Nonlinear Models for Josephson junction: sine-Gordon equation

The Josephson junction is shown in the figure 1 consist of two superconductors separate by thin oxide layers. In 1962 Josephson discover that the Cooper electron pairs can be tunnel from the superconductor S1 to S2.

 
Figure 1:
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The Electrodynamics of long Josephson junction is gouverned by the sine-Gordon equation:

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Figure 2: Kink
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Other solution of the sine-Gordon equation are called Breathers .