Schrodinger方程與色散波方程之交互作用的研究

Abstract

這計畫主要是針對Schrodinger 方程與其他色散波方程之交互作用的研究，這類 的色散波包含有著名的KdV 方程、長波-短波方程還有耦合型的Schrodinger 方程。這 些方程在非線性光學、量子力學、電漿物理、水波方程都有重要應用，近年來則以在 凝態物理之應用最為出色。在這為期三年之計畫，我們將逐步研究底下之問題： (1)存在性問題： 關於這問題我們將利用兩種不同的方法來研究：WKB 分析與調和分析。 WKB 分析 始終是研究Schrodinger 這類方程的局部平滑解並瞭解其流體結構的最好方法，這對於 研究其半古典極限(Semiclassical limit)或零色散極限(Zero dispersion limit)的最 佳方法。其次調合分析的方法，主要是運用Fourier 變換的限制性估計。並引進J. Bourgain, C. Kenig、T. Tao 等人的最近研究方法。 (2)自我相似解與解之漸近行為： 研究自我相似最自然、最直觀的方法是量綱分析，我們將由量綱分析(dimensional analysis)的角度去判斷是否有可能存在自我相似解，之後再由典型的方法將之化為常 微分方程來證明其存在性，而後再探討其漸近行為。最後，再由量綱分析的手法，來 得到衰減估計(decay estimate)。 (3)奇異極限(singular limit) 根據研究Schrodinger 方程之半古典極限之經驗，我們依然可以探討這類方程之 零色散極限(Zero dispersion limit)，從中研究其交互作用，亦即耦合型方程之特色， 並藉以判別與單純的Schrodinger 方程之差異。

This project is devoted to the study of the Schrodinger equation coupled with the other dispersive equations. Among them are the KdV equation, short wave long wave interaction equations, and the coupled Schrodinger equations which occur in nonlinear optics, quantum mechanics, plasma physics and water waves. It even occurs in Bose-Einstein condensation.In this three years』 project we plan to treat the following problems. (1)Existence problem： We will prove the local smooth solution by the WKB analysis and the quasilinear symmetric hyperbolic system. This method is well adapted to the semiclassical limit or the zero-dispersion limit. However, we are also interested in J. Bourgain, C. Kenig and T. Taos』 work treating the PDE problem by Harmonic analysis. (2)Self-similar solution and the asymptotic behavior The best and natural way to study the self-similar solution is by dimensional analysis. We will apply the dimensional analysis to testify the existence of the self-similar solution, then prove rigorously the existence by investigating the associated ODE. Furthermore, we will employ the self-similar solution to prove the global solution whenever the initial data is closed to the self-similar solution. Finally we still use the dimensional analysis to obtain the asymptotic behavior and the decay estimates. (3)singular limit Based on the previous experience of the semiclassical limit of the Schrodinger type equations, we can still study the zero-dispersion limit of the coupled equations, but we need to see the effect of the coupling in the singular limit process.