The system of two coupled non-linear 2nd order
differential equations that represent the Josephson’s Junction phenomenon on
superconductors is presented here, as adapted from Hairer et al
.
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The equations describe the electron-pair interactions
between two superconducting elements that are separated by a thin insulator,
under the influence of an applied current source. The obvious first step in attempting
to numerically integrate these equations is to decouple it.
Coupled equations can be decoupled in a variety of ways.
Perhaps the easiest one is the brute-force method, where in a few algebraic
sleights are performed to eliminate one the variables in either equation. In
the subject case, multiplying the second equation by "alpha" and adding it to the first to eliminate one of the 2nd
derivative variables. Subsequently, this equation can be substituted into the
first equation to obtain the decoupled versions.
The
algebraic sleight works best with a limited set of equations (usually up to two
at the most). As the number of equations increases, the algebraic decoupling
process becomes rather cumbersome. An alternative is to use the matrix formulation
instead. The Josephson equations then become:
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Since the matrix that requires the inversion is a constant,
only a single inversion is required in the integration process.
And the two 2nd order ODE’s can be considered as
being successfully decoupled. Numerical integration is just a step away, once
the 2nd order ODE’s are converted to 1st order ODE’s.
Plotted below is the numerical solution of the equations, performed
using a Rosenbrock technique. The initial conditions and values for the
constants from Hairer et al were used for the analysis.
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