Analytical analysis of ground states on 0-$\pi$ long Josephson junctions

We investigate analytically a long Josephson 0-$\pi$-junction with several 0 and $\pi$ facets which are comparable to the Josephson penetration length $\lambda_J$. Such junctions can be fabricated exploiting (a) the d-wave order parameter symmetry of cuprate superconductors; (b) the spacial oscillations of the order parameter in superconductor-insulator-ferromagnet-superconductor structures with different thicknesses of ferromagnetic layer to produce 0 or $\pi$ coupling or (c) the structure of the corresponding sine-Gordon equations and substituting the phase $\pi$-discontinuities by the artificial current injectors. We investigate analytically the possible ground states in such a system and show that there is a critical facet length $a_c$, which separates the states with half-integer flux quanta (semifluxons) from the trivial ``flat phase state'' without magnetic flux. We analyze different branches of the bifurcation diagram, derive a system of transcendental equations which can be effectively solved to find the crossover distance $a_c$ (bifurcation point) and present the solutions for different number of facets and the edge facets length. We show that the edge facets may drastically affect the state of the system.