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Half integer Josephson vortices in 0-$\pi$-junctions, discussed theoreticallyand observed experimentally, spontaneously appear at the point where theJosephson phase is $\pi$-discontinuous. The creation of \emph{arbitrary}discontinuities of the Josephson phase has been demonstrated recently. Here westudy fractional vortices formed at an arbitrary $\kappa$-discontinuity,discuss their stability and possible ground states. The two stable states arenot mirror symmetric. Furthermore, the possible ground states formed at two$\kappa$-discontinuities separated by a distance $a$ are investigated, and theenergy and the regions of stability of each ground state are calculated. Wealso show that the ground states may strongly depend on the distance $a$between the discontinuities. There is a crossover distance $a_c$ such that for$a<a_c$ and for $a>a_c$ the ground states may be qualitatively different.Comment: 7 figures, submitted to PRB In v.2 one figure is added, and refs are  updated In v.3 major revision, many issues fixe

physical review. b, condensed matter and materials physics

Ground states of one and two fractional vortices in long Josephson<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>0</mml:mn><mml:mtext>−</mml:mtext><mml:mi>κ</mml:mi></mml:mrow></mml:math>junctions

Ground states of one and two fractional vortices in long Josephson0−κjunctions