Some analytical solutions for magnetic flux distribution in long Josephson junction with second harmonic in the current phase relation

Abstract

In the current work long Josephson junctions are being studied. Magnetic flux distribution is the physical measure for phase difference of the wave functions in the superconducting layers of the junction. The current phase relation, in most cases, can be considered as an odd strict 2π-periodic function, and hence, it can be presented in Fourier series of sinuses. It is well-known from the physical experiment that with a sufficient degree of precision, a number of physical systems are reliably described with the contribution of only first two harmonics. The adequate mathematical model for the distribution of the magnetic flux is then the double sine-Gordon equation with Neumann boundary conditions at the ends of the junction. Even in the stationary case, the boundary problem is highly nonlinear and the only tool for its comprehensive study is numerical methods. The aim of the present work is to show that in the case of zero external current, the stationary equation turns out to be a fully integrable model, derived from a variation principle with a cosine potential. In our work analytical solutions for the magnetic flux distributions described in the terms of Jacobi elliptic sinuses are derived. Analytical studies in this case serve to further numerically study of this multiparametric nonlinear boundary problem, which is so important in the applied nanophysics.In the current work long Josephson junctions are being studied. Magnetic flux distribution is the physical measure for phase difference of the wave functions in the superconducting layers of the junction. The current phase relation, in most cases, can be considered as an odd strict 2π-periodic function, and hence, it can be presented in Fourier series of sinuses. It is well-known from the physical experiment that with a sufficient degree of precision, a number of physical systems are reliably described with the contribution of only first two harmonics. The adequate mathematical model for the distribution of the magnetic flux is then the double sine-Gordon equation with Neumann boundary conditions at the ends of the junction. Even in the stationary case, the boundary problem is highly nonlinear and the only tool for its comprehensive study is numerical methods. The aim of the present work is to show that in the case of zero external current, the stationary equation turns out to be a fully integrable model,...