The classical problem of the Josephson junction of arbitrary length W in thepresence of externally applied magnetic fields (H) and transport currents (J)is reconsidered from the point of view of stability theory. In particular, wederive the complete infinite set of exact analytical solutions for the phasedifference that describe the current-carrying states of the junction witharbitrary W and an arbitrary mode of the injection of J. These solutions areparameterized by two natural parameters: the constants of integration. Theboundaries of their stability regions in the parametric plane are determined bya corresponding infinite set of exact functional equations. Being mapped to thephysical plane (H,J), these boundaries yield the dependence of the criticaltransport current Jc on H. Contrary to a wide-spread belief, the exactanalytical dependence Jc=Jc(H) proves to be multivalued even for arbitrarilysmall W. What is more, the exact solution reveals the existence of unquantizedJosephson vortices carrying fractional flux and located near one of thejunction edges, provided that J is sufficiently close to Jc for certain finitevalues of H. This conclusion (as well as other exact analytical results) isillustrated by a graphical analysis of typical cases.
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