Analysis of linear and nonlinear wave phenomena and the inverse problem

线性和非线性波动现象及其反问题分析

基本信息

批准号：
13640219
负责人：
MOCHIZUKI Kiyoshi
金额：
$ 2.05万
依托单位：
CHUO UNIVERSITY (2002-2003)Tokyo Metropolitan University (2001)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640219/
关键词：
generalized KPP equation Inverse spectral problem semilinear wave equation dissipative wave equation scattering amplitude large time asymptotics Sturm-Liouville operator Dirac operator シュレディンガー方程式 Sturm-Liouville作用素 Dirac作用素 Schr\"od

项目摘要

In this project, we are mainly concerned with the analysis of wave phenomena governed by some basic partial differential equations in Applied Mathematics and Physics, and also by some model equations appearing in Chemistry and Biology. Summarizing the results obtained by the investigators in the period 200 1-2003, we can say the objective of this project is accomplished fruitfully.The head investigator published 5 papers analyzing linear and nonlinear waves and nonlinear diffusions. The topics include the following:(1)Nonlinear parabolic equations: Large time asymptotics of solutions for generalized KPP equation are studied. Precise formula obtained here is expected to have many applications in Biology.(2)Inverse spectral problem: Inverse problem to determine the potential from some spectral data is studied. We obtained a uniqueness results for Sturm-Liouville operator and for Dirac operator on firiete interval. Our problem 'to determine the potential from interior spectral data' is a new formulation and is expected to be applied to many other operators.(3)Semilinear wave equations: Asymptotic for wave equation with nonlinear dissipation is studied. We require that the dissipation is inhomogeneous in' space and time and sufficient conditions are obtained for solutions to be asymptoically free.(4)Inverse scattering problem: First the direct scattering theory is established for wave equation with first time derivative term, and then the coefficient of the term is shown to be reconstructed from the scattering amplitude with a fixed energy.Each investigator developed many interesting results on the related fields.

在这个项目中，我们主要关注对应用数学和物理学中一些基本偏微分方程的波浪现象的分析，以及化学和生物学中出现的某些模型方程。总结了研究人员在200 1-2003期间获得的结果，我们可以说该项目的目的是有效完成的。首席研究员发表了5篇分析线性和非线性波和非线性扩散的论文。主题包括以下内容：（1）非线性抛物线方程：研究通用KPP方程的溶液的较大时间渐近学。预计此处获得的精确公式将在生物学中具有许多应用。（2）逆频谱问题：逆问题确定来自某些光谱数据的潜力。我们为Sturm-Liouville运营商和Dirac Operator在Firiete间隔中获得了独特的结果。我们的问题“确定内部光谱数据的潜力”是一种新的公式，预计将应用于许多其他操作员。（3）半连续波方程：对具有非线性耗散的波动方程渐近。我们要求耗散在“空间和时间”中是不均匀的，并且获得了足够的条件，以使解决方案在渐近上无均匀。（4）逆散射问题：首先，以首次衍生术语为波动方程建立了直接散射理论，然后建立了直接散射理论，然后建立了直接散射理论。该术语的系数被证明是通过固定能量从散射幅度重建的。每个研究者在相关场上产生了许多有趣的结果。