Uniqueness of 1D Generalized Bi-Schrödinger Flow

Abstract

We consider the initial value problem for the generalized bi-Schrödinger flow equation for maps from the one-dimensional flat torus into a compact locally Hermitian symmetric space. The governing equation, which is satisfied by sections of the pull-back bundle induced from the flow, is a fourth-order extension of the 1D Schrödinger map flow equation with loss of derivatives. As the author’s previous work, time-local existence of a smooth solution has been obtained by an intrinsic approach based on the geometric energy method. In the present paper, as a continuation of the work, we establish the uniqueness of the solution. To show the uniqueness, we adopt an extrinsic approach to compare two solutions via an isometric embedding into an ambient Euclidean space. We introduce an energy modifying the classical \(H^2\)-energy for the difference of two solutions, which enables us to eliminate the difficulty of the loss of derivatives. In order to achieve this, we establish some useful tools of computation exploiting the geometric structure of the locally Hermitian symmetric space in the ambient Euclidean space.