\documentstyle{article} \newlength{\defaultparindent} \setlength{\defaultparindent}{\parindent} \input{tcilatex} \begin{document} \begin{center} {\bf Todor L. Boyadjiev} {\small Faculty of Mathematics and Computer Science,} {\small University of Sofia.} {\small e-mail: todorlb@fmi.uni-sofia.bg} {\bf Bifurcations in a long inhomogeneous Josephson junctions} \end{center} The bifurcations of the solutions of the nonlinear boundary value problem $$-\varphi _{xx}+j_{D}(x)\sin \varphi +\gamma =0,\quad x\in [-R,R], \label{1}$$ $$\varphi _{x}(\pm R)=0, \label{2}$$ at change of the parameters $\gamma$ and $h_{B}$ are studied numerically. Here $j_{D} (x)$ is a given continuous function. These solutions describe the possible stationary distributions of the magnetic flux $\varphi (x)$ in a long inhomogeneous Josephson junctions. Each solution of the problems (1), (2) generates a Sturm-Liouville problem $$-\psi _{xx}+\left[ q\,(x)-\lambda \right] \;\psi =0,\quad x\in [-R,R], \label{3}$$ $$\psi _{x}(\pm R)=0, \label{4}$$ $$\int\limits_{-R}^{R}\psi ^{2}(x)\;dx=1,$$ where the potential $q$ is defined as $q(x,\gamma ,h_{B})=j_{D}(x)\cos \;\varphi \,(x,\gamma ,h_{B}),$ The stability condition is $\lambda _{\min } (\gamma ,h_{B} )>0$ . To find directly the bifurcation curve $\lambda _{\min } (\gamma ,h_{B} )=0$ we consider the system (1) - (5) for a given l and given parameter $\gamma$ or $h_{B}$ as a nonlinear eigenvalue problem with spectral parameter $h_{B}$ or $\gamma$ respectively. The continuous analog of Newton method for solving this eigenvalue problem is used. The numerical results are compared to known experimental data. \end{document} 