IVC Calculation Problem for Josephson Junction Stacks.On Asymptotic Construction near the Breakpoint

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Solving the system of essentially nonlinear differential equations for different we obtain thecurrent-voltage characteristics (IVC) for a system of Josephson junctions (JJ) as a hysteresisloop. When the current approaches on the back way the breakpoint the voltage ()falls sharply to zero. In addition, in numerical modelling (non-periodic boundary conditions(NPBC)) IVC multiple branching is observed near . It is interesting to study this phenomenonanalytically developing asymptotic methods. There had been developed simple “asymptotic”formulas suitable for calculation of all IVC points except near to . A numerical-analyticalmethod allowing to shorten IVC calculation time essentially was proposed. This method showedgood results in IVC multiple branching calculation in particular. All calculations were performedusing the REDUCE system. We succeeded first to calculate analytically all points of IVC. Anapproximate solution at the breakpoint region (periodic boundary conditions (PBC)) has beendeveloped using the Bogolyubov-Krylov method.

1. Introduction The definition of the singular points of the current voltage characteristics together with the estimation of the width of their influence region provide adequate input for physical experiments aiming at studying the finite JJ stacks [1-3]. The mathematical problem of IVC calculation for the stack of Josephson junction [4] asks for the solution of the following essentially nonlinear system: Here is derivative of Solving the Cauchy problem for different (right way) and we obtain IVC of Josephson junctions as hysteresis loop. The points ( with corresponding to the right way, form the right branch of the loop. And the points (, with corresponding to the back way, form the back branch of the loop. () is the total voltage of the stack. For the system (1) is solved with zero initial data. For each next found are used as initial data. The coefficients , ′ of the system (1) are elements of the matrix . In the case of PBC is symmetric square matrix of order : And in the case of NPBC is symmetric square tridiagonal matrix of order : (3) The dynamics of phase differences () had been simulated [4] by solving the equation system (1) using the fourth order Runge-Kutta method. The aim was to shorten the time of IVC calculation, and it was achieved. First the number of the system equations was reduced to one in the case of PBC and halved in the case of NPBC. The longtime “asymptotics” formulas were developed. The mixed numerical-analytical method was suggested: all points of IVC are calculated using the asymptotics except points of the little arc of the back branch of the loop calculated numerically [5]. And Fig. 5 shows that the Runge-Kutta method of the fourth order accuracy can be replaced by the simplest explicit cross-scheme of the second order accuracy. This shortens the time of IVC calculation complimentarily. 2. On Long-Time “Asymptotics” Construction In the case of PBC IVC calculation for a stack of intrinsic Josephson junctions is reduced to solving a unique equation [5] Solving this equation with given initial data is equivalent to solving the following integral equation . (4) The simple iteration method starting from zero gives on the second steparctg(/)) where And the total voltage of the stack of Josephson junctions is calculated as In Fig. 1, Fig. 2, Fig. 3 the solid curve is the IVC back branch calculated using the Runge-Kutta method. In Fig. 1 the pictures of the back way of the hysteresis loop for 9 Josephson junctions are shown. The circles refer to “asymptotic” (using (5)) calculations respectively. Here we put 50, the step in the Runge-Kutta method Figure 1. IVC back branch, = 9. PBC. The circles were calculated using (5) Figure 2. IVC back branch, = 9. PBC. The circles refer to he mixed numerical-analytical calculation In Fig. 2 the circles refer to calculation performed by the following mixed analytical- numerical method. The right branch of the hysteresis loop and the back branch on the interval 0.4 < < 1.45 were computed using (5). The rest ones were computed numerically. The REDUCE system [6] was used in all calculations performed. We proved [7] that in the case of NPBC with = 1 the problem of hysteresis loop calculation reduces to solving the following system of integral equations: And is integer part of. The system (6) was solved using simple iterations starting at zero. The results obtained on the second step are regarded as long-time “asymptotics” of the system (6) solution. We constructed “asymptotics” following to [8]. In the introduction N. Levinson declared: “A method is given for showing that formal “approximate solutions” of nonlinear differential equations are in fact the leading terms in an asymptotic representation of actual solutions”. The Fig. 3, Fig. 4 confirm this declaration. At the same time these pictures show efficiency of the suggested mixed numerical-analytical method. This time ∆ = 0.001. Fig. 3 refers to numerical calculation: all points of IVC were calculated using the Runge- Kutta method. Fig. 4 refers to calculation performed by the mixed numerical-analytical method: the right branch of IVC and the part of the back branch (1.45 > > 0.4) were calculated analytically (using the long-time “asymptotics” of (6) system solution). The rest points (0.4 > > 0.2) were calculated numerically using the Runge-Kutta method. Figure 3. Multiple branching of IVC back branch, = 8. NPBC. All points of IVC were calculated using the Runge-Kutta method Figure 4. Multiple branching of IVC back branch, = 8. NPBC. All points of IVC were calculated using “asymptotics” formulas except points of the interval on the picture calculated using the Runge-Kutta method Figure 5. Multiple branching of IVC back branch, = 8. NPBC. All points of IVC were calculated using “asymptotics” formulas except points of the interval on the picture calculated using simplest cross-scheme of the second order accuracy Figure 6. IVC back branch, = 9. PBC. In difference with Fig. 1 two circles ( = 0.3, = 0.25) were calculated by (7), obtained by Bogolyubov-Krylov method The Fig. 5 is similar to Fig. 4, but here the simplest explicit cross-scheme of the second order accuracy instead of the fourth order accuracy Runge-Kutta method was used. Remark that in both cases ∆ = 0.1 was chosen. 3. The Bogolyubov-Krylov Method Employment For studying multiple branching in current-voltage characteristics analytically we dream to prolong “asymptotics” till This moment we succeeded only to calculate analytically the whole hysteresis loop in the case of PBC. We developed approximate solution for the equation with given initial data in the case of small , , using the Bogolyubov-Krylov method [9]. We consider where and are determined by the first approximation equations the first equation of (5) gives and is determined of the third order polynomial equation This equation has at least one real root. When , are found,By such a way we could correct Fig. 1. In Fig. 6 the back branches of the hysteresis loop for the case of PBC are presented. The solid curve is the same as in Fig. 1. The circles on the solid curve were found analytically using mixed analytical method: all points of the hysteresis loop were calculated using (5) - except two points ( = 0.3, 0.25) calculated using (7).

S I Serdyukova

Laboratory of Information Technologies Joint Institute for Nuclear Research

Author for correspondence.
Email: sis@jinr.ru
6 Joliot-Curie str., Dubna, Moscow region, 141980, Russian Federation

Serdyukova S. I. - professor, Doctor of Physical and Mathematical Sciences, leading researcher of Scientific Department of Computational Physics of Information Technology Laboratory of Joint Institute for Nuclear Research, Dubna

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