Nonlinear Schrodinger eqnations on riemannian manitolds

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The purpose of our research was :(1) to reduce the vortex filament equation to the nonlinear Schrodinger equation.(2) by the reducing, to prove the existence of solutions.(3) application of them.We got the following results corresponding to each.(1) We analyzed the problem on three-dimensional homogeneous spaces, which heavily reflect the differential geometric aspect of the equation. As the result, we have shown that the vortex filament equation corresponds to a nonlinear Schrodinger equation similar to the case of the euclidean space.(2) We proved the existence of the solutions of the nonlinear Schrodinger equation. It implies that there is a short time solution of the vortex filament equation.(3) In the above procedure, we have shown that the short time existence of the solutions of the nonlinear Schrodinger equation is stable under adding certain terms. On the uniqueness of the solution, we have proved that it always holds.