Classical and Quantum integrable systems focusing on the Korteweg-de Vries's theory

除了Liouville的可积性概念外，它还存在另一个可积性概念，称为Hirota可积性，它源于求解一些非线性偏微分方程组的所谓的Hirota双线性形式方法。研究这两个定义如何匹配是很有趣的：据我们所知，还没有发现只有Hirota的可积性而没有Liouville的可积性的情况，但也许还有更多的东西有待发现。另外，Hirota的方法本身对我非常有趣：它允许从称为Hirota的tau函数的辅助函数构造一个一般的、非线性的MSS(M孤子解)。一些作者(如A.Dimakis和D.F.Muller-Hoissen)指出，这种热带化过程只是将波解转化为线轨迹，从而简化了问题。利用热带几何和Hirota方法可以得到许多关于非线性系统如何工作的建议，把以前的想法推广到其他不同于Korteveg-de Vries的可积系统可能是有趣的。

Along with Liouville's, it do exist also another concept of integrability, called Hirota's. It derives from the so called Hirota's bilinear form method of solving some nonlinear partial differential equations. It would be interesting to study how the two definitions match together: as far as we know, there have not been found yet cases in with there is just Hirota's integrability without the Liouville's one, but maybe there is mores still to be found.Also Hirota's method itself is very interesting to me: it permits the construction of a general, nonlinear, MSS (M Soliton Solution) from an auxiliary function called Hirota's tau-function. During the preparation of my dissertation, I studied KdV's tau-functions under the schemes of tropical geometry. It has been pointed out (by some authors such as A. Dimakis and d F. Muller-Hoissen) that this procedure of tropicalization just turns wave solution into line trajectories simplifying the problem. The sum of exponentials from which one builds up the tau-function becomes dominated by just one exponential and the whole structure develops into a linear one in a certain domain.So, using the tropical geometry along with Hirota's method, one can obtain many suggestions about how a nonlinear system works, it may be intriguing to extend the previous idea to other integrable systems different form the Korteveg-de Vries' one.