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%\title{Numerical Investigation of Bifurcations in Inhomogeneous Josephson Junctions}
\title{A Free Boundary Value Problem for Long Inhomogeneous Josephson
Junctions}
\bigskip
\author{
M. D. Todorov\\
\small\it Institute of Applied Mathematics and Informatics\\[-1.mm]
\small\it Technical University of Sofia
\\ \\
T. L. Boyadjiev\\
\small\it Faculty of Mathematics and Informatics\\[-1.mm]
\small\it Sofia University 'St.Kl.Ohridski'
}
\date{ }
\begin{document}
\maketitle
\bigskip
A method for calculating the bifurcations of a magnetic
field in long inhomogeneous Josephson junctions is proposed. Since the length
of the
junctions is a variable quantity the nonlinear spectral problem as a
problem
with free boundary is interpreted. We solve the following differential system
consisting
of the perturbed Sin--Gordon equation, stability condition (Sturm--Liouville
problem) both with boundary conditions of Neumann type and norm condition for
the eigenfunctions, namely:
\begin{eqnarray*}
&&-\varphi^{''}+j_D(x) \sin \varphi + \gamma =0\\
&&\varphi^{\prime}(-R)=\varphi^{\prime}(R)=h_B\\
&&-\psi^{''}+\left(j_D(x) \cos \varphi -\lambda\right) \psi =0\\
&&\psi^{\prime}(-R)=\psi^{\prime}(R)=0\\
&&\int_{-R}^R \psi^2 dx = 1\>,
\end{eqnarray*}
\noindent where $\varphi$ is magnetic flux, $j_D$ - the amplitude of the
Josephson junction, $\gamma$ - the external current, $h_B$ - the
boundary magnitude of magnetic field.
The above system is linearized appropriately after that a continuous analogue
of the Newton method with optimal iterative step and
a spline scheme of fourth order of approximation on an uniform mesh is used.
\end{document}