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Eiji Onodera

Publications and source records attributed to Eiji Onodera.

14 recordsLinked to original sources
arXiv2026

Local well-posedness for fourth-order nonlinear dispersive systems on the one-dimensional torus under structural conditions

We study local well-posedness of the initial value problem for a class of fourth-order nonlinear dispersive systems on the one-dimensional torus. The main difficulty comes from the loss of derivatives in the nonlinear terms. By introducing suitable structural conditions that compensate for derivative loss through cancellation mechanisms, and constructing modified energies via gauge-type transformations combined with a diagonalization procedure, we establish local well-posedness in high-regularity Sobolev spaces. The assumptions in the present result relax those in previous works, at the expense of requiring higher regularity. In particular, our structural conditions extend the range of admissible nonlinearities beyond the scalar case and provide a unified framework for treating multi-component systems.

math.AP↗
arXiv2024

Local well-posedness for a fourth-order nonlinear dispersive system on the 1D torus

This paper is concerned with the initial value problem for a system of one-dimensional fourth-order dispersive partial differential equations on the torus with nonlinearity involving derivatives up to second order. This paper gives sufficient conditions on the coefficients of the system for the initial value problem to be time-locally well-posed in Sobolev spaces with high regularity. The proof is based on the energy method combined with the idea of a gauge transformation and the technique of Bona-Smith type parabolic regularization. The sufficient conditions can been found in connection with geometric analysis on a fourth-order geometric dispersive partial differential equation for curve flows on a compact locally Hermitian symmetric space.

math.AP↗
arXiv2024

Local well-posedness of the initial value problem for a fourth-order nonlinear dispersive system on the real line

This paper investigates the initial value problem for a system of one-dimensional fourth-order dispersive partial differential-integral equations with nonlinearity involving derivatives up to second order. Examples of the system arise in relation with nonlinear science and geometric analysis. Applying the energy method based on the idea of a gauge transformation and Bona-Smith approximation technique, we prove that the initial value problem is time-locally well-posed on the real line for initial data in a Sobolev space with high regularity.

math.AP↗
arXiv2024

Structure of a fourth-order dispersive flow equation through the generalized Hasimoto transformation

This paper focuses on a one-dimensional fourth-order nonlinear dispersive partial differential equation for curve flows on a Kähler manifold. The equation arises as a fourth-order extension of the one-dimensional Schrödinger flow equation, with physical and geometrical backgrounds. First, this paper presents a framework that can transform the equation into a system of fourth-order nonlinear dispersive partial differential-integral equations for complex-valued functions. This is achieved by developing the so-called generalized Hasimoto transformation, which enables us to handle general higher-dimensional compact Kähler manifolds. Second, this paper demonstrates the computations to obtain the explicit expression of the derived system for three examples of the compact Kähler manifolds, dealing with the complex Grassmannian as an example in detail.

math.DG↗
arXiv2020

Uniqueness of 1D Generalized Bi-Schrödinger Flow

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 $H^2$-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.

math.AP↗
arXiv2016

Local existence of a fourth order dispersive curve flow on locally hermitian symmetric spaces and the application

This paper is concerned with a fourth order nonlinear dispersive partial differential equation for closed curve flow on a Kähler manifold. The main results is that the initial value problem has a solution locally in time if the Kähler manifold is a compact locally hermitian symmetric space. The proof is based on the geometric energy method combined with a nice gauge transformation to eliminate the 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 problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.

math.AP↗
arXiv2016

A fourth-order dispersive flow equation for closed curves on compact Riemann surfaces

A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main results, time-local existence and the uniqueness of a solution to the initial value problem is established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the absence of the local smoothing effect. The proof is based on the geometric energy method combined with a kind of gauge transformation to eliminate the loss of derivatives. Specifically, to show the uniqueness of the solution, the detailed geometric analysis of the solvable structure for the equation is presented.

math.AP↗
arXiv2013

A fourth-order dispersive flow into Kähler manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.

math.AP↗
arXiv2009

A Remark on the global existence of a third order dispersive flow into locally Hermitian symmetric spaces

We prove global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation we consider generalizes two-sphere-valued completely integrable systems modelling the motion of vortex filament. Unlike one-dimensional Schrödinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an energy which is not necessarily preserved in time but does not blow up in finite time.

math.AP↗
arXiv2008

A third order dispersive flow for closed curves into almost Hermitian manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.

math.AP↗
arXiv2008

A third-order dispersive flow for closed curves into Kähler manifolds

This paper is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into Kähler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of vortex filament. We prove the local existence theorem by using geometric analysis and classical energy method.

math.AP↗
arXiv2008

The initial value problem for a third-order dispersive flow into compact almost Hermitian manifolds

We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This equation causes the so-called loss of one-derivative since the target manifold is not supposed to be a Kähler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially.

math.AP↗
arXiv2008

Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces

We study the structure of differential equations of one-dimensional dispersive flows into compact Riemann surfaces. These equations geometrically generalize two-sphere valued systems modeling the motion of vortex filament. We define a generalized Hasimoto transform by constructing a good moving frame, and reduce the equation with values in the induced bundle to a complex valued equation which is easy to handle. We also discuss the relationship between our reduction and the theory of linear dispersive partial differential equations.

math.AP↗