第一部份： 我們致力於改進並推廣在D. Cao and E. S. Noussair
[Multiplicity of positive and nodal solutions for nonlinear elliptic problems in R^N, Ann. Inst. H. Poincare Anal. Nonlineire 13 (1996) 567-588] 這篇文章中的結果。得到的變號一次解之正部與負部分別集中靠近在權函數的極大點處。經由一個動態系統的探討，可得到一個變號函數的連續體並建立了巴萊-斯麥爾序列。我們因此有了變號一次解之個數的新下界。
第二部份：我們研究一個長波、短波交互作用的模型，自然出現在流體動力學及電漿物理之薛丁格-KdV 系統的流體力學結構及半古典分析。
第三部份：我們主要是探討由馬克斯韋爾系統均質化導出的記憶效應。其記憶核可由渥特拉積分方程來描述，並可藉由楊氏測度來確認。由動力方程化可允許我們去獲得能量密度的均質化以及相關的守恒律。

Part I
In part one, we are devoted to improve and generalize the results on D. Cao and E. S. Noussair [Multiplicity of positive and nodal solutions for nonlinear elliptic problems in R^N, Ann. Inst. H. Poincar´e Anal. Non
Lineair´e 13 (1996) 567–588]. The 2–nodal solutions we obtain have their positive and negative parts concentrate near the set of maximum points of weight functions. Via a dynamical systems approach, we get a continuum
of sign–changing functions to exhibit a Palais–Smale sequence. We establish new lower bounds for the number of 2–nodal solutions.
Part II
We study the semiclassical limit of the Schr¨odinger-Korteweg-de Vries system which appears naturally in fluid dynamics and plasma physics as a model of interaction between a short wave and a long wave.
Part III
We study the memory effect induced by homogenization of the Maxwell system for conducting media. The memory kernel is described by the Volterra integral equation. It can be characterized explicitly in terms of Young’s measure and the kinetic formulation allows us to obtain the
homogenization of the energy density and the associated conservation law with the Poynting vector.

Part I
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