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The kink solution in the long Josephson junction is studied. The perturbation scheme of constructing the fluxon solution in curved junction is formulated. The prediction from the perturbation scheme is compared with the prediction that follows from the numerical studies of the complete field model.

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Purpose: The purpose of this report is to present the similarities and differences between modified sine-Gordon models used in description of the curved Josephson junctions. The leading dynamical variable in this system is a gauge invariant phase difference of the macroscopic wave functions of the superconducting electrodes that form the junction. Findings: The main finding of this article is the observation that in the model used in description of junctions with quickly varying curvatures the significant part of the kink energy is confined in the curved regions of the junction. Research limitations/implications: The paper is limited to the description of the dynamics of fluxions in the long Josephson junctions. These junctions due to small transverse sizes (smaller than the Josephson penetration depth) can be considered as a one dimensional systems. Practical implications: It seems that junctions with appropriate geometry will find applications in future electronic devices. It is expected that curved Josephson junctions can be used in order to store a binary data. Originality/value: The main idea of the paper is to use a Riemann geometry in order to describe the influence of the curvature on the kink motion in the junction.