The Stability Of The Gross-pitaevskii Equation In The Neighborhood Of Steady State Solutions

We consider the Gross-Pitaevskii equation: -i(?)v/(?)t+△v + λv(1- |v|~2) = 0 ina bounded domain Ω with the specific boundary condition. It is shown that thesolution to the mixed problem exists and is globally well-posed. Furthermore in a non-simple connected domain Ω, this solution is stable in the sense of Lyapunov when the initial data is given in the neighborhoods of some non-trivial stationary solution with large parameter A.