Periodic solutions of a resistive model for nonlocal Josephson dynamics
A novel method is developed for constructing periodic solutions of a model equation describing nonlocal Josephson electrodynamics. This method consists of reducing the equation to a system of linear ordinary differential equations through a sequence …
Authors: Yoshimasa Matsuno
P erio dic solutions of a resistiv e m o del for nonlo cal Josephson dynamics Y oshimasa Matsuno Division of Applie d Mathematic al Scienc e, Gr aduate Scho ol of Scienc e and Engine ering Y amaguchi U niversity, Ub e 755-8611 , Jap an E-mail: matsuno@y amaguc hi- u.ac.jp Abstract A no v el metho d is dev elop ed for constructing p erio dic solutions of a mo del equation describing nonlo cal Josephson elec tro dynamics. This metho d consists of reducing the equation to a system of linear ordinary differen tial equations through a sequence of non- linear transformations. T he p erio dic solutions are then obtained b y a standard pro cedure whic h are repres en ted in terms o f trigonometric functions. It is found that the large time asymptotic o f the solution exhibits a steady profile whic h do es not dep end on initia l conditions. P A CS num b ers: 02.30.Ik, 05.45.Yv, 74 .50.+r 1. In tro duction The recen t studies on Jo sephs on tunnel junctions with high-temp erature sup erconductors rev eal that the nonlo cal nature of Josephson electrodynamics becomes dominan t when the Josephson p enetration depth λ J is shorter than the London p enetration depth λ L . In particular, if w e consider a thin lay er b et w een t wo sup erconductors, the phase diffe rence φ ( x, t ) across the Josephson junction is described by the follo wing mo del equation [1-5] ω − 2 J φ tt + ω − 2 J η φ t = − sin φ + λ 2 J π λ L Z ∞ −∞ K 0 | x − x ′ | λ L φ x ′ x ′ ( x ′ , t ) dx ′ + γ . (1) Here, K 0 is the mo dified Bessel function o f order zero, ω J is the Josephson plasma fre- quency , η is a p ositiv e parameter in vers ely prop or t io nal to the resistance of a unit area of the tunneling junction, γ is a bias curren t densit y across the j unction normalized b y the Josephson critical curren t density and the subscripts t and x ′ app ended to φ denote 1 partial differentiation. When the characteristic space scale l of φ is extremely larg e com- pared w ith λ L , the k ernel K 0 has an approx imate expression K 0 ( x ) ∼ π δ ( x ) where δ ( x ) is D irac’s delta function. Then, the equation (1 ) reduces to the p erturb ed sine-Gordon equation [6]. In the opp osite limit l << λ L , o ne can use the asymptotic of the k ernel K 0 ( | x | ) ∼ − ln | x | . In a ddition, if we restrict our consideration t o the o v erdamp ed case η >> 1 as w ell as the zero bias curren t γ = 0, then unlik e the p erturb ed sine-Gordon equation, t he equation (1 ) b ecomes an in tegro differen tial (or nonlo cal) equation. It can b e written in a n a ppropriate dimensionless for m as φ t = − sin φ + H φ x , H φ x = 1 π P Z ∞ −∞ φ x ′ ( x ′ , t ) x ′ − x dx ′ , (2) where H is the Hilbert transform op erator. Equation (2) ma y be termed a resistiv e mo del for nonlo cal Josephson electro dynamics [7]. Note tha t the equation (2) has b een prop osed for the first time in searc hing in tegrable nonlinear equations with dissipation [8]. The general multikink solutions of the equation ( 2) ha v e b een o btained and their prop erties hav e b een in v estigated in detail [8]. In this paper, we report some new res ults concerning p erio dic s olutions of the e quation (2). Sp ecifically , w e show that the equation (2 ) can b e transformed to a finite- dimensional nonlinear dynamical system throug h a dep enden t v ar ia ble transformatio n. W e then lin- earize the system of equations to deriv e a first-order system of line ar o r dinary differen tia l equations (OD Es). Its initial v alue problem can b e solv ed explicitly to obtain p erio dic solutions. It is sho wn that the la r g e time asymototic of the p erio dic solution relaxes to a steady pro file indep enden t of init ia l conditions. 2. Exact metho d of solution 2.1. A nonline a r dynamic al system W e seek p erio dic solution of (2) of the form φ = i ln f ∗ f , f = N Y j =1 1 β sin β ( x − x j ) , (3) where x j = x j ( t ) are complex functions of t whose imaginary parts are all p ositiv e, β is a p ositiv e parameter, N is an arbitrary p ositive inte ger and f ∗ denotes the complex 2 conjugate expression of f . Using a form ula fo r the Hilbert tra nsform, one has H φ x = − (ln f ∗ f ) x . Substitution of this express ion and (3) in to (2) giv es the follo wing bilinear equation for f and f ∗ : i ( f ∗ t f − f ∗ f t ) = i 2 ( f 2 − f ∗ 2 ) − f ∗ x f − f ∗ f x . (4) W e divide (4) by f ∗ f , substitute f from (3) and t hen ev aluat e t he residue at x = x j on b oth sides to obtain a system o f nonlinear ODEs fo r x j ˙ x j = − 1 2 β Q N l =1 sin β ( x j − x ∗ l ) Q N l =1 ( l 6 = j ) sin β ( x j − x l ) + i, j = 1 , 2 , ..., N , (5) where an ov erdot denotes differen tiation with resp ect to t . Note that a dynamical system corresp onding to the m ultikink solutio n is deriv ed simply from (5) b y taking the limit β → 0 [8]. As in the m ultikink case, it will b e demonstrated from (5) that the imaginary part of x j remains p ositiv e if it is p ositiv e at an initial time. Before pro ceeding, it is conv enien t to introduce some notations: z = e 2 iβ x , ξ j = e 2 iβ x j , η j = e 2 iβ x ∗ j , j = 1 , 2 , ..., N , (6) s 1 = N X j =1 x j , s 2 = N X j
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