文档介绍： Scaling Mathematical Mo delling In tegrable Systems Initial Profile t=.2 t=.4
0 0 0 − − − 0 2 4 6 0 2 4 6 0 2 4 6 t=.5 t=.6 t=1
0 0 0 − − − 0 2 4 6 0 2 4 6 0 2 4 6 D H Sattinger Univ ersit y of Minnesota Preface This pro ject grew out of a con v ersation with Willi J ager in Heidelb erg on a visit in Jan uary I discussed with Willi m yin v estigations with Mariana Haragus no w Haragus Courcelle at Stuttgart Univ ersit at on the v alidit yof the Kort w eg deV ries appro ximation to the Euler equations of gra vit yw a v es Man y of pletely in tegrable systems of partial di eren tial equa tions arise as formal singular asymptotic appro ximations in some manner or other to plicated nonlinear disp ersiv e w a v e equations It is a curious fact that the metho ds of asm yptotic expansion and c hoice of scaling should lead so often pletely in tegrable mo dels This artifact has nev er been prop erly explained These deriv ations are generally formal in nature and with a few ex ceptions ha v e nev er b een prop erly justi ed with mathematical rigor Willi prop osed that I giv e a series of Deutsche Mathematische V er einigung lectures on the sub ject of scaling and mathematical mo delling My hop e here is to presen t an in tro duction to disp ersiv e w a v es and the deriv ation of some of the asso pletely in tegrable systems raise a n um b er of issues that ha v e not b een fully dealt with in tro duce some metho d ologies including the use of n umerical animation of w eakly nonlinear disp er siv e w a v e mo dels using Matlab and pro vide prehensiv e bibliograph y y I w ould lik e to thank m y colleagues and studen ts for man men ts and suggestions in the preparation of these notes esp ecially Sam Al b ert Jerry Bona Mariana Haragus Courcelle Yi Li P ete