Two-dimensional generalizations of the Korteweg-de Vries equation in the vortex dynamics of ultraclean high-temperature superconductors

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In an ultraclean type-II superconductor in the absence of pinning and the Hall force, the nonlinear Korteweg-de Vries (KdV) equation governs the evolution of the first-order electrodynamic field corrections for one-dimensional vortex motion. Here two-dimensional (2D) vortex motion is investigated in the ultraclean regime, in the long-wavelength, small-amplitude limit. The effects of nonlocal vortex interaction and vortex inertia are included, using London theory. Using different orderings for the transverse coordinate and associated velocity component, 2D generalizations of the KdV equation are obtained, including the Kadomtsev-Petviashvili equation. These equations possess soliton solutions, stable to 2D perturbations. These results provide the basis of modelling solitonic behavior in more complex type-II superconductors.

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