Nonlinear diffusion in hard and soft superconductors

We discuss the diffusion of magnetic flux in a field-cooled ("hard") superconducting slab in a creep regime in which E ∝ |J|^σ J. Bryksin and Dorogovtsev recently discussed flux diffusion in a pinningless ("soft") superconductor in which E ∝ |B|J. This problem is closely related to the flux-creep one with σ=1, and provides additional insight into the possible types of behaviour. We list a series of possible long-term asymptotic solutions of a scaling form, which are either analytically exact or accurately calculated. We check numerically that the relevant long-term solution is approached after various initial conditions. Amongst other conclusions we find S=d(In|M|)/d(Int)→-1/σ or -1/2σ, after application and removal of an additional field, according to whether the disturbance has reached the sample centre or not. A relaxing sandpile model appears to have wide validity when creep is only a small correction to the critical state, as when σ→∞.