Uniqueness of 1D Generalized Bi-Schrödinger Flow

2020/05/21 by Eiji Onodera, Onodera, Eiji
Mathematics · Physics and Astronomy · #35G61 #35Q35 #35Q40 #35Q55 #53C21 #53C44 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2005.10575

openalex publication_date 2020/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish the uniqueness of a smooth generalized bi-Schrödinger flow 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 nonlinear dispersive partial differential equation with loss of derivatives. 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 H²-energy for the difference of two solutions, the detailed estimate of which enables us to eliminate the difficulty of the loss of derivatives. In particular, we demonstrate how to decide the form of the modification by exploiting the geometric structure of the locally Hermitian symmetric space.

Citations

• Harmonic Mappings of Riemannian Manifolds

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