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

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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.