QUARTERLY OF APPLIED MATHEMATICSVOLUME LVII, NUMBER 2JUNE 1999, PAGES 355-367

ASYMPTOTIC ANALYSIS OF THE ONE-DIMENSIONALGINZBURG-LANDAU EQUATIONS NEAR SELF-DUALITY

By

Y. ALMOG

Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science,Rehovot, Israel

Abstract. It is known that when the Ginzburg-Landau parameter k = l/\/2 the one-dimensional Ginzburg-Landau equations exhibit self-duality and may be reduced into apair of first-order ODE. The present asymptotic analysis initially focuses on infinitesamples of superconductors for which |/c— l/\/2| < 1. It is shown that when the value ofthe applied magnetic field at infinity lies between k and l/y/2 a superconducting solutionexists. It is later shown, for arbitrary values of n, that no solution, other than the normalstate, can exist for applied magnetic field values that lie outside the above interval.

1. Introduction. Solutions for the one-dimensional Ginzburg-Landau equations,which model superconductivity, have been discussed mostly from the point of view oflinear bifurcation theory. Millman and Keller [12] provided the framework for obtainingthe applied magnetic field for which the bifurcation from the normal state occurs. Asan example, they have discussed the one-dimensional case on a finite interval. Chapman[5] provided several other examples, among them the one-dimensional case on an infinitesample.

We shall subsequently refer to superconducting materials for which the Ginzburg-Landau parameter k < l/\/2 as type I superconductors and to those superconductorsfor which n > l/\/2 as type II. Chapman [5] demonstrated for the above-mentionedexample that when the applied magnetic field h is equal to k a superconducting branchbifurcates from the normal state solution. This superconducting solution, for the lin-earized Ginzburg-Landau equations, can exist only for h < n (h > k) for type II (typeI) superconductors. Chapman also showed, in a follow-up contribution [6], that the su-perconducting solution is stable (unstable) for type I (type II) superconductors, demon-strating that the value of what has been termed the upper critical field (at which thesolution ceases to be normal when the applied magnetic field is gradually decreased) isexactly k.

Linear bifurcation theory is an effective tool, when seeking the value of the parameterat which bifurcation occurs and the solutions themselves near the bifurcation. It cannot,

Received January 9, 1997.1991 Mathematics Subject Classification. Primary 82D55, 58F14.

©1999 Brown University

356 Y. ALMOG

however, predict highly nonlinear effects. For instance, we cannot be sure that no solutionother than the normal one exists when the applied field is larger than the upper criticalfield (for type II superconductors). Furthermore, there is no guarantee that any solution,other than the normal state, would still exist, if we decrease the applied magnetic fieldfurther to values that are not close to the upper critical field.

In order to properly address the questions above, which all involve nonlinear effects, itis necessary to discuss solutions of the nonlinear Ginzburg-Landau equations (in contrastto linear bifurcation theory which investigates solutions of the linearized system). Sucha family of solutions can be obtained when the Ginzburg-Landau equations exhibit whathas been termed self-duality [1], [2], These solutions can be obtained, however, only fora specific value of k, the Ginzburg-Landau parameter (i.e., k = l/\/2). In the presentwork we look first at superconductors with k close to \j\f2. We shall be able to obtain inthis way the range of applied magnetic field values for which superconducting solutionsexist. Motivated by the results of this asymptotic analysis, we shall be able to prove anonexistence theorem valid for arbitrary values of k.

Consider a superconducting material given in an infinite domain. The magnetic fieldH is taken to be directed along the z axis and the magnetic vector potential A to bedirected along the y axis. We assume that all functions depend solely on x, hence,Hs — or simply, H = = A'2. (Since A2 is the only nonzero component of Awe shall subsequently denote it by A.) The Ginzburg-Landau equations may then bewritten in their one-dimensional form [5]:

%- = - -0 + A2xf), (1.1a)K

A" = tfA, (1.1b)

where is the superconducting order parameter (the gauge has been chosen so that ipwould be real). The boundary conditions satisfied by xjj and A are

if)' 0 as |x| —> oo, (1.1c)A' —* h as |x| —> oo, (1.Id)

where h is the applied magnetic field at infinity. Note that at least one solution for (1.1)always exists, namely

V - 0; A = hx + c. (1.2a,b)

We shall refer to (1.2) as the normal solution. Any other solution of (1.1) would be calleda superconducting solution. We shall be interested only in solutions for which ip > 0.

When k = 1 /a/2 (1.1) exhibit what has been termed self-duality, so that for h = l/sj2,solutions of (1.1) are also solutions of the following pair of first-order ODEs:

y/2A'= 1-V2, (1.3a)

\/2ip' = -ipA. (1.3b)

ONE-DIMENSIONAL GINZBURG-LANDAU EQUATIONS 357

[There may be, of course, different solutions for (1.1).] A family of solutions can be foundfor (1.3)

*=/"" (L4)Jip2(0) V2ip2{ip2 - c - logip2)1/2

where c may be any number greater than 1. Chapman [5] plots ip and A' versus x forvarious values of c. It is easy to show that the value of ip2{0) increases as c —> 1. Forc = 1 (1.4) does not satisfy anymore (1.1c) and (l.ld) but instead satisfies differentboundary conditions for x —> oo and x —> — oo, i.e., while the boundary condition atx —> oo remains unaltered, we must have ip = 1, ^4 = 0 as x—> —oo.

In the next section we provide an asymptotic approximation for superconducting so-lutions of (1.1) for 0 < \k — l/V^I < 1- Motivated by the results of Sec. 2 we prove inSec. 3 that superconducting solutions of (1.1) cannot exist when the applied magneticfield h does not lie between k and 1 /y/2. In the last section we examine the significanceof our results to time-dependent analyses.

2. Asymptotic solutions near self-duality. We first define a new parameter

£=l-2 (2.1)

and rewrite (1.1) in the form

(2 + e)ip" = ip3 — ip + ipA2, (2.2a)

A" = ip2A, (2.2b)V/ — > 0, (2.2c)

|x| —>oo

A' > h. = —— + h\£. (2.2d)|s|-»oo \/2

Our goal is to approximate the solution of the above problem for e 1. Since for e = 0the solution is known [and is given by (1.4)] we introduce the expansion

ip = ipo + £ipi + £2ip2 + 0(e3), (2.3a)

A = Aq + sA\ + £2 A2 + O(s^). (2.3b)

As was pointed out in the preceding section, we have a family of solutions for the casee = 0. We expect, however, in view of the results of linear bifurcation theory [5], to havea unique superconducting solution for £ ^ 0. Thus, we expect that the O(e) balancewould be solvable only for a specific ipo, or equivalently, in view of (1.4), for a specificvalue of c. The O(e) balance

2V4' = (3^ - 1 + A2)^ + 2^0A0A1 - <, (2.4a)

^1 = "00^1 + 2'0o^oV'i! (2-4b)ip' > 0, (2.4c)

|x|—^CX)

A' ♦ hx (2.4d)|x|—>oc

358 Y. ALMOG

is a fourth-order linear system with nonconstant coefficients. In order to solve (2.4) weintroduce the following pair of functions:

f = A'--j=( 1-V2), (2.5a)

g = V2ip' + ipA. (2.5b)Using the above definitions we can rewrite (2.2a) and (2.2b) in the alternative form

\[2g' — y/2 ftp — Ag = —etp", (2.6a)/' - i>g = o. (2.6b)

Upon expanding / and g into power series of e,

/ = /o + e/i + e2h + C(e3), (2.7a)

/ — 9o + 92 + 0(e3) (2.7b)

and noticing that, as a result of (1.3), /o and go must identically vanish, we obtain thefollowing fourth-order system which is equivalent to (2.4):

A[ + V2i>oVi=/i, (2.8a)\/2xj>[ + tpiAo + ipoAi = g\, (2.8b)

y/2g[ - V2fitp0 - AQgi = -ip'0', (2.8c)

fx ~ o9i = 0. (2.8d)In view of (2.8a) and the asymptotic behavior for large x of tpo [utilizing (1.4) it is easy

to show that ip0 ~ 0(e~x~/A)\, f\ must satisfy the following boundary value problem:

/" - rth = (2-9a)h (2.9b)

| a: |—>oo

Integrating the product of (2.9a) with Aq between x and oo (x > 0) we obtain

fl\ 1,2i i2M)

Hence, for x > 0, f\ is expressible in the form

1 f°° h f°° 1/, = - /l(j J ipldx+-^=A0J ~^dx. (2.11)For x < 0 we simply have f\(x) = f\(—x).

Clearly, f\ is continuous at x = 0. However, to be a valid solution of (2.9) f[ needsto be continuous at x — 0 as well. Equivalently, since fi is even, we may require f[ = 0.This requirement leads to the following solvability condition:

2 .V2 f0 dxhi = —V 2 • (2-12)4 r30 V-n rjr

Jo (A'0)2 UX

The right-hand side of (2.12) is a function of c, or alternatively of ^o(0) = tp- As will besubsequently demonstrated [cf. (2.22)], if (2.12) holds, solutions for (2.4) can be found.

ONE-DIMENSIONAL GINZBURG-LANDAU EQUATIONS 359

Hence, (2.12) serves as both a necessary and sufficient condition for the solvability of

(2.4).When ip <C 1 the approximations

ipo = ijje-x2/4 + 0(ij?), (2.13a)

A'o = ^ + °W (2.13b)

may be substituted into (2.12) to obtain

h, = -^= + 0&2). (2.14)

Since by (2.1) k = 4= — 4\/2£ + 0(e2), it follows that

h = k + 0(ip2, e2)

which is in accordance with the results of linear bifurcation theory [5].By (1.3a) and (1.4) A'0(x,rt) is a decreasing function of ip. Therefore, hi must be an

increasing function of tjj. We conclude that a solution to (2.4) can exist only when

7= < hi < 0.4\/2

For type I (type II) superconductors we may express the above condition in the alter-native form k < h < l/\/2 {1/V2 < h < k). While the lower bound for hi is expected inview of linear bifurcation theory results [5], the upper bound cannot be detected by anylinear theory since it involves highly nonlinear effects. It is, however, intuitively expectedsince for h = \/\/2 the free energies of the superconducting state and the normal stateare equal.

Figure 1 displays the dependence of ip on — 4\/2/ii according to (2.12). It can be seenthat as hi —> — —> 0 and that as hi —* 0, tp —> 1. A monotonic dependence of ipon hi is observed.

Once the derivation of the relation (2.12) between the applied magnetic field and theleading-order solution is complete, the energy of the solutions may be approximated.The Ginzburg-Landau equations (1.1) are the Euler-Lagrange equations associated withthe energy functional

E = J f2 +g2 + (v/2 — 2 h)A' + ^h2 — + e(ip')2 dx (2.15)

reflecting the difference between the energy of (ip,A) and the normal state (1.2). Forthe critical case k = l/%/2, h = l/\/2 the energy vanishes identically. For nearly criticalcases it is of 0{e).

Introducing the expansion

E^eEi+0{£2) (2.16)

together with (2.3) and (2.7) we obtain

Ei =/OO

\\f2hiipl + (^o)2] dx. (2.17)-OO

360 Y. ALMOG

"0 0.5

J I I I I I I I L

Integration by parts then yields

Ei = /oo 1v/2/ii^o + tV>o(1 - V'o)-oo L ^

0 independen

/OO

{ipl)ndx.-OO

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

—4y/2hi

Fig. 1. Variation of ip, the maximal superconducting order parame-ter, with —4y/2hi, representing the applied magnetic field

dx. (2.18)

We shall first demonstrate that E\ > 0 independently of h\'s value. Define

In =

Then,

/— 1 I\ 1 I\1 4/^ Vg(l - ipl)-2dx > 4^=1/„

and hence by (2.18), we obtain-vOO

AEi > ' V IZ^n=1±n

ONE-DIMENSIONAL GINZBURG-LANDAU EQUATIONS 361

Application of Holder's inequality (with p = n/(n — l),q = n) yields

T <" T~^ T^/n . T 7"l/nh S h in+l, in S ln+i h ■

Hence, ^In < hln+i from which positivity of E\ follows.Recalling that e is positive for type I superconductors and negative for type II, and

that the energy of the normal state vanishes, we conclude that (ipQ,Ao) can serve asa global minimizer for E only for type II superconductors. This is, however, a highlyexpected result since, in general, superconducting solutions were found to be stable onlywhen k > l/y/2 [6].

Dorsey [9] approximated the energy of the solution for the asymptotic problem, whenip —» 1 and A —> 0 as x —> —oo, near self-duality. The energy of such a solutionhas been termed "the surface tension" since it presents the surface energy of a nor-mal/superconducting interface. We shall now demonstrate the relation between Ei andDorsey's [9] approximation.

We first derive an upper bound for ipo- Observing that Aq > ^4q(0);e for x > 0 wesubstitute in (1.3b) to derive the following crude estimate:

ipo < exp | —jt1 - mW} >

from which we find

fJoipodx < -

2tt 1/2

(2.19).l-Vg(O)Next, it is easy to show that on some finite interval 0 < x < I, for sufficiently small

values of 1 — V'o(O), we have

^ K(1-^2)2 - [1_^2(0)]2

where K is independent of V'o(O). Hence,

kiJJ 0/0 (l-V'o2)2 - [1 -V^g(O)]2Combining (2.12), (2.17), (2.19), and (2.20) we then have

dx > y- (2.20)

lim E\h-*0 /OO

{tp'ofdx. (2.21)-oo

Comparison with [9] reveals that E\e tends, as hi —► 0, to be twice the approximatedvalue, near self-duality, of the surface tension. The numerical calculation in [9] then finds

lim Ei = 0.388.h i—>0

We conclude this section by obtaining the next order term, demonstrating that ourasymptotic scheme works for higher-order terms as well. To this end we first obtain thegeneral solution of (2.4)

= Ci K°1 + C2 H°] + M . (2.22)yij IVo. .Vi h_ _Vip_

362 Y. ALMOG

The first term on the right-hand side appears very often when applying perturbationexpansions (cf. [11]). The functions v and ipih appearing in the second term are definedas follows:

1,2

l (A'")2 ^ (2'23a)

i>ih = -

1o

V'o■vip'o (2.23b)

.V2A'0The last term on the right-hand side of (2.22) is a particular solution of (2.4) and isexpressible in the form

A\p = (ui + u2v)A'0, (2.24a)

(2'24b)

in which

Ui — — I " i ^ (2.24c)

U2 =

y/2AoA?afi

V2vA0A'0fi (2.24d,Jo V'o

In the same manner in which we have obtained (2.9) we obtain the following boundaryvalue problem for fa:

?2 - ^0/2 = (V>i5i)' + -^ipol^Mi + Aigi - (2.25a)

fa > 0. (2.25b)|x|—>OC

Note that, unlike /1, fa is not necessarily even. We therefore obtain two solvabilityconditions: The first one ensures continuity of faa,tx = 0,

/OO[A0(ipi9iY ~ ipo(V2fiipi + Aigi - rp")] dx = 0, (2.26)

-OO

and the second condition guarantees the continuity of f2 at x = 0,

/ {/ {A°^l9lY ~ ^0(^2/1^1 + Aigi -ip")\0dx"^ dx'

+ / —

wherein

/°° 1 T 1-jT (^ifi)'+ -^l>o(V2fiil)X +A1g1 -V")(2.27)

dx' = 0,

[u]e(x) = i[u(x) +U(-X)],

[u]e(z) = |[w(ar) - w(-^)]

are, respectively, the even and the odd part of the function u.Substituting (2.22) into (2.26) we note that the contribution of the second and the

third terms on the right-hand side of (2.22), which are both even, to the left-hand side

ONE-DIMENSIONAL GINZBURG-LANDAU EQUATIONS 363

of (2.26) vanishes. We are therefore left with the condition

/OO[AoWoaiY - <(V2M'0 + a'o9i - 43))] dx = o.

-oo

It is not difficult to show that the above condition is satisfied for arbitrary C\. Byinvoking symmetry conditions once again, it is readily seen that (2.27) is satisfied forarbitrary C\ as well. Hence, C\ remains undetermined. This indeterminacy is highlyexpected in view of the translational invariance of (2.2), i.e., if i^{x) is a solution then sois rp(x + C\e) for any C\ and since

i){x + C\e) = tjj(x) + Ci£ip'(x) + 0(e2),

C\ must be arbitrary.

3. Nonexistence of solutions. In the preceding section we have demonstrated fortype I (type II) superconductors that superconducting solutions may exist only for n <h < l/\/2 (l/\/2 < h < k). However, this result was derived only for k values such that|k — l/v^l < I- In the following we wish to demonstrate that the validity of this resultmay be extended to arbitrary values of k. More explicitly, our goal would be to provethe following claim.

Theorem 1. Suppose that n < l/y/2 (k > 1/V2) and eithera. h > l/\/2 (0 < h < l/v/2)

orb. 0 < h < k {h > k).

Then no solution of (1.1), for which 0 < V f°r all x, can exist.

The case h = 0 has been excluded, since ip = 1, A = 0 is then a solution.Our first step would be to state the following lemma [8].

Lemma 1. Any solution of (1.1) has the following properties:a. A' > 0 Vrr,b. — 1 < ijj < 1 \/x.c. Suppose that for some xo we have: ip(xo) > 0, ip'(x0) < 0. Then,

ip{x) >0 Vx > xq => ip'(x) <0 Vx > xq-

d. ip < e~Kx2 for some positive K, and hence A' ~ h + 0(e~Kx2) as |x| —> 00.

Note that the arguments invoked by Chapman et al. [8] to show that V must bepositive are not valid in our case because the boundary conditions at —00 in the problemthey study are different than ours. Furthermore, ip is n°t necessarily positive in ourcase. The results of linear bifurcation theory [5] clearly indicate that solutions, for whichip < 0 on some subsets of the real line, must exist. However, since solutions for whichip is negative are not of any practical interest (and since they were found unstable bylinear stability analysis [6]), we shall confine the discussion to the "physical" solutionsfor which if> is nonnegative.

Our next step would be to prove the following symmetry property.

364 Y. ALMOG

Lemma 2. Any solution of (1.1) for which i/j > 0 Vx is symmetric, i.e., V'(0) = 0 <;=>A(0) = 0.

Note that the choice x = 0 is arbitrary in view of the translational invariance of (1.1).Proof. Suppose for a contradiction that there exists a solution (tpi,Ai) of (1.1) for

which A-i(0) = 0,V4(0) < 0. Then, the pair (^2,^2) defined by faix) — xpi(-x),A2{x) = —A\(—x) is also a solution of (1.1) for which tp'2 > 0.

Following [8] we define

(A')2; R = A2

and use the Hamiltonian relation

+ (A')2 - + </>2 - i'2A2 = h2 (3.1)

together with the identities

,9 duV = dR'

. ,9 Ru ( d2u \ ^w=-g(w) to'R>0

to obtain the equation

R (d2u \ du 1 I 1 / du \ du du^\dR2) + dtf = u{2\dRj ~dR+ dR

Define u\ = (A[)2,U2 = (A'2)2; hence, ipl = ^-,^2 = IZr- Obviously «i(0) = ^2(0),^■(0) = ^ff(O), and «i(oo) = 1x2(00). However, since Vi(0) < 0 and 1P2W > 0> ui andu2 satisfy initially different equations. Explicitly, u\ satisfies the equation

R1'2 d2ux ' ' * * ' 1 V2k dR2

whereas u2 satisfies

i?1/2 d2U2

^TJr2

1 (du 1 \ 2 du\ ^du\ , ,! S ~*R+R4R+h I I } (3.2)

- 1 du2\2 du2 du2 2-~ldR ~dR+RdR+h -.H"2. (3.3,

While (3.2) is valid for any positive value of R, (3.3) remains valid as long as ip'2 > 0. In.2

view of Lemma Id, at some Ro > 0 we must have = 0. Setting $ = u2 — u\, for0 < R < Rq we have > 0; hence, since $(0) = ^ = 0 both $(i?o) and jfi(Ro) mustbe positive.

Following [8] once again, we write the equation for <3?, for R > Ro, in the followingform:

R (d2U\ d2u2 \ d2& c?<I> 1K2\~dR2 + 1R2 J dR? + ("' 'dR + um

1 f du2 \2 du2 du2 ,2\dR ~dR+RdR+hl ^* = 0.dR

ONE-DIMENSIONAL GINZBURG-LANDAU EQUATIONS 365

The precise form of the coefficient (• • •) is irrelevant here, since, recalling that

d2ui d?u2 . . „+ <0 for R > 0,dR2 dR2

1 1 f du2\2 du2 i ^du2"j 2 \dR) dR dR

du2 _ R (d2u2\2 du2~dR ~~ a? \~d& ) ~dR>

we see that <I> cannot have a positive maximum for R > Ro, which clearly contradicts<3>(oo) = 0.

It is now clear that A(0) = 0 =£- ip' = 0. Suppose now that tjj' = 0 but ^4(0) < 0. ByLemma la, at some xq, A must vanish. By Lemma lc, ^'(xq) must be negative, which isimpossible as we have just proved. If ,A(0) > 0 and ip' = 0 we apply the transformation

ip{x) —> i/>(—x); A(x) —> — x)

and proceed as before. □It is interesting to note here that when studying (1.1) on a finite interval, nonsymmetric

solutions can exist [3], [4] though the problem is symmetric. The arguments applied inthe proof of Lemma 2 are not valid on finite intervals since the problem is not translationinvariant there.

Lemma 2 allows us to prove the following result.

Lemma 3. Consider the function

G = — +ipA.K

For type I (type II) superconductors, G must be nonpositive (nonnegative) Vx > 0.

Proof. It is easy to show that G satisfies the relation

G" - V>'kA+ — G'- 2/c2t/>2 — kA— + kA'tp

G = (1 - 2K2)ip3A.

Obviously for type I superconductors, G cannot have a positive maximum, whereas fortype II it cannot have a negative minimum. The lemma is now proved since by Lemma2, G(0) = 0 and by Lemma Id, G(oo) = 0. □

An immediate consequence of Lemma 2 is that when k = 1/%/2, G must identicallyvanish. Hence every solution of (1.1) must be a solution of (1.3).

Proof of Theorem 1. We shall consider only type I superconductors. The proof fortype II superconductors is almost identical. We first prove part (a) of the theorem.

Suppose for a contradiction that although h > l/\/2, a solution (ip,A) exists for (1.1).Following [7] we define the function

F = V2 - 1 + V2A'.By Lemma 1 we have

G' > K(ipF + AG). (3.4)

The Hamiltonian relation (3.1) may be represented in terms of the functions F and G inthe form

F{il? - 1 - V2A') -2G - i>A) = -2 fh2 - i

366 Y. ALMOG

Since G(0) = 0, ip < 1, and A! > 0 we must have F(0) > 0, and hence, by (3.4), G'(0) > 0,which clearly contradicts Lemma 3.

We now prove part (b) by invoking informal arguments. We first note that by LemmaId, when x —> oo, ip is approximately, up to an exponentially small error, a solution of

ip" — n2[(hx + a)2 - l\ip = 0 (3.5)

in which a = lima;_00(J4 — hx). We seek an asymptotic approximation for the solutionof (3.5) as x —> oo. To this end we set the WKB expansion [13]

ip = exp{u>i + w2 + u>3 H } (3.6)

wherein wn ~ o(wn-i). Substitution into (3.5) yields

12 fl-KWi = T^unx , W2 = —nax, W3 = ;— logo;,z h

from which we obtain

G=(-^ + 0(x'2)),''

Obviously h must be greater than k. otherwise Lemma 3 is violated once again. □We note that in both cases, for h > l/\/2 and for h < k (for type I), Lemma 3 is

violated. However, in the first case the negativity of G is violated near x = 0, whereasin the latter one it is violated as x —► 00.

4. Concluding remarks. In the following we comment on the significance of theprevious results to time-dependent analysis.

The linear stability of the solutions to (1.1) has been studied by Chapman [6] in thecase ip ^ I- His results clearly show that superconducting solutions are stable for typeII superconductors and unstable for type I. Expecting that these results would be stillvalid for solutions for which tp is not necessarily small, it seems plausible that for typeI superconductors, the solutions obtained in Sec. 2 are not of any practical importancesince they are unstable. On the other hand, we expect to have stable solutions fortype II superconductors when l/\/2 < h < n. Our energy calculations support thisassumption: the superconducting solutions were found to have lower (higher) energythan the normal state for type II (type I) superconductors. The results of Sec. 3 clearlyshow that such "mixed state" solutions do not exist when either h < 1 /\/2 or h > k.Linear stability analysis [6] shows that for h > k the normal solution (1.2) is stable;however, for 0 < h < l/\/2 no stable solution of (1.1) exists. In the latter case, thelong-time solution of the time-dependent Ginzburg-Landau equation cannot be steady.

If we look at the leading-order solution of (2.2), ipo we see that as hi —> 0, ip —»1. Looking at Chapman's [5] plots of ipo we see that as ip becomes closer to 1 thedomain in which tpo is very close to 1 becomes larger. This domain constitutes a purelysuperconducting region around x = 0. In addition, two transition layers and two normalregions, as x —> ±00, exist. The solution inside the purely superconducting region is(almost) ip = 1, A = 0. In the normal region it is approximately (1.2) and the transitions

ONE-DIMENSIONAL GINZBURG-LANDAU EQUATIONS 367

layer, formed around x = ±a:o(/ii) (at which ip = 1/2 say), approximately satisfies thefollowing boundary conditions:

-0 —> 1, ^4. —>■ 0 as sign(a?o)a: — xq —> — oo,ip' —> 1, A' —i> as sign(xo)a; — x$ —► oo.

The qualitative features of the solution of (1.1a) and (1.1b) together with the aboveboundary conditions, have been thoroughly discussed in [8]. For type II superconductorswhen h > 1/\f2 we expect that the transition layers would save as wavefronts propagatingtowards ±oo. The velocity of propagation should be determined by the free-boundarymodel obtained by Chapman [7].

For large but finite intervals we expect that the solution would reach a steady statewhen the wave fronts hit the edges. Yet, the solution is expected to be of a completelydifferent nature than the solutions obtained in this work or from any linearized solution[12]. It is more likely to be wholly superconducting in the bulk of the material, whereasnear the edges a boundary layer is expected. Such a solution was provided in [10].

Acknowledgments. Dr. S. J. Chapman is acknowledged for suggesting to the authorthe idea of performing the asymptotic analysis in Sec. 2. The author wishes to thankProfessor J. Rubinstein for a lot of helpful comments.

References[1] M. S. Berger, Creation and breaking of self-duality symmetry—a modern aspect of calculus of

variations, Contemp. Math. 17, Amer. Math. Soc., Providence, RI, 1983, pp. 379-394[2] E. Bogomolnyi, The stability of classical solutions, Sov. Jour. Nucl. Phys. 24, 449-454 (1976)[3] C. Bolley and B. Helffer, Rigorous results on Ginzburg-Landau models in a film submitted to an

exterior parallel magnetic field I, Nonlinear Stud. 3, 1-29 (1996)[4] C. Bolley and B. Helffer, Rigorous results on Ginzburg-Landau models in a film submitted to an

exterior parallel magnetic field II, Nonlinear Stud. 3, 121-152 (1996)[5] S. J. Chapman, Nucleation of superconductivity in decreasing fields I, EJAM 5, 449-468 (1994)[6] S. J. Chapman, Nucleation of superconductivity in decreasing fields II, EJAM 5, 469-494 (1994)[7] S. J. Chapman, Asymptotic analysis of the Ginzburg-Landau model of superconductivity, reduction

to a free boundary model, Quart. Appl. Math. 53, 601-627 (1995)[8] S. J. Chapman, S. D. Howison, J. B. Mcleod, and J. R. Ockendon, Normal/Superconducting tran-

sition in Landau-Ginzburg Theory, Proc. Roy. Soc. Edinburgh 119A, 117-124 (1991)[9] A. T. Dorsey, Dynamics of interfaces in superconductors, Ann. Phys. 233, 248-269 (1994)[10] V. A. Ginzburg and L. D. Landau, On the theory of superconductivity, Soviet Phys. JETP 20,

1064-1082 (1950)[11] J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Applied Mathematical

Sciences, vol. 34, Springer-Verlag, New York, 1981[12] M. H. Millman and J. B. Keller, Perturbation theory of nonlinear boundary value problem, J. Math.

Phys. 10, 342-361 (1969)[13] J. D. Murray, Asymptotic Analysis, second edition, Applied Mathematical Sciences, vol. 48,

Springer-Verlag, New York, 1984