Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

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Abstract

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.

MSC

primary
53C44
secondary
35Q35
35Q40
35Q55
35G61
53C21

Keywords

Fourth-order dispersive partial differential equation
Time-local existence
Loss of derivatives
Energy method
Locally Hermitian symmetric space
Bi-Schrödinger flow

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