London's limit for the lattice superconductor

A stability problem for the current state of the strong coupling superconductor has been considered within the lattice Ginzburg-Landau model. The critical current problem for a thin superconductor film is solved within the London limit taking into account the crystal lattice symmetry. The current dependence on the order parameter modulus is computed for the superconductor film for various coupling parameter magnitudes. The field penetration problem is shown to be described in this case by the one-dimensional sine-Gordon equation. The field distribution around the vortex is described at the same time by the two-dimensional elliptic sine-Gordon equation.