Analysis of linear and nonlinear wave phenomena and the inverse problem

在这个项目中，我们主要关注应用数学和物理中的一些基本偏微分方程以及化学和生物学中出现的一些模型方程所支配的波动现象的分析。总结2001 -2003年期间的研究成果，可以说本课题的研究目标是卓有成效的，主要研究者发表了5篇分析线性和非线性波以及非线性扩散的论文。（1）非线性抛物型方程：研究了广义KPP方程解的大时间渐近性。由此得到的精确公式在生物学中有着广泛的应用。（2）谱反问题：研究了由某些谱数据确定位势的反问题。得到了Sturm-Liouville算子和Dirac算子在firiete区间上的唯一性结果。我们的问题“确定潜在的内部光谱数据”是一个新的配方，预计将被应用到许多其他运营商。（3）半线性波动方程：研究了具有非线性耗散的波动方程的渐近性。我们要求耗散在空间和时间上是非齐次的，并得到解是渐近自由的充分条件。（4）逆散射问题：首先对含一阶导数项的波动方程建立了直接散射理论，然后证明了该项的系数可以由具有固定能量的散射振幅重构，在相关领域中，研究者们发展了许多有趣的结果。

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.