Vortex dynamics equation in type-II superconductors in a temperature gradient

in this work we determined a vortex dynamics equation in a temperature gradient in the frame of the time dependent ginzburg-landau equation. in this sense, we derived a local solvability condition, which governs the vortex dynamics. also, we calculated the explicit form for the force coefficients, which are the keys for the understanding of the balance equation due to vortex interactions with the environment.