This summer I'm going to learn a mini-course about soliton theory ("Soliton equations and symmetric functions" in LHSM (Russian summer school in mathematics). The web-page of this course is https://mccme.ru/dubna/2019/courses/rozhkovskaya.html (warning: it's in Russian, so here's my translation of its program:

  1. Examples of soliton equations, their solutions. Properties of solitons.

  2. Definition and properties of Hirota derivatives.

  3. KdV and KP equations in terms of Hirota derivatives.

  4. Bilinear form of the KP hierarchy.

  5. Symmetric functions: main definitions, properties of elementary, complete and power sum symmetric functions.

  6. Interpretation of the bilinear form of KP hierarchy in terms of symmetric functions.

  7. (If it'll be enough time) Some words about the action of fermions on the symmetric functions and solutions of KP hierarchy).

This program has intrigued me (despite my main interest is representation theory and related topics) and now I'm looking for a book in which the soliton theory will be outlined according to this program (i.e. using Hirota derivatives from very beginning), maybe even without any mentions of symmetric functions..

I saw Kasman's book but he doesn't say too much about Hirota derivatives and his style seems simplistic to me...

I'll be grateful for anyone who'll give me some references. It'll be great if these books will contain an "algebraic approach" to the subject or some connections with representation theory... But any other reference will also be greeted with admiration!

My question's already been posted on math.SE: https://math.stackexchange.com/questions/3286654/literature-about-solitons-and-hirota-derivatives. But I'd no response, so I decided to post it here, because I thought that it's applicable for this site. If it isn't so, please tell me. I'll delete it.

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    $\begingroup$ I notice you cross-posted within 3 hours (and now have similar answers on both sites); you should really wait at least one day before cross-posting. $\endgroup$ – Carlo Beenakker Jul 8 '19 at 14:44
  • $\begingroup$ @CarloBeenakker Thank you for the comment! OK, I understood. Should I now delete this answered question? When I'm trying to do this site tells me that they don't recommend deleting questions with answers... $\endgroup$ – kotlinski Jul 8 '19 at 14:47
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    $\begingroup$ it's no big deal, you can leave this as it is, just keep it in mind for future postings. $\endgroup$ – Carlo Beenakker Jul 8 '19 at 14:55

The canonical reference is The Direct Method in Soliton Theory, Cambridge UP, 2009, by Ryogo Hirota himself.


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