Josephson Effect and Isomonodromic Deformations
JJosephson Effect and Isomonodromic Deformations
Yulia BibiloOctober 5, 2018
Abstract
We consider some properties of double confluent Heun equation related to the Joseph-son Effect. In particular, we prove that adjacency points of phased-locked areas on aparameter plane can be described via poles of Bessel solution of Painleve 3 equation.
The main object in this work is a system of two linear differential equations with twoirregular singularities of the following form dxdζ = (cid:32) − a ζ − bζ − a iωζ iωζ (cid:33) x, (1)where ζ is variable over the Riemann sphere ¯ C , x ( ζ ) ∈ C is an unknown vector-function,and a, b, ω are scalar (and real) parameters that do not depend on ζ . The system (1)is equivalent to a non-linear differential equation which models the Josephson effect insuperconductivity. Adjacency points of so-called Arnold tongues (phased-locked areasthat are level sets of a rotation number) correspond to values of parameters when thesystem (2) has the trivial monodromy data (i.e. identity monodromy matrices and identityStokes matrices) [1].We consider an isomonodromic family dxdζ = (cid:18) A ζ + A ζ + A (cid:19) x, (2)(here ζ ∈ ¯ C , x ( ζ ) ∈ C is an unknown vector-function, and A , A , A are coefficientmatrices that may depend on a deformation parameter) with the trivial monodromy data.We are looking for values of deformation parameter such that the isomonodromic familycoincides with the original system, i.e. it has the same structure as the linear system (1)describing the Josephson effect.The isomonodromy condition in the case of the family (2) is equivalent to Painleve 3equation. Thus the coefficients of the isomonodromic family can be described explicitlyin terms of a solution of Painleve 3 equation. a r X i v : . [ m a t h . C A ] J un Results on rotation number
In this section we provide a brief review of results on dynamical systems modeling theJosephson effect. See [1 ,
19] for detailed information and open problems.Following articles [5 , ,
19] we consider a family of nonlinear differential equations dφdt = − sin( φ ) + B + A cos( ωt ) , (3)where A , B , and ω are scalar real parameters. It is a modeling equation for the Josephsoneffect in superconductivity. A rotation number and its phase locking domains were studiedin articles [8 , , , , , A, B, ω andit gives some description of periodic trajectories.It is possible to transform the nonlinear equation (3) into DCHE that is a lineardifferential equation of the second order. It can be done in several steps [1 , τ = ωt in (3) we get dφdτ = − sin( φ ) ω + Bω + Aω cos( τ ) , this equation is defined on T = S × S , i.e. ( φ, τ ) ∈ R / π Z . The graphs of its solutionsare the orbits of the vector field (cid:26) ˙ φ = − sin( φ ) ω + Bω + Aω cos( τ )˙ τ = 1 . (4)According to [1 ,
19] equation (4) has the following form after the transformationΦ = e iφ , ζ = e iτ d Φ dζ = (cid:18) Bωζ + A ω + A ωζ (cid:19) Φ − iωζ (Φ − , the unknown function Φ is related to the unknown vector-function x = (cid:18) x x (cid:19) of thelinear system dxdz = (cid:18) − A ωz − Bωz − A ω iωz iωz (cid:19) x (5)as Φ = x /x . If we denote parameters as b = Bω , a = Aω , (6)we get the system (1), which is the object of our studies.It is proved in [19] that E ( z ) = e µz x ( z ) satisfies DCHE d Edz + (cid:18) az + b + 1 z − a (cid:19) dEdz + (cid:18)(cid:18) ω − a (cid:19) z − a ( b + 1) z (cid:19) E = 0 (7)if and only if x ( z ) is the second component of the solution x of the system (1).Rotation number (see [1 , ,
19] for details) is a mapping ρ ( bω, aω ; ω ) = ρ ( B, A ; ω ) = lim k → + ∞ φ (2 πk ) k , where φ is a general solution of the first equation of the system (4). ts level sets { ρ ( bω, aω ; ω ) = r } with non-empty interior are called phase lockingdomains or Arnold tongues . Arnold tongues have symmetries ( b, a ) → ( b, − a ), ( b, a ) → ( − b, a ) [5]. Thus we may focus on the case when a > b ≥
0, and ω >
0. There aremany significant results on Arnold tongues.The boundary of each Arnold tongue consists of two analytic curves { b = ω g r, ± ( a ) } [9], where functions { g r, ± } have Bessel asymptotics [5]: (cid:26) g r, − = r + 1 /ωJ r ( a ) + o ( a − / ) ,g r, + = r − /ωJ r ( a ) + o ( a − / ) , a → + ∞ , a ∈ R . Figure 1: Arnold tongues (phased locked areas) on the parameter plane ( b, a ). Each Arnold tongue is an infinite chain of bounded domains going to infinity in thevertical direction [5], in that chain each two subsequent domain are separated by onepoint. If a = 0, they can be calculated explicitly. The other separation points, lie outsidethe horizontal b -axis, are called the adjacency points [1 , b = k ∈ Z on ( b, a )-planedue to the lemma: Lemma 1 ( [ , Lemma 3.3 ] ) A pair of parameters ( b, a ) corresponds to an adjacencypoint if and only if b ∈ Z , a ∈ R \{ } and the linear system (1) is (locally) analyticallyequivalent to its formal normal form in the neighborhood of z = 0 . In [19 ,
3] borders of phased lock areas were described via ’conjugated’ DCHE - if ithas a polynomial solution, the pair of parameters is a point on a border of some Arnoldtongue.
We start with the system (1). One of the goals of the following computations is to findvalues of parameters a, b, ω such that the linear system (1) has trivial monodromy data.First we apply a transformation ζ = z/a to the system (1) and get dxdz = (cid:18) − a z − bz −
12 12 iωz iωz (cid:19) x. (8) he systems (1) and (8) have the trivial monodromy and Stokes matrices for the samevalues of parameters. The system (8) is a particular case of a linear problem related toPainleve 3 equation where deformation parameter is t = a . A linear system of the form dxdz = (cid:18) A z + A z + A (cid:19) x, (9)with two irregular singularities at z = 0 and z = ∞ is a particular case of a meromorphicsystem over the Riemann sphere. Both singularities of (9) have Poincare ranks equal 1 andwe suppose that they are non-resonant, i.e. each of matrices A and A has two non-equaleigenvalues. In a neighborhood of a non-resonant irregular singularity the linear systemis formally equivalent to a diagonal one (and if Stokes matrices are identity matrices, itis locally analytically equivalent to a differential system of linear equations with diagonalcoefficient matrix [11]).A family dxdz = (cid:18) A ( t ) z + A ( t ) z + A ( t ) (cid:19) x, t ∈ D , (10)( t is a parameter and D is a deformation space) is said to be isomonodromic if1. it includes the original linear system, i.e. there is t ∈ D such that A ( t ) = A , A ( t ) = A , and A ( t ) = A ;2. monodromy representation and Stokes matrices are constant.An isomonodromic family is also called isomonodromic deformation. Theorem 1
A family of linear differential equations (10) is isomonodromic if and onlyif there is a meromorphic differential 1-form Ω such that Ω | fixed t = (cid:18) A ( t ) z + A ( t ) z + A ( t ) (cid:19) dz, (11) d Ω = Ω ∧ Ω . (12)Here condition (12) means that the differential form is integrable in the Frobenius sense.The isomonodromic deformation (10) is a particular case of Jimbo-Miwa-Ueno isomon-odromic deformation (see [14 ,
15] and also [11]) of a meromorphic system over a Rie-mann sphere (and it can be also considered as a Malgrange’s isomonodromic deformation[17 , t is a vector that includes singular points and diagonalentries of coefficient matrices of local normal forms of the meromorphic system. (10) hasonly two singular points so their deformation can be described as meromorphic transfor-mation of variable z .If one knows an analytic fundamental solution Y ( z, t ) of isomonodromic deformation(10), 1-form from the theorem 1 can be defined in terms of Y ( z, t ). In case when t is ascalar parameter, it has the form Ω = ∂Y∂z dz + ∂Y∂t dt , t ∈ D . .3 Lax pair Next we consider an isomonodromic deformation introduced by M. Jimbo in [6] andstudy its integrability conditions (12). The deformation space is D = (cid:94)C \{ } . ∂Y∂z = (cid:18) − tz A ( t ) + 1 z B ( t ) + (cid:18) −
00 0 (cid:19)(cid:19) Y, (13)where G ( t ) in holomorphically invertible matrix on D . ∂Y∂t = 1 z A ( t ) Y, (14)where coefficient matrices are A ( t ) = G ( t ) (cid:18)
00 0 (cid:19) G − ( t ) , (15) B ( t ) = (cid:18) − b ∗∗ (cid:19) , (16)˜ B ( t ) = G − ( t ) B ( t ) G ( t ) = (cid:18) − b ∗∗ (cid:19) . (17)Here ∗ is a placeholder for unspecified entries of matrices.The choice of a deformation parameter t is determined by formal normal forms of (13). System (13) has the fundamental solution near z = ∞ Y ∞ j ( z, t ) ∼ ( I + O ( z − ))( z − ) J ∞ e − z
00 0 , z → ∞ in L ∞ j , (18)where L ∞ j = { z ∈ C : − πj + π < arg (cid:16) − z (cid:17) < − πj + 5 π } ,J ∞ = (cid:18) − b
00 0 (cid:19) , and Y ∞ ( z, t ) = Y ∞ ( z, t ) S ∞ , S ∞ = (cid:18) s ∞ (cid:19) , (19) Y ∞ ( z, t ) = Y ∞ ( ze π √− , t ) e πJ ∞ = Y ∞ ( z, t ) S ∞ , S ∞ = (cid:18) s ∞ (cid:19) . Here S ∞ and S ∞ are Stokes matrices, s ∞ , s ∞ are constants.Local solutions of the system (8) near z = 0 are the following. Y j ( z, t ) ∼ G ( t )( I + O ( z )) z J e − t z −
00 0 , z → ∞ in L j , (20) There is another frequently mentioned isomonodromic deformation that leads to Painleve 3 equation[15 , here L j ( t ) = { z ∈ C : − πj + π < arg (cid:18) − t z − (cid:19) < − πj + 5 π } ,J = (cid:18) − b
00 0 (cid:19) , and Y ( z, t ) S = Y ( z, t ) , S = (cid:18) s (cid:19) , (21) Y ( z, t ) = Y ( ze π √− , t ) e πJ S = Y ( z, t ) , S = (cid:18) s (cid:19) ,S , S are Stokes matrices, s ∞ , s ∞ are constants.So monodromy matrix is G = e π √− J S S , (22)where G appears in Y ( ze πi , t ) G = Y ( z, t ), here Y ( ze πi , t ) is an analytic continuationof Y ( z, t ) along a small loop around z = 0. Thus matrix parameter b is constant if (13)is isomodromic.A formal normal form of the system (13) near z = 0 is ∂ ¯ Y∂z = (cid:18)(cid:18) − t
00 0 (cid:19) z + J z (cid:19) ¯ Y , ¯ Y ( z, t ) ˆ F ( z, t ) = Y ( z, t ) , ˆ F ( z, t ) is a formal Tailor series in z .If we fix a path σ between 0 and ∞ such that z ∈ L ∞ as z → ∞ , and z ∈ L as z → Y ∞ ( z, t ) = Y ( z, t ) C, matrix C is called connection matrix, it is calculated in [6]. Frobenious integrability condition (12) is equivalent to a system of non-linear differentialequations on functions that are entries of matrices in 13. If we eliminate all unknownfunctions except the fraction y ( t ) = − ( B ( t )) / ( A ( t )) , (23)we get a second order non-linear differential equation ˜ P y (cid:48)(cid:48) = ( y (cid:48) ) y − y (cid:48) y + 14 y t − b y t −
14 1 y + 1 t (cid:18) b − (cid:19) , t ∈ D . (24)In general, differential equation ˜ P has the form y (cid:48)(cid:48) = ( y (cid:48) ) y − y (cid:48) t + ˜ γ y t + ˜ α y t + ˜ δy + ˜ βt , t ∈ D ( ˜ P ( ˜ α, ˜ β, ˜ γ, ˜ δ )) nd may be transformed to Painleve 3 equation using the transformation of the unknownfunction y ( t ) = w ( t ) / √ t and the transformation of the independent variable √ t = τ : w (cid:48)(cid:48) = ( w (cid:48) ) w − w (cid:48) τ + w − b w τ − w + (2 b −
2) 1 τ , w (cid:48) = dw ( τ ) dτ . (25)Let us recall that Painleve 3 equation is w (cid:48)(cid:48) = ( w (cid:48) ) w − w (cid:48) τ + α w τ + β τ + γw + δ w . ( P ( α, β, γ, δ ))The parameters of Painleve 3 equation are expressed in terms of b in the following way α = − b, β = 2 b − , γ = 1 , δ = − . (26)Furthermore, the equation (25) is normalized Painleve 3 equation. A solution of Painleve 3 equation is a transcendental function, but for some particularvalues of parameters there are rational, algebraic solutions, and solutions expressed viaspecial functions. If (26) hold, there is a solution expressed via Bessel functions of thefirst and the second kind.We will describe Bessel solutions of Painleve 3 equation following [4].
Theorem 2 If a = γ , c = − δ , and a β + c ( α − a ) = 0 , then w ( τ ) = − du ( s ) ds a u ( s ) is asolution of Painleve 3 equation P ( α, β, γ, δ ) , where τ = λs , λ = ( ac ) − , and u ( s ) = s ν ( C J ν ( s ) + C Y ν ( s )) , ν = α a . The theorem 2 is based on the fact that a solution of Riccati equation satisfies Painleve3 with a special choice of coefficients. The obtained Riccati equation may be linearizedand then solved in terms of Bessel functions. According to the theorem 2 Painleve 3 equation in form (24) that was obtained fromisomonodromy condition has the following Bessel solution (for any b ∈ C ) w ( τ ) = du ( s ) /dsu ( s ) | s = τ . (We take a = − c = − λ = 1, and ν = b to apply theorem 2.)It means that y ( t ) = w ( √ t ) √ t (27)is a solution of ˜ P in the particular form (24).After applying Backlund transformation one gets ’Bessel-type’ solutions of normal-ized Painleve 3 equation that has integer parameters but we will focus on (27). (see([4 , chapter 12.13]), [7]). Theorem 3 If ˆ α + (cid:15) ˆ β = 4 k + 2 , (cid:15) = 1 , and k ∈ Z , then normalized Painleve 3equation P ( ˆ α, ˆ β, , − has one parametric Bessel solution. Defined τ is not related to the variable τ used in section 2. P ( α, β, γ, δ ) is normalized if γ = 1, δ = − Bessel functions J ν ( s ) and Y ν ( s ) are two linearly independent solutions of Bessel equation d yds + s dyds + (1 − ν s ) y = 0. .7 Isomonodromic family defined by Bessel solution Generally it is a complicated problem to find values of parameters in the linear system(1) (and (8)) when its monodromy and Stokes operators are trivial. In this section weare looking for values of t when the isomonodromic family (13) coincides with the system(8). If the isomonodromic family (13) has the trivial monodromy, it gives us a way to findvalues of parameters that correspond to adjacency points.We are interested to find parameter t = t ∗ such that A ( t ∗ ) = A ( t ∗ ) = 0and B ( t ∗ ) = B ( t ∗ ) (cid:54) = 0 . First we suppose that t = t ∗ is a pole of Bessel solution y ( t ) defined above.Let us denote A ( t ) = h ( t ) / ( −
2) and parametrize G ( t ) in the following way G ( t ) = (cid:32) g ( t ) h ( t ) g ( t ) g ( t ) g ( t )+ g ( t ) h ( t ) g ( t ) (cid:33) , where g ( t ), g ( t ), and h ( t ) are meromorphic functions on D . Let us also denote b ( t ) = B ( t ) , b ( t ) = B ( t ) , functions b ( t ), b ( t ) are also meromorphic over D . The integra-bility condition (12) is equivalent to non-linear differential equations (28), (30), and (31)on these functions.An equation g (cid:48) ( t ) g ( t ) = 12 t y ( t ) + g (cid:48) ( t ) g ( t ) (28)leads to g ( t ) g ( t ) = e (cid:82) t y ( t ) dt . Let us rewrite y ( t ) / u defined in the theorem 2: y ( t )2 = du ( √ t ) d ( √ t ) √ t ωu ( √ t ) = du ( √ t ) dt √ t √ t u ( √ t ) = dudt tu . Then g ( t ) g ( t ) = e (cid:82) dudt ttu dt = u ( √ t ) ˜ C , (29)where ˜ C is a constant. Thus, if u ( √ t ∗ ) = 0 and g ( t ∗ ) (cid:54) = 0, then g ( t ∗ ) = 0.From the equation − h ( t )4 − b (cid:48) ( t ) = 0 (30)and the definition of y ( t ) y ( t ) = b ( t ) / ( h ( t )2 )we get b ( t ) = e (cid:82) − dty ( t ) and h ( t ) = 2 y ( t ) e (cid:82) − dty ( t ) . unction y ( t ) has a simple zero at t = t ∗ then b ( t ) has no pole at t = t ∗ and h ( t ∗ ) = 0, b ( t ∗ ) = b (here b is a constant). Entries A ( t ∗ ) = A ( t ∗ ) = A ( t ∗ ) = 0as h ( t ∗ ) = 0, and A ( t ∗ ) = 12 . Integrability conditions do not give other restrictions on g ( t ) except for g ( t ) (cid:54)≡ b ( t ) b ( t ) = b ( t ) g ( t ) g ( t ) h ( t ) + g ( t )( − bg ( t ) + b ( t ) g ( t )) g ( t ) . (31)The second term is zero at t = t ∗ (It follows from (29)). For the first one we get b ( t ) g ( t ) g ( t ) h ( t ) = 12 ω y ( t ) g ( t ) g ( t ) = dudt tu u ˜ C = dudt t ˜ C , (32)where ˜ C is a constant from (29). It means that b ( t ) has no pole at t = t ∗ and b ( t ∗ ) (cid:54) = 0as du ( √ t ∗ ) dt (cid:54) = 0.Finally we get dxdz = (cid:32) − t ∗ z − bz − b ( t ∗ ) zb ( t ∗ ) z (cid:33) x. (33) After the transformation of the independent variable z = a ∗ ζ, ( a ∗ ) = t ∗ , (34)applied to the system (33) we get dxdζ = (cid:32) − a ∗ ζ − bζ − a ∗ b ( t ∗ ) ζb ( t ∗ ) ζ (cid:33) x, where b ( t ∗ ) = b , b ( t ∗ ) = d (cid:0) s l ( J l ( s ) + y Y l ( s )) (cid:1) ds s | s = a ∗ b . Then for the fixed t ∗ (and a ∗ ) we apply a constant in z gauge transformation q = (cid:18) d d − (cid:19) x , d ∈ C \{ } that does not change monodromy data of the previous system dqdz = (cid:32) − t ∗ z − bz − b ( t ∗ ) d zb ( t ∗ ) zd (cid:33) q. (35)If constant d satisfies the condition b ( t ∗ ) d = b ( t ∗ ) d , the system (35) is symmetric.Then we may define parameter ω = ω ∗ using the following equation b ( t ∗ ) b ( t ∗ ) = − ω ∗ ) . (36) inally, the received linear system (35) has the same form as the original system (1)does.Function u ( s ) = s b ( J b ( s ) + y Y b ( s )) (37)appears in solution (27) and it has infinitely many real positive zeroes a ∗ , a ∗ , . . . , a ∗ k , . . . (38)(they are poles of the considered Bessel solution (27) of Painleve 3 equation). So we getcorrespondingly to (38) a sequence ω ∗ , ω ∗ , . . . , ω ∗ k , . . . (39)such that the system (1) with described parameters (or (35)) has identity monodromyand Stokes matrices. Theorem 4
If there is t ∈ D such that the system (13) with t = t has the trivialmonodromy data, then there are infinitely many adjacency points ( b, a ∗ ) , ( b, a ∗ ) , . . . for ω ∗ , ω ∗ , . . . correspondingly, where { a ∗ k } are real positive zeroes of (37) , { ω ∗ k } are determinedby (36) . Constants y , b are determined by the initial system (13) with t = t . Theorem 5 If ( b, a ) is an adjacency point for ω , then ( b, a ∗ k ) is an adjacency pointfor ω ∗ k , where a ∗ k is zero of u ( s ) = s b ( J b ( s ) + y Y b ( s )) , where y = − J b ( a ) Y b ( a ) , (40) and ω ∗ k = ω a duds | s = a a ∗ k duds | s = a ∗ k . (41) Figure 2: Adjacency points defined by theorem 5 in 3-dimensional space of parameters ω, b, a . Note 1
Any adjacency point may be taken as an initial point ( b, a ) in the theorem . Thus any adjacency point can be received via theorem and theorem for appropriateinitial conditions (constants y , b or an adjacency point ( b, a ) ). .9 Generalizations and open problems In the previous calculations we did not utilize the condition b ∈ Z . It means theorems 4and 5 hold also in case of non-adjacency points on the parameter plane – systems of theform (1) with parameters (40), (41) have the same monodromy and Stokes matrices as theinitial system (1) with parameters a , b, ω . This idea works due the following statementabout boundaries of Arnold tongues in the plane of parameters. Theorem 6 [ ] A point in the parameter space of equation (3) (or system (4) ) liesin the boundary of a phase-lock area, if and only if the monodromy of the correspondingDCHE (7) has multiple eigenvalue. In that case the monodromy either has Jordan celltype, or is the identity.
In general, if a family (13) is isomonodromic, a new family received after the transfor-mation z = √ tζ (42)is still isomonodromic. Indeed monodromy representation depends on the homotopic class[ σ ] of a loop σ , thus the transformation (42) does not change monodromy matrices as itdoes not change homotopic classes of loops around z = 0 (and z = ∞ .) Also, since theStokes matrices (19), (21) of the system are constant in z and t , then they are constantin ζ and t after the transformation (42).We get a theorem that gives some information about boundaries of Arnold tongues. Theorem 7
Suppose that point ( b, a ) belongs to a boundary of some Arnold tongue for ω = ω , then ( b, a ∗ k ) defined in (40) belongs to the boundary of a Arnold tongue for ω = ω ∗ k defined in (41) ; and monodromy eigenvalues of corresponding DCHEs are preserved. Instead of Bessel solution we may consider rational solutions. Moving singularities ofPainleve 3 equation are first order poles and residues of solution are either 1 or −
1. Incase when residue is 1 functions that are entries of coefficient matrices in (13) have nopoles.Painleve 3 equation also has rational solutions:
Theorem 8
If parameters of normalized Painleve 3 equation P ( ˆ α, ˆ β, − , satisfycondition ˆ α + (cid:15) ˆ β = 4 k , (cid:15) = 1 , k ∈ Z , then there is a rational solution of P ( ˆ α, ˆ β, − , . Parameters (26) satisfy conditions of theorem 8 in the case b ∈ Z . Rational solutionsare listed in [4 , References [1] A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, and I. V. Schurov
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