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Instability of the vortex solution in the complex Ginzburg–Landau equation
Nonlinear Analysis 45 (2001) 11 – 17
www.elsevier.nl/locate/na
Instability of the vortex solution in the complex Ginzburg–Landau equation Tai-Chia Lin ∗ Department of Mathematics, Chung-Cheng University, Minghsiung Chiayi Taiwan 621, People’s Republic of China Received 29 June 1998; accepted 25 June 1999
Keywords: Complex Ginzburg–Landau equation; Dynamical instability; Topological stability
1. Introduction In this paper, we investigate the complex Ginzburg–Landau equation as follows: ut = 5u + u + |u|2 u
for x ∈ R2 ; t ¿ 0;
(1.1)
where ; ; are complex-valued constants and u = u(x; t) is a complex-valued function. Here 5 is the Laplacian operator in R2 . The complex Ginzburg–Landau equation is raised in 8uid dynamics (cf. [14]). This equation is also familiar to students of superconductivity from whence it derives its name (cf. [7]). Suppose that = r ; = r and = r are nonzero real-valued constants. Then Eq. (1.1) becomes a parabolic equation ut = r 5u + r u + r |u|2 u
for x ∈ R2 ; t ¿ 0:
(1.2)
Assume that r r ¡ 0. Replacing u by u, where is a suitable constant, and using spatial and time scaling, we may transform (1.2) into two equations as follows: ut = 5u + (1 − |u|2 )u
for x ∈ R2 ; t ¿ 0
(1.3)
and ut = −5u + (1 − |u|2 )u
for x ∈ R2 ; t ¿ 0:
∗
Fax: +886-5-2720497. E-mail address: [email protected] (T.-C. Lin).
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 3 2 6 - 0
(1.4)
12
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
Eqs. (1.3) and (1.4) are determined by the sign of r ; r and r . Moreover, Eq. (1.3) is a well-known model to investigate the vortex dynamics in superconductors (cf. [2,13]). The steady-state equation of (1.3) is 5u + (1 − |u|2 )u = 0
for x ∈ R2 :
(1.5)
Let U+ = f+ (R)ei be a solution of (1.5), where f+ (R) is a real-valued function and (R; ) is the polar coordinate on R2 . Then f+ satisFes an ordinary diGerential equation as follows. f +
1 1 f − 2 f + (1 − f2 )f = 0 R R
for R ¿ 0:
(1.6)
From [1,3,4], we learn that Theorem A. Eq. (1:6) has a unique solution f+ (R) such that (i) f+ (R) ¿ 0 is strictly monotone increasing on R ¿ 0, (ii) f+ (R) → 1 as R +∞, → ∞ (iii) f+ (R) = aR + k=1 A2k+1 (a)R2k+1 ; for R ¿ 0, where a ¿ 0; A2k+1 ’s are constants. The solution U+ is called the symmetric vortex solution. By Theorem A, U+ has a single vortex at the origin with winding number one. From [13], such a vortex is topologically stable. Moreover, Neu [13] also provides a strong numerical evidence that the topological stability of vortices indicates their dynamical stability as solutions of Eq. (1.3). To study the dynamical stability of the solution U+ , we consider the linearized operator of (1.5) with respect to U+ as follows: 2 L+ w = −5w − (1 − f+ )w + 2(U+ · w)U+ ;
(1.7)
where U+ · w = 12 (U+ wJ + UJ + w) and J is the complex conjugate. Suppose the domain of L+ is H 1 (R2 ; C). From [15], we learn that the spectrum of L+ is [0; + ∞) and 0 is not an eigenvalue of L+ . However, assume that the domain of L+ is H01 (B1= (0); C), where 0 ¡ 1 and B1= (0) is the disk on R2 with center at the origin and radius 1=. Then the eigenvalues of L+ are all positive (cf. [6,10]). Moreover, from [8,9], we obtain more information about the eigenvalues of L+ as follows. Theorem B. Let 1 and 2 be the 3rst and the second eigenvalues of the operator L+ over H01 (B1= (0); C). Then (i) 0 ¡ 1 = O(2 (log 1=)−1 ) as → 0, and the multiplicity of 1 is two, (ii) There exist positive constants c; 1 such that 2 ≥ c2 for 0 ¡ ≤ 1 . Now we consider the steady-state equation of (1.4) as − 5u + (1 − |u|2 )u = 0
for x ∈ R2 :
(1.8)
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
13
Let U− = f− (R)ei be a solution of (1.8). Then f− satisFes an ordinary diGerential equation as follows: 1 1 f − 2 f − (1 − f2 )f = 0 R R From [12], we learn that f +
for R ¿ 0:
(1.9)
Theorem C. Eq. (1:9) has a solution f− (R) such that (i) f− (R) ¿ 0; ∀R ¿ 0 and f− has precisely one zero, (ii) f− (0) = 0 and f− (R) → 0 as R → +∞. In Section 2, we check that the solution U− has a single vortex at the origin with winding number one. From [13], such a vortex is topologically stable. However, we will prove that U− is dynamically unstable with respect to Eq. (1.4). Therefore, the topological stability may not imply the dynamical stability as solutions in Eq. (1.4). To prove the dynamical instability of U− , we consider the linearized operator of (1.8) with respect to U− as 2 )w − 2(U− · w)U− : L− w = −5w + (1 − f− −
(1.10)
H01 (B1= (0); C),
where 0 ¡ 1 and B1= (0) is a disk Assume that the domain of L is with the center at the origin and radius 1=. Then we prove that Theorem I. There exists a negative eigenvalue 0 of L− over H01 (B1= (0); C) such that 0 ≤ −c0 as → 0, where c0 is a positive constant independent of . Hence U− is dynamically unstable with respect to Eq. (1.4). In the rest of this paper, we will describe the qualitative behavior of f− in Section 2. Then we prove Theorem I in Section 3. 2. Preliminaries In this section, we study the qualitative behavior of the vortex solution U− =f− (R)ei of (1.8) such that f− satisFes Theorem C in Section 1. Then f− (R) satisFes 1 1 f − 2 f + f3 − f = 0; R R f(0) = 0; f(∞) = 0: f +
∀R ¿ 0;
(2.1)
By [11], f− is analytic on R ¿ 0. Hence by induction, we have f− (R) = R +
∞
P2k+1 ()R2k+1 ;
∀R ¿ 0;
(2.2)
k=1
where is a positive constant and P2k+1 ’s are constants depending on . Thus U− has a single vortex at the origin with winding number one. In [13], such a vortex is topologically stable.
14
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
Now we estimate f− (R) as R → ∞. To do this, we need the following lemma. ˜ Lemma II. There exists f(R) a solution of 1 1 f + f − 2 f − 2 f = 0; ∀R ¿ 0; R R where is any positive constant. As R → ∞, we have 1 ˜ : f(R) = R−1=2 e− R 1 + O R
(2.3)
(2.4)
˜ Proof. Let f(R) = R−1=2 e− R !(R) be a solution of (2.3). Then !(R) satisFes 1 ! − 2 ! + 2 ! = 0; ∀R ¿ 0: (2.5) 4R Then it is easy to transform (2.5) into an integral equation and we can solve it by the standard iteration method (cf. [5, pp. 199 –212]). Hence (2.5) has a solution !(R) = 1 + O(1=R) as R → ∞. Therefore, we complete the proof of Lemma II. From Lemma II with = 1, there exists f1 (R) a solution of 1 1 f1 + f1 − 2 f1 − f1 = 0; ∀R ¿ 0; R R such that 1 as R → ∞: f1 (R) = R−1=2 e−R 1 + O R In addition, by Lemma II with = 12 , there exists f2 (R) a solution of f2 + such that
1 1 1 f − f2 − f2 = 0; R 2 R2 2
∀R ¿ 0;
√ 1 f2 (R) = R−1=2 e− (1=2)R 1 + O R
By Theorem C and (2.1), we obtain 1 1 3 f− + f− − 2 f− − f− = −f− ≤ 0; R R f− (∞) = 0 and
(2.7)
(2.8)
as R → ∞:
∀R ¿ 0;
1 1 1 1 3 ≥0 f − f− − f − = f − − f − R − R2 2 2 f− (∞) = 0;
f− +
(2.6)
for R ≥ R1 ;
(2.9)
(2.10)
(2.11)
where R1 is a positive constant. From (2.7) and (2.9), there exist R2 ≥ R1 and positive constants 1 and 2 such that f− (R2 ) = 1 f1 (R2 ) = 2 f2 (R2 ):
(2.12)
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
15
Hence by (2.6) and (2.10), we have (f− − 1 f1 ) +
1 1 (f− − 1 f1 ) − 2 (f− − 1 f1 ) R R
− (f− − 1 f1 ) ≤ 0
for R ≥ R2 :
(2.13)
Moreover, (2.8) and (2.11) imply that 1 (2 f2 − f− ) R 1 1 − 2 (2 f2 − f− ) − (2 f2 − f− ) ≤ 0 R 2
(2 f2 − f− ) +
for R ≥ R2 :
(2.14)
Thus by (2.12) – (2.14) and the maximum principle, we obtain 1 f1 ≤ f− ≤ 2 f2
for R ≥ R2 :
(2.15)
Therefore by (2.7), (2.9) and (2.15), we have shown that Lemma III. There exist positive constants 1 and 2 such that √ 1 R−1=2 e−R ≤ f− ≤ 2 R−1=2 e− (1=2)R as R → ∞: In the next section, we will use Lemma III to prove Theorem I.
3. Proof of Theorem I To prove Theorem I, we consider a function w = h(R)ei ∈ H01 (B1= (0); C), where h is deFned by f− (R) if 0 ≤ R ≤ 1 − 1; h(R) = (3.1) f− ( 1 − 1)( 1 − R) if 1 − 1 ≤ R ≤ 1 : Then the quadratic form L− w; w becomes 1= 1 − 2 2 2 L w; w = 2 R(h )2 + h2 + R(1 − f− )h2 − 2Rf− h dR; R 0
(3.2)
where · ; · is the L2 inner product. From (3.1) and Lemma III, we have 1= 1 2 2 2 R(h )2 + h2 + R(1 − f− )h2 − 2Rf− h dR R 0 1=−1 1 2 2 2 2 4 = R(f− ) + f− + R(1 − f− )f− − 2Rf− dR + O() R 0 as → 0:
(3.3)
16
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
By Theorem C, f (1= − 1) ¡ 0 as small enough. Hence by (2.1) and integration by part, we obtain 1=−1 1 2 2 2 2 4 R(f− ) + f− + R(1 − f− )f− − 2Rf− dR R 0 1=−1 1 1 4 = f− Rf− dR − 1 f− −1 −2 0 1=−1 4 ¡ −2 Rf− dR: (3.4) 0
From Lemma III and the deFnition of h, there exists # a positive constant such that 1=−1 1= 4 Rf− dR ≥ # Rh2 dR; (3.5) 0
for 0 ¡ ≤
0
1 2.
Therefore, by (3.2) – (3.5), we complete the proof of Theorem I.
Final Remark. In [12], McLeod et al. classify all solutions of Eq. (1.9). Each solution of (1.9) satisFes one of the following conditions: √ (a) If a maximum of |f| does not exceed 2, then f has no further zeros and f → ±1 as R → ∞, √ (b) If all maxima of |f| exceed 2 (and only a Fnite number can do so), then f → 0 as R → ∞. For (b), Lemma III can be used and Theorem I is also true for this case. On the other hand, Lemma III does not hold for (a). However, the proof of Theorem I is still 1= 1= valid for (a) because f → 0 as R → ∞, 0 Rf2 dR ∼ −2 and 0 Rf4 dR ∼ −2 as → 0. Therefore, we conclude that all symmetric vortex solutions f(R)ei of (1.8) are dynamically unstable with respect to Eq. (1.4) as is suNciently small. Acknowledgements The author is partially supported by National Science Center in Taiwan. He sincerely thanks the referee for the helpful suggestions. References [1] X. Chen, C.M. Elliott, T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg– Landau equation, Proc. Roy. Soc. Edinburgh 124A (1994) 1075–1088. [2] W. E, Dynamics of vortices in Ginzburg–Landau theories with applications to superconductivity, Physica D 77 (1994) 383– 404. [3] P. Hagan, Spiral waves in reaction diGusion equations, SIAM J. Appl. Math. 42 (4) (1982) 762–786. Q [4] R.M. HervQe, M. HervQe, Etude qualitative des solutions rQeelles d’une eQ quation diGQerentielle liQee aR l’Qequation de Ginzburg–Landau, Ann. Inst. Henri PoincarQe 11 (4) (1994) 427– 440. [5] F. John, Ordinary DiGerential Equations, Lecture Notes of Courant Institute, 1965.
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
17
[6] E.H. Lieb, M. Loss, Symmetry of the Ginzburg–Landau Minimizer in a disc, Math. Res. Lett. 1 (6) (1994) 701–715. [7] E.M. Lifshitz, L.P. Pitaevstii, Statistical Physics, Vol. 2, Pergamon, Oxford, 1980. [8] T.C. Lin, The stability of the radial solution to the Ginzburg–Landau equation, Comm. PDE 22 (1997) 619–632. [9] T.C. Lin, Spectrum of the linearized operator for the Ginzburg–Landau equation and its application, preprint. [10] P. Mironescu, On the stability of radial solutions of the Ginzburg–Landau equation, J. Funct. Anal. 130 (2) (1995) 334–344. [11] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966. [12] J.B. McLeod, C.A. Stuart, W.C. Troy, An exact reduction of Maxwell’s equation, in: N.G. Lloyd, W.M. Ni, L.A. Peletier, J. Serrin (Eds.), Progress in Nonlinear DiGerential Equations: Nonlinear DiGusion Equations and Their Equilibrium States, Vol. 3, BirkhTauser, Boston 1992, pp. 391– 405. [13] J.C. Neu, Vortices in complex scalar Felds, Physica D 43 (1990) 385– 406. [14] A.C. Newell, J.A. Whitehead, J. Fluid Mech. 38 (1969) 279. [15] M.I. Weinstein, J. Xin, Dynamic stability of vortex solutions of Ginzburg–Landau and nonlinear SchrTodinger equations, Comm. Math. Phys. 180 (1996) 389– 428.
www.elsevier.nl/locate/na
Instability of the vortex solution in the complex Ginzburg–Landau equation Tai-Chia Lin ∗ Department of Mathematics, Chung-Cheng University, Minghsiung Chiayi Taiwan 621, People’s Republic of China Received 29 June 1998; accepted 25 June 1999
Keywords: Complex Ginzburg–Landau equation; Dynamical instability; Topological stability
1. Introduction In this paper, we investigate the complex Ginzburg–Landau equation as follows: ut = 5u + u + |u|2 u
for x ∈ R2 ; t ¿ 0;
(1.1)
where ; ; are complex-valued constants and u = u(x; t) is a complex-valued function. Here 5 is the Laplacian operator in R2 . The complex Ginzburg–Landau equation is raised in 8uid dynamics (cf. [14]). This equation is also familiar to students of superconductivity from whence it derives its name (cf. [7]). Suppose that = r ; = r and = r are nonzero real-valued constants. Then Eq. (1.1) becomes a parabolic equation ut = r 5u + r u + r |u|2 u
for x ∈ R2 ; t ¿ 0:
(1.2)
Assume that r r ¡ 0. Replacing u by u, where is a suitable constant, and using spatial and time scaling, we may transform (1.2) into two equations as follows: ut = 5u + (1 − |u|2 )u
for x ∈ R2 ; t ¿ 0
(1.3)
and ut = −5u + (1 − |u|2 )u
for x ∈ R2 ; t ¿ 0:
∗
Fax: +886-5-2720497. E-mail address: [email protected] (T.-C. Lin).
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 3 2 6 - 0
(1.4)
12
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
Eqs. (1.3) and (1.4) are determined by the sign of r ; r and r . Moreover, Eq. (1.3) is a well-known model to investigate the vortex dynamics in superconductors (cf. [2,13]). The steady-state equation of (1.3) is 5u + (1 − |u|2 )u = 0
for x ∈ R2 :
(1.5)
Let U+ = f+ (R)ei be a solution of (1.5), where f+ (R) is a real-valued function and (R; ) is the polar coordinate on R2 . Then f+ satisFes an ordinary diGerential equation as follows. f +
1 1 f − 2 f + (1 − f2 )f = 0 R R
for R ¿ 0:
(1.6)
From [1,3,4], we learn that Theorem A. Eq. (1:6) has a unique solution f+ (R) such that (i) f+ (R) ¿ 0 is strictly monotone increasing on R ¿ 0, (ii) f+ (R) → 1 as R +∞, → ∞ (iii) f+ (R) = aR + k=1 A2k+1 (a)R2k+1 ; for R ¿ 0, where a ¿ 0; A2k+1 ’s are constants. The solution U+ is called the symmetric vortex solution. By Theorem A, U+ has a single vortex at the origin with winding number one. From [13], such a vortex is topologically stable. Moreover, Neu [13] also provides a strong numerical evidence that the topological stability of vortices indicates their dynamical stability as solutions of Eq. (1.3). To study the dynamical stability of the solution U+ , we consider the linearized operator of (1.5) with respect to U+ as follows: 2 L+ w = −5w − (1 − f+ )w + 2(U+ · w)U+ ;
(1.7)
where U+ · w = 12 (U+ wJ + UJ + w) and J is the complex conjugate. Suppose the domain of L+ is H 1 (R2 ; C). From [15], we learn that the spectrum of L+ is [0; + ∞) and 0 is not an eigenvalue of L+ . However, assume that the domain of L+ is H01 (B1= (0); C), where 0 ¡ 1 and B1= (0) is the disk on R2 with center at the origin and radius 1=. Then the eigenvalues of L+ are all positive (cf. [6,10]). Moreover, from [8,9], we obtain more information about the eigenvalues of L+ as follows. Theorem B. Let 1 and 2 be the 3rst and the second eigenvalues of the operator L+ over H01 (B1= (0); C). Then (i) 0 ¡ 1 = O(2 (log 1=)−1 ) as → 0, and the multiplicity of 1 is two, (ii) There exist positive constants c; 1 such that 2 ≥ c2 for 0 ¡ ≤ 1 . Now we consider the steady-state equation of (1.4) as − 5u + (1 − |u|2 )u = 0
for x ∈ R2 :
(1.8)
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
13
Let U− = f− (R)ei be a solution of (1.8). Then f− satisFes an ordinary diGerential equation as follows: 1 1 f − 2 f − (1 − f2 )f = 0 R R From [12], we learn that f +
for R ¿ 0:
(1.9)
Theorem C. Eq. (1:9) has a solution f− (R) such that (i) f− (R) ¿ 0; ∀R ¿ 0 and f− has precisely one zero, (ii) f− (0) = 0 and f− (R) → 0 as R → +∞. In Section 2, we check that the solution U− has a single vortex at the origin with winding number one. From [13], such a vortex is topologically stable. However, we will prove that U− is dynamically unstable with respect to Eq. (1.4). Therefore, the topological stability may not imply the dynamical stability as solutions in Eq. (1.4). To prove the dynamical instability of U− , we consider the linearized operator of (1.8) with respect to U− as 2 )w − 2(U− · w)U− : L− w = −5w + (1 − f− −
(1.10)
H01 (B1= (0); C),
where 0 ¡ 1 and B1= (0) is a disk Assume that the domain of L is with the center at the origin and radius 1=. Then we prove that Theorem I. There exists a negative eigenvalue 0 of L− over H01 (B1= (0); C) such that 0 ≤ −c0 as → 0, where c0 is a positive constant independent of . Hence U− is dynamically unstable with respect to Eq. (1.4). In the rest of this paper, we will describe the qualitative behavior of f− in Section 2. Then we prove Theorem I in Section 3. 2. Preliminaries In this section, we study the qualitative behavior of the vortex solution U− =f− (R)ei of (1.8) such that f− satisFes Theorem C in Section 1. Then f− (R) satisFes 1 1 f − 2 f + f3 − f = 0; R R f(0) = 0; f(∞) = 0: f +
∀R ¿ 0;
(2.1)
By [11], f− is analytic on R ¿ 0. Hence by induction, we have f− (R) = R +
∞
P2k+1 ()R2k+1 ;
∀R ¿ 0;
(2.2)
k=1
where is a positive constant and P2k+1 ’s are constants depending on . Thus U− has a single vortex at the origin with winding number one. In [13], such a vortex is topologically stable.
14
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
Now we estimate f− (R) as R → ∞. To do this, we need the following lemma. ˜ Lemma II. There exists f(R) a solution of 1 1 f + f − 2 f − 2 f = 0; ∀R ¿ 0; R R where is any positive constant. As R → ∞, we have 1 ˜ : f(R) = R−1=2 e− R 1 + O R
(2.3)
(2.4)
˜ Proof. Let f(R) = R−1=2 e− R !(R) be a solution of (2.3). Then !(R) satisFes 1 ! − 2 ! + 2 ! = 0; ∀R ¿ 0: (2.5) 4R Then it is easy to transform (2.5) into an integral equation and we can solve it by the standard iteration method (cf. [5, pp. 199 –212]). Hence (2.5) has a solution !(R) = 1 + O(1=R) as R → ∞. Therefore, we complete the proof of Lemma II. From Lemma II with = 1, there exists f1 (R) a solution of 1 1 f1 + f1 − 2 f1 − f1 = 0; ∀R ¿ 0; R R such that 1 as R → ∞: f1 (R) = R−1=2 e−R 1 + O R In addition, by Lemma II with = 12 , there exists f2 (R) a solution of f2 + such that
1 1 1 f − f2 − f2 = 0; R 2 R2 2
∀R ¿ 0;
√ 1 f2 (R) = R−1=2 e− (1=2)R 1 + O R
By Theorem C and (2.1), we obtain 1 1 3 f− + f− − 2 f− − f− = −f− ≤ 0; R R f− (∞) = 0 and
(2.7)
(2.8)
as R → ∞:
∀R ¿ 0;
1 1 1 1 3 ≥0 f − f− − f − = f − − f − R − R2 2 2 f− (∞) = 0;
f− +
(2.6)
for R ≥ R1 ;
(2.9)
(2.10)
(2.11)
where R1 is a positive constant. From (2.7) and (2.9), there exist R2 ≥ R1 and positive constants 1 and 2 such that f− (R2 ) = 1 f1 (R2 ) = 2 f2 (R2 ):
(2.12)
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
15
Hence by (2.6) and (2.10), we have (f− − 1 f1 ) +
1 1 (f− − 1 f1 ) − 2 (f− − 1 f1 ) R R
− (f− − 1 f1 ) ≤ 0
for R ≥ R2 :
(2.13)
Moreover, (2.8) and (2.11) imply that 1 (2 f2 − f− ) R 1 1 − 2 (2 f2 − f− ) − (2 f2 − f− ) ≤ 0 R 2
(2 f2 − f− ) +
for R ≥ R2 :
(2.14)
Thus by (2.12) – (2.14) and the maximum principle, we obtain 1 f1 ≤ f− ≤ 2 f2
for R ≥ R2 :
(2.15)
Therefore by (2.7), (2.9) and (2.15), we have shown that Lemma III. There exist positive constants 1 and 2 such that √ 1 R−1=2 e−R ≤ f− ≤ 2 R−1=2 e− (1=2)R as R → ∞: In the next section, we will use Lemma III to prove Theorem I.
3. Proof of Theorem I To prove Theorem I, we consider a function w = h(R)ei ∈ H01 (B1= (0); C), where h is deFned by f− (R) if 0 ≤ R ≤ 1 − 1; h(R) = (3.1) f− ( 1 − 1)( 1 − R) if 1 − 1 ≤ R ≤ 1 : Then the quadratic form L− w; w becomes 1= 1 − 2 2 2 L w; w = 2 R(h )2 + h2 + R(1 − f− )h2 − 2Rf− h dR; R 0
(3.2)
where · ; · is the L2 inner product. From (3.1) and Lemma III, we have 1= 1 2 2 2 R(h )2 + h2 + R(1 − f− )h2 − 2Rf− h dR R 0 1=−1 1 2 2 2 2 4 = R(f− ) + f− + R(1 − f− )f− − 2Rf− dR + O() R 0 as → 0:
(3.3)
16
T.-C. Lin / Nonlinear Analysis 45 (2001) 11 – 17
By Theorem C, f (1= − 1) ¡ 0 as small enough. Hence by (2.1) and integration by part, we obtain 1=−1 1 2 2 2 2 4 R(f− ) + f− + R(1 − f− )f− − 2Rf− dR R 0 1=−1 1 1 4 = f− Rf− dR − 1 f− −1 −2 0 1=−1 4 ¡ −2 Rf− dR: (3.4) 0
From Lemma III and the deFnition of h, there exists # a positive constant such that 1=−1 1= 4 Rf− dR ≥ # Rh2 dR; (3.5) 0
for 0 ¡ ≤
0
1 2.
Therefore, by (3.2) – (3.5), we complete the proof of Theorem I.
Final Remark. In [12], McLeod et al. classify all solutions of Eq. (1.9). Each solution of (1.9) satisFes one of the following conditions: √ (a) If a maximum of |f| does not exceed 2, then f has no further zeros and f → ±1 as R → ∞, √ (b) If all maxima of |f| exceed 2 (and only a Fnite number can do so), then f → 0 as R → ∞. For (b), Lemma III can be used and Theorem I is also true for this case. On the other hand, Lemma III does not hold for (a). However, the proof of Theorem I is still 1= 1= valid for (a) because f → 0 as R → ∞, 0 Rf2 dR ∼ −2 and 0 Rf4 dR ∼ −2 as → 0. Therefore, we conclude that all symmetric vortex solutions f(R)ei of (1.8) are dynamically unstable with respect to Eq. (1.4) as is suNciently small. Acknowledgements The author is partially supported by National Science Center in Taiwan. He sincerely thanks the referee for the helpful suggestions. References [1] X. Chen, C.M. Elliott, T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg– Landau equation, Proc. Roy. Soc. Edinburgh 124A (1994) 1075–1088. [2] W. E, Dynamics of vortices in Ginzburg–Landau theories with applications to superconductivity, Physica D 77 (1994) 383– 404. [3] P. Hagan, Spiral waves in reaction diGusion equations, SIAM J. Appl. Math. 42 (4) (1982) 762–786. Q [4] R.M. HervQe, M. HervQe, Etude qualitative des solutions rQeelles d’une eQ quation diGQerentielle liQee aR l’Qequation de Ginzburg–Landau, Ann. Inst. Henri PoincarQe 11 (4) (1994) 427– 440. [5] F. John, Ordinary DiGerential Equations, Lecture Notes of Courant Institute, 1965.
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[6] E.H. Lieb, M. Loss, Symmetry of the Ginzburg–Landau Minimizer in a disc, Math. Res. Lett. 1 (6) (1994) 701–715. [7] E.M. Lifshitz, L.P. Pitaevstii, Statistical Physics, Vol. 2, Pergamon, Oxford, 1980. [8] T.C. Lin, The stability of the radial solution to the Ginzburg–Landau equation, Comm. PDE 22 (1997) 619–632. [9] T.C. Lin, Spectrum of the linearized operator for the Ginzburg–Landau equation and its application, preprint. [10] P. Mironescu, On the stability of radial solutions of the Ginzburg–Landau equation, J. Funct. Anal. 130 (2) (1995) 334–344. [11] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1966. [12] J.B. McLeod, C.A. Stuart, W.C. Troy, An exact reduction of Maxwell’s equation, in: N.G. Lloyd, W.M. Ni, L.A. Peletier, J. Serrin (Eds.), Progress in Nonlinear DiGerential Equations: Nonlinear DiGusion Equations and Their Equilibrium States, Vol. 3, BirkhTauser, Boston 1992, pp. 391– 405. [13] J.C. Neu, Vortices in complex scalar Felds, Physica D 43 (1990) 385– 406. [14] A.C. Newell, J.A. Whitehead, J. Fluid Mech. 38 (1969) 279. [15] M.I. Weinstein, J. Xin, Dynamic stability of vortex solutions of Ginzburg–Landau and nonlinear SchrTodinger equations, Comm. Math. Phys. 180 (1996) 389– 428.