Nearly five decades have passed since the introduction of the inverse scattering transform (IST). More than a century passed since solitons were first discovered by John Scott Russell in his famous experiment. Even older than both is the field of integrable systems, which emerged with the development of Hamiltonian mechanics.
Despite the popularity and importance of the field, the theory of integrable systems remains one of the most difficult mathematical theories to learn. The difficulty stems in the exciting fact that there is still an excess of open fundamental problems in the theory. There are dozens of books on the subject, yet a welcoming and simple introduction to the field is not easily found.
The goal of this codex is to make the theory of integrable systems more accessible and easy to learn. Being practical, it is difficult to imagine that this manuscript will turn into a complete and rigorous treatment of integrable systems, as to date such a treatment does not even exist. Nevertheless, I will do my best to keep the discussion rigorous and concise. My most important goal here is to introduce the subject in the same way I would have of liked it to be taught to me.