On small perturbations of stationary states in a nonlinear nonlocal model of a Josephson junction

20 February

1995

PHYSICS

LETTERS

A

Physics J_etters A 198 ( 1995) 105-l 12

On small perturbations of stationary states in a nonlinear nonlocal model of a Josephson junction G.L. Alfimovapl, V.P. Silinb a EV Lukin Research Institute of Physical Problems, Zelenograd, 103460 Moscow, Russian Federation h EN. Lebedev Physical Institute of the Russian Academy of Sciences, 117924 Moscow, Russian Federation Received 24 November 1994; accepted for publication 23 December Communicated by V.M. Agranovich

1994

Abstract The problem of weak perturbations of steady states in a Josephson junction is considered in the strong coupling limit, when the Josephson length AJ is smaller than the London depth AL. This problem reduces to a nonlocal eigenvalue problem with periodic or localized potential. An infinite set of eigenfrequencies and corresponding eigenmodes of perturbations is found in explicit form for two different kinds of steady-state periodic structures and for a localized formation, corresponding to an isolated magnetic Josephson vortex.

1. One of the basic equations

of Josephson junction

theory [l]

is the well-known

sine-Gordon

equation

urr-u,,+sinu=O.

(1)

It describes the distribution of the phase difference on a Josephson junction in the case when the Josephson length A; is considered to be much greater than the London penetration depth AL. The simplest objects covered by this equation are equilibrium vortex structures corresponding to its different steady-state solutions u( t, X) - u(x). It is well known that there are three different kinds of these solutions: (a) Periodic solutions, which are of the form u,(x,k)

=z-+2arccos[dn(x,k)].

Here dn(x, k) is Jacobi P = 2K( k), where

elliptic

(2) function

with modulus

aI2 K(k)

=

is a complete

elliptic integral of the first kind.

’ E-mail address: [email protected]. Elsevier Science B.V. SSDIO375-9601(95)00018-6

k, 0 < k <

1 [2].

The period

of the solution

is

G.L.Aifmov, V.P.Silin /Physics LettersA 198 (1995)105-112

106

(b) Rotational formula

(periodic

ur(x, k) = 2arccos[

mod 27r) solutions

-sn(x/k,

which are determined

over the period ( -kK(

k) , kK( k) ) by the

k)],

(3)

where sn(x/k, k)) is the Jacobi elliptic sinus [ 21, 0 < k < 1. (c) The 2r-kink solution has the form ubnk( x, k) = 4 arctan( e-“)

(4)

and can be obtained from solutions (2), (3) when k --+ 1 (P -+ w). Solutions (2) and (3) correspond to steady-state periodic vortex chains situated along the junction whereas solution (4) describes one isolated magnetic vortex (fluxon). The classical results on small-amplitude perturbation analysis for these formations are well known. Let us recall those results for the case of periodic solutions (2). Substituting into Eq. ( 1) the perturbed periodic solution (2) in the form u(x, t) = u,,(x, k) + A(x, k) e”‘, we obtain that this problem reduces to the analysis of the Lam6 equation, AxX - [2k*sn*(x,

k) - l]A=

PA,

p=c*.

(5)

It is known [ 3 ] that the set of periodic solutions with period P = 4K( k) for this eigenvalue problem consists of two families of Lam6 functions At (x, k) , i = 0, 1,2, . . . and A; (x, k) , j = 1,2, . . . with corresponding eigenvalues p;(k) and p.7 (k) . The eigenfunctions AT(x, k) are even and the A,;( x, k) are odd * . Every eigenfunction Ai (x, k) , A;( x, k) has exactly n zeros in the half-open interval [ 0, $P) (semiperiod). and pj (k) corresponding to the eigenfunctions A: (x, k) and A; (x, k) are ordered, &(k)

> ,$(k)

> p;(k)

/-4(k) >/4(k)

>/4(k)

&( k)

> ... .

namely G( x, k) , AT (x, k) and Ai (x, k) , have explicit forms (the Lamt polynomials),

Three eigenfunctions, Ag(x,k)

> . . .t

The eigenvalues

=dn(x,k),

As(x, k) = sn(x, k),

p;(k)

= 1 -k*,

,4(k)

= -k*.

AF(x, k) = cn(x, k),

Other eigenfunctions, Ai (x, k) , A;( x, k), n > 2, are transcendental infinite series only. In this case the following equality holds [ 31,

p;(k) =/-4,(k),

p;(k)

Lam6 functions

= 0,

and can be represented

by

n 2 2.

So the spectrum of problem (5) is represented by three simple eigenvalues and an infinite number of double eigenvalues. The zero eigenvalue ,uc (k) = 0 corresponds to the translation mode AT(x, k) = (d/ dx) IcP(x, k) . Furthermore there is the only positive eigenvalue &(k) for arbitrary k, 0 < k < 1. This implies the instability of the periodic solutions (2) for any value of period P. The solutions A:( x, k), A; (x, k) and ,uF( k) , pT( k), i 2 2, j > 1 determine unstable oscillations around the steady state (2). The similar problem for the rotational solutions (3) has the same spectrum structure because it reduces to the same Lam6 equation (5) by means of substitutions. But in this case all eigenvalues of the corresponding eigenvalue problem turn out to be negative, except the only zero eigenvalue, which corresponds to the translation * The correspondence 4,

I (x,k)

= E.:“‘+‘(x),

between these notations for the Lam6 functions and the notations of n;m(x,k)

= E.~;m(x).

n;m+l (x,k)

= Ecy+‘(x).

[ 31 is as follows: A&(x, k) =

Ecf”( x),

GLAljimov,

V.P. Silin /Physics

Letters A 198 (199s) 105412

107

mode. This implies the marginal stability of rotational solutions (3) with respect to periodic modulations of spatial period P = 2kK( k), which is equal to the period of the solution u,( X, k) . The statement on the marginal stability of the 2z--kink solution (4) can be obtained by means of passing to the limit when k --) 1 (P ---fCO) in the spectrum of (5). 2. However, nowadays examples of Josephson junctions are under consideration in the literature for which the sine-Gordon approximation ceases to be true [4-61. They are the Josephson junctions between superconductors for which the Josephson length AJ is smaller than the London depth AL of the magnetic field penetration into the superconductor. For example, such a situation can arise in the case of high temperature superconductors. To describe the Josephson junction in this case the following dimensionless equation for the phase difference u(x, t) on the Josephson junction was suggested (see Refs. [5-7]), u,, - H[ u,] + sin z4= 0.

(6)

Here H[ $1 is a Hilbert transform,

-co

where the integral is the principal value of the Cauchy integral. Eq. (6) has a very simple form and is free from external parameters; in Ref. [ 61 it was called sine-Hilbert equation. Steady-state solutions for Eq. (6) satisfy the equation sinu = H[u,].

(7)

This equation (up to sign) arises in the dislocation theory for the description of the Peierls dislocation [ 81. Three different kinds of its solutions are known, they correspond exactly to three kinds of steady-states solutions of sine-Gordon equation: (a) Periodic solutions with period P = 2r/tanh (Y, $(x,a)

=7r+2arctan

(Y> 0 is a parameter. (b) Rotational (periodic

sin( x tanh cr) sinhcu

.

mod 27r) solutions which are determined

over the period ( - ;P, i P) by the formula

tan[ i sinh(p)x] z&(x,/3) = T + 2arctan

tanh($)

’

(9)

Here P = 21r/ sinh p and p > 0 is a parameter. (c) The 2r-kink solution i&k(X)

=r+farctanX,

(10)

which can be obtained from solutions (8) and (9) by means of transitions (Y -+ 0 or p + 0. Similar to solutions (2) and (3)) solutions (8) and (9) describe the periodic vortex magnetic structures in the Josephson junction. Solution ( 10) describes a solitary magnetic vortex (fluxon). The main purpose of this paper is to investigate the problem of small amplitude perturbations for the steadystate structures (8)-( 10). In Sections 4, 5 and 6 the corresponding nonlocal eigenvalue problems are considered. An infinite set of eigenfrequencies and corresponding eigenmodes of perturbations is found in explicit form for steady-state periodic structures (8) and (9). It is shown that it is a complete solution of the problem in

108

G.L.Aljimov, V.P. Silk /Physics

Letters A 198 (1995) 105-112

a class of excitations which conserve the basic spatial period P of steady state. Furthermore, a complete analytical solution is found for the problem on weak perturbations of the isolated magnetic Josephson vortex ( 10). 3. Let us recall now some properties of the Hilbert transform [9] and give some useful formulas. Let L(P) be a Hilbert space of periodic functions with period P = 25-/w, where the trigonometric basis {sin nwx, cosmwx, n= 1,2 ,..., m=0,1,2 ,... } exists. Then the following relations hold, (i) H[sinwx] (ii) H

= coswx,

y&u 1= $H[u]

[

E ‘Flu,

and the operator N is self-adjoint Let us denote

s, =

sin nwx

formulas

H[S2n+11

=C2n+1

-

H[C2n+11

= -S2n+1

w 3 0,

(11)

(12)

UEL(P)!

in L(P).

c, =

cos 2wx - cash 2cy ’

Then the following

H[ cos wx] = - sin ox,

w > 0,

cos nwx

cos 2wx - cash 2a

are valid,

,-(2n+l)a

cosha

n=0,1,2

CI,

,...)

(13a)

,-(2n+l)n

-

wsinhn

H[ Szn] = CZ,, - e-2na Co, NC2nl

= 42n

n=0,1,2

SI,

(13b)

,...)

n= 1,2,..., n=0,1,2

- &S2,

(13c)

(13d)

,....

A deduction of formulas ( 13a), ( 13b), ( 13d) for the case n = 0 is given in Ref. [ 81. To check formulas ( 13) for n # 0 it is sufficient to expand both right- and left-hand sides of (13) in Fourier series and to use the relations ( 11) . 4. Let us turn to the stability solution (8) in the form

2.4(x,t> = i&(x, a) + $(x, we obtain the following

analysis

of the periodic solutions

(8). Substituting

into Eq. (6) the perturbed

a>e”‘,

eigenvalue

problem, 4 sinh2 (Y

cos 2wx - cash 2a >

* = A*,

(14)

where w = tanha, A = u2. Some particular solutions of this problem were represented in Ref. [ lo]. Complete information tions of problem (14) periodic with period P = 27r/w is contained in the following statement. Statement. The complete

set of periodic solutions

of period P = 27r/w for the problem

(w E tanha!), &(&a)

=

1 cos 2wx - cash 2cu’

A;(a)

= w2,

(14)

about solu-

is as follows

G.L.Aljimov,

@(&a> = qq(x,a)

=

+;(&a)

=

&(x,a)

=

cm wx sin wx

cos 20.x - cash 2a ’

=

=o,

’ AC,(a) = A;(o)

= -1,

cosnwx-2e-2ncos(n-2)wx+e-4”cos(n-4)wx cos 2wx - cash 2a

,

sin nox - 2 e-2cr sin( n - 2)wx + eA4a sin( n - 4) wx

cos 2wx - cash 2a

n=3,4,...

109

- 2 e-*a

cos 2wx - cash 2a sin 2wx

Letters A 198 (1995) 105-112

A; (a) = -1 + a*,

cos 2wx - cash 2cu ’

(clnc(x,Ly)= $$(x,n)

/q(a)

cos 20x - cash 2ff ’

cos 2wx( 1 + e-4a)

V.P. Silin /Physics

/$(a)

= A;(a) = -[ 1+ (n - 2)0],

.

(15)

Applying formulas (13) it is easy to check that all $~(x,(Y), h:(a), n = O,l,. . . as well as $~
sin( 21+ 1) wx = - sinh 2cu( e-2a’ + e-2a(‘+1))qQ~ - sinh 2cu c

e-2a(‘-i)

I&+, + $,&+3,

i=l

I= 0,l

.,

92,.

(16) 1

cos(21+

1)wx = -sinh2a(e-2”‘-e-2n(1’1))~~

l=O,1,2

.

1

- sinh2cr)‘e-*“(‘-‘)cG~~+, U i=l

(...,

+ $I&+,, (17)

I sin 21wx = - sinh 2a c

e-2a(‘-i)

I& + $&+2,

1=1,2,...,

(18)

i=l i-1

21wx = - e-2a1 tanh 2cu (sinh 2ar& + 4 e4a &)

cos

I= 1,2,..., I=--.

&

- sinh 2a c e-2a(’ I=1

--I

(19) (sinh* 2@6 - i e2a @ ) .

(20)

Here the following notations are used: $i E (G-,” (x, (w), @h 3 @,Y,, ( X,(Y). In formulas (16)-( 19) the sums are considered to be zero if a lower limit of summation exceeds an upper limit. Let one more eigenfunction 6 (x) exist which does not belong to the set of functions (15) and is orthogonal to all these functions. However, it follows from formulas (16)-(20) that r+?(x) is orthogonal to all functions of the basis {sinnox,cos mwx, n = 1,2,. . ., m = 0, 1,2,. . .} and consequently 1,8(x) z 0.

110

G.L.Aljimov,V.P.Silin/PhysicsLettersA 198(1995) 105412

Observation. Each of the functions $z( x, a). Qi(x, [ 0, i P) (semiperiod). See the Appendix for the proof.

(Y) has exactly

n zeros on the half-opened

interval

The presence of one positive eigenvalue A = h:(a) in the spectrum of problem (14) permits us to conclude that the periodic solutions (8) are unstable for an arbitrary value of the period P. The zero eigenvalue A = hT(cu) = 0 corresponds to the translation mode e(x) = I&(X,(W). Those solutions were found earlier in Ref. [ lo]. It is necessary to underline a remarkable similarity between the eigenvalue problems in the local and nonlocal cases. In both cases there exists the only positive eigenvalue corresponding to the node-free unstable mode and the only zero eigenvalue corresponding to the translation mode. In either case the spectrum of the problem is represented by three simple eigenvalues and an infinite number of double eigenvalues. Moreover there is a one-to-one correspondence between the eigenfunctions of the Lame equation ,$( x, k), A;( x, k), and the eigenfunctions $t (x, cu), I# (x, (Y), i = 0, 1, . . ., j = 1,2,. . .. The functions i=O,l,..., j= 1,2 ,... corresponding to each other have the same parity and the same number of zeros. However, in contrast to the local case, all solution of period P of the nonlocal problem can be written in closed form and have simple expressions ( 15). Let us note that it is possible to rewrite the system of the functions (15) in a more compact equivalent form, replacing functions +i (x, a), +i(x, (u), n 2 2 by the complex functions +T (x, a), $; (x, a), where +e-4”)t&(x,~)l

@(x,(y)

= ~e**[@(x,a)

&ill

$:(X,(Y)

= ie2a[+i(x,(Y)

*i$~(x,c~)l

=

sin( wx F icu) sin( wx f icu) ’

sin( wx F ia) = sin(wxficu) e*i(“-2)oxY

5. Let us consider the stability problem for the rotational (9) solutions this case the corresponding eigenvalue problem has the form

-&$I

cash p +

-

sinh* /3 cos 2yx - cash /3 >

It is easy to check that problem p=2a,

ij =xlj,

equation

and eigenfunctions

(14) by means of the substitutions

(23)

of (22) in terms of functions

q&-M>

= tii!X~cosh*(~PL

$3,

G(P)

= 0,

$;(.O)

= rc/;(xcosh*($P>,

$1,

G(P)

= -+(coshj?

- l),

&(x,P)

=cf/;(=osh*(

%$$>,

ii;(p)

= -$(coshp+

1),

&x,P)

= $:(xcosh*($?L

$?>,

&(x,P>

i(n - 2) sinhp].

(6). In

(22)

x = Acosh*( i/?) - sinh*( $p).

This allows one to express all eigenvalues

= -[coshp+

of the sine-Hilbert

(21)

y= isinhp.

(22) can be reduced to problem

Y=xcosh*($?),

XC,(p) = x”,(p)

n = 3,4, . . .

= rl/;(xcosh*($),

$3),

n = 2,3,.

( 15),

. ., (24)

Formulas (24) represent all solutions of period P = 27r/y for eigenvalue problem (22). The eigenvalue A;(p) = 0 corresponds to the node-free translation mode $6 (x, p) = dii,( x, p)/ dx, other eigenvalues are negative. From this follows the statement on the marginal stability of rotational solutions with respect to modulations of period P (see Ref. [ lo] ).

G.L.Aljimov,

V.P. Silin/PhysicsLettersA

6.Lastly, let us discuss the problem of small amplitude case the corresponding eigenvalue problem has the form

198(1995)

perturbations

105-112

111

for the 2n--kink solution

(10). In this

(25) The solutions

of this problem can be easily obtained by passing to the limit LY-+ 0 in formulas

cl/o(x)= -

( 15) and (21),

1 A0 =o,

1 +x*’

XT i @(x)=xe

fiqx

,

Aq=-(l+q),

430.

This implies that the spectrum of problem (25) is represented by one zero eigenvalue to the translation mode $0 (x) and the continuous spectrum A < - 1.

Aa = 0, which corresponds

The authors are grateful to Professors V.M. Eleonsky and N.E. Kulagin for useful discussions. One of the authors (V.P.S) was supported by the Scientific Council for High Temperature Superconductivity Problems (project No. 93015). G.L.A was supported by grants No. 94-01-01504(a) of the Russian Foundation for Basic Researches and No. R0500 of the International Science Foundation.

Appendix Let us prove that the functions @i = @i (x, (u) , I,/J~E I); (x, a) have, on the half-open interval 2n-/w, exactly n zeroes. For n = 0, 1,2,. . . it is easy to check this statement in a straightforward n = 3,4, , . . let us consider the complex-valued function I,$:,

Icl,‘<-w>= +e*“[q$(x,a)

*i@i(x,a)]

[0,iP) , P = fashion.

For

sin(wx - ia) ei(n_2)wx _ -explip(x,a)l, sin( wx + icu)

=

where x

&X,(Y) =7r+(n--2)wx+2wsinh2a

J

d5

cash 2a - cos 2~5’

0

When x increases from x = 0 up to x = 27r/w, cp(x, (Y) increases monotonically from qo= 7r up to rp = (2n+ 1) T. As this takes place the representative point (cr,’(x, a) moves monotonically in the complex plane along the circle Iz( = 1 andmakencompleteturns.ThevaluesRe+~(x,cu) = ie2”+;(x,cr) andImcCI,+(x,a) = ie20&(x,cr) vanish alternately twice per one turn. Taking into account the evenness of tig (x, a) and oddness of @F(x, a) with respect to x we obtain the desired result.

References 111A. Barone and G. Patemo, Physics and applications of the Josephson effect (Wiley, New York, 1982) 121 M. Abramowitz and IA. Stegun, Applied mathematics series 55, Handbook of mathematical functions (National Bureau of Standards, 1964). r31 H. Bateman and A. Erdelyi, eds., Higher transcendental functions, Vol. 3 (McGraw-Hill, Vol. 5 ( 1992) p. 230. 141 YuM. Aliev, VP Silin and S.A. Utyupin, Superconductivity, [Sl A. Gurevich, Phys. Rev. B 46 (1992) 3187.

New York, 1955).

112

G.L.Aljintov, V.P. Silin /Physics

Letters A I98 (1995) 105-112

161 Yu.M. Aliev and VP Silin, J. Exp. Theor. Phys. 77 ( 1993) 142. [7] R.G. Mints and LB. Snapiro. Phys. Rev. B 49 ( 1994) 6188. 18 1 A. Seeger. Theorie der Gitterfehlstellen, in: Handbuch der Physik, Vol. 7, part I Kristallphysik (Springer, [ 91 H. Bateman and A. Erdelyi, eds., Tables of integral transforms, Vol. 2 (McGraw-Hill, New York, 1954). [IO] G.L. Altimov and VP Silin, J. Exp. Theor. Phys. 79 (1994) 369.

Berlin, 1955) p. 383