The vortex filament equation and a semilinear Schrödinger equation in a Hermitian symmetric space
We consider the initial value problem of the vortex filament equation on \(\mathbb{R}^3\): \[ \gamma_t=\gamma_x\times\gamma_{xx} \] with \(|\gamma_x|\equiv 1\) and \(\times\) denoting the exterior product. We prove the existence and the uniqueness of a classical solution for the initial value problem and generalize it to the case of curves in 3-dimensional space forms. We also consider related semilinear Schrödinger equations for curves in Kähler manifolds. It is remarkable that we need symmetric spaces as manifolds for infinite time existence of solutions.
scientific article; zbMATH DE number 1014611