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

除了Liouville的概念，还有另一个可积性的概念，叫做Hirota的。它来源于Hirota的双线性形式的方法来解一些非线性偏微分方程。研究这两种定义如何匹配是很有趣的就我们所知，还没有发现只有Hirota的可积性没有Liouville的可积性的情况，但也许还有更多的情况有待发现。Hirota的方法本身对我来说也很有趣：它允许从一个叫做Hirota的tau函数的辅助函数中构造一个一般的、非线性的MSS （M孤子解）。在论文的准备过程中，我研究了热带几何格式下的KdV的tau函数。有人指出（一些作者如A. Dimakis和d . F. Muller-Hoissen），这种热带化过程只是把波解变成线轨迹，简化了问题。建立tau函数的指数和变成了一个指数，整个结构在某一特定区域发展成线性结构。因此，使用热带几何和Hirota的方法，人们可以得到许多关于非线性系统如何工作的建议，将之前的想法扩展到其他不同于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.