetric vortices 415 And plugging this into (15) yields y() d C e t (16) kv(t)k H + kv t (t)k H 1 C 1 e t for t min ft T g, where C 1 is some xed constant which depends on and C, but is independent of. We now dene T as in (7), choosing " such that (17) C 1 " < min n 8 C 1 o Then, if T min ft T g, clearly the lemma follows. On the other hand, if T > min ft T g, we claim that T T. It thus follows that min ft T g = T and, once more, the lemma follows easily. To prove T T,we argue by contradiction. If not, we would have T >T and therefore min ft T g = T. Letting t = T in (16) would yield kv(t )k H + kv t (T )k H 1 C 1 e T <C 1 e T = C 1 " by the denition of T. However, this is impossible by the choice (17) since it would contradict the denition of T in (14). Now, we may prove our main result. Theorem 1. Let (a ) be a vortex such that (18) he (a )(v 1 ) v 1 i < for some v 1 H 1 (R ). Then, there exist constants " >, C >, so that for any small > there exists a family of solutions v (t) of the Maxwell-Higgs system (6) such that the vortex number of W () is zero, and kv ()k H + kv t ()k H 1 C but sup kv (t)k H 1 + kv t (t)k L " ftc j ln jg

8 416 L. Almeida and Y. Guo Proof. By [4, Theorem 1.3], there exists a dominant growing mode v e!t of the linearized Maxwell-Higgs system with! > and v H (R ). We normalize v such that (19) kv k H 1 + k!v k L =1 Moreover, we assume that kv k H + k!v k H 1 = r<1 Now we solve the Maxwell-Higgs system with a family of initial data vj t= = v and v t j t= =!v. Notice that the vortex number (charge) of a + W is the same as that of a. We denote the corresponding H solutions by v (t). They can be written as () v (t) =e!t v + L(t ; ) N (v ) d where L is the solution operator for the linearized Maxwell-Higgs system, and N (v )= i t t ) N(v ) Let! <!, and n T =sup s for all t [ s] (1) kv (t) ; v e!t k H 1 + kv t (t) ;!v e!t k 1 e!t o Using the triangle inequality, we see that for t T, kv (t)k H 1 + kv t (t)k kv e!t k H 1 + k!v e!t k + kv (t) ; v e!t k H 1 + kv t (t) ;!v e!t k () 3 e!t (kv k H 1 + k!v k ) = 3 e!t For any " >, from the sharp linear estimate for L as in [4, Theorems 1.3 and 1.5], we know that the solutions of the linearized Ginzburg- Landau equation grow no faster than e (!+")t. By Lemma 1 with =!,

9 Dynamical instability of symmetric vortices 417 C = max f3= rg, there exist constants C 1 " > such that for t min ft T g, kv (t)k H + kv t (t)k C 1 e!t C 1 " From the Sobolev embedding, kv (t)k 1 C kv (t)k H Ce!t CC 1 " for t min ft T g, where e!t = ". From the linearized estimate () together with the linear estimate in [4, Theorem 1.5], for t min ft T g, (3) kv (t) ; e!t v k H 1 + kv t (t) ;!e!t v k C C C C e (3=)!(t; ) i t t ) + kn(v )k d e (3=)!(t; ) (kv ()k 1 kv ()k H 1 + kv ()k 1 kv ()k ) d e (3=)!(t; ) kv ()k 1 kv ()k H 1 d e (3=)!(t; ) (e! )(e! ) d = C e (3=)!t e (1=)! d C (e!t ) where C is a constant. Notice that both Lemma 1 and (3) remain valid with the same constants C 1 and C, respectively, for all smaller ", as long as we take the corresponding T in (7). In particular, if necessary we can x " suciently small so that (4) C " 1 Now, clearly T T. Otherwise, we would have T < T, and from (19), (3), () and (4) it would follow that kv (T )k H 1 + kv t (T )k e!t +(C e!t ) e!t < 3 e!t

10 418 L. Almeida and Y. Guo which would contradict the denition of T in (1). Once we know T T, using (7), (19) and (3) again, we see that kv (T )k H 1 + kv t (T )k e!t (kv k H 1 + k!v k ) ;kv (T ) ; v e!t k H 1 and the proof is complete. ;kv t (T ) ;!v e!t k e!t ; C " e!t " > Remark. In [4] and [1], we had already proved that when jnj > 1, such negative directions exist for large. Very recently, Gustafson and Sigal ([6]) proved that for all > 1 and jnj > 1, there exists a v 1 such that he (a )(v 1 ) v 1 i <. Thus, we obtain as an easy corollary of Theorem 1, the dynamic instability of symmetric vortices when jnj > 1 and >1. Corollary 1. For all > 1, radially symmetric solutions (a ) of Ginzburg-Landau equations (i.e. solutions of (1) of the form (3)) with jnj > 1, are dynamically unstable in H 1 norm, in the sense of Theorem 1. Acknowledgement. We thank the referee for helpful comments. References. [1] Almeida, L., Bethuel, F., Guo, Y., A remark on the instability of symmetric vortices with large coupling constant. Comm. Pure Appl. Math. L (1997), [] Berger M. S., Chen Y. Y., Symmetric vortices for the Ginzberg-Landau equations of superconductivity and the nonlinear desingularization phenomenon. J. Funct. Anal. 8 (1989),

11 Dynamical instability of symmetric vortices 419 [3] Bogomol'nyi, E. B., The stability of classical solutions. Sov. J. Nucl. Phys. 4 (1976), [4] Guo Y., Instability of the symmetric vortices with large charge and coupling constant. Comm. Pure Appl. Math. 49 (1996), [5] Guo, Y., Strauss, W., Relativistic unstable periodic BGK waves. Comm. Pure Appl. Math. 8 (1999), [6] Gustafson, S., Sigal, I. M., The stability of magnetic vortices. Comm. Math. Phys. 1 (), [7] Jae, A., Taubes, C. H.,Vortices and Monopoles. Birkhauser, 198. [8] Nielson, H. B., Olesen, P., Vortex-line models for dual strings. Nucl. Phys. B 61 (1973), [9] Plohr, B. J., Thesis. Princeton University, 198. Recibido 17 de septiembre de Revisado 16 de febrero de. Lus Almeida Laboratoire J. A. Dieudonne UMR CNRS 661 Universite de Nice Sophia-Antipolis Parc Valrose 618 Nice Cedex, FRANCE luis@math.unice.fr and Yan Guo Lefschetz Center for Dynamical Systems Division of Applied Mathematics Brown University Providence, RI 91, U.S.A. guoy@cfm.brown.edu