Integrable Hierarchies 2026

• Lax pairs and examples of integrable systems [BBT07, secs.2.4-2.9]
• Lax pairs for tops ([BBT07, sec.3.1] examples 1 and 2)
• Integrable hierarchy in classical mechanics ([BBT07, sec.3] eqs. (3.37), (3.45))
• Free bosons, free fermions, -function definition, bilinear identity [Mor94, secs.4.2, 4.3]. Vertex operators [BBT07, sec.9.3]. Grassmanian [BBT07, sec.9.6] and determinant formula ([BBT07] eq. (9.61))
• Boson-fermion correspondence and pseudodifferential operators [BBT07, secs.9.7-9.8]. Kadomtsev-Petviashvili (KP) hierarchy ([BBT07] eq. (9.74))
• Adler trace for pseudodifferential operators [BBT07, sec.10.1]. KP equation [BBT07, sec.10.2]
• KP solutions: solitons (see fig. 1), rational, periodic
• Korteweg-De Vries hierarchy. Reduction from to affine . Solutions

• Big yellow book [BBT07], you don't need everything here
• Small Kyoto school book [MJD00]. The goal of the course is more-or-less to understand the set of topics covered here
• Eternal classic [Mor94]. Quite unique text covering a unique set of topics. You don't need everything here, but you'll need more in the future (see e.g. last Matrix Models course page)
• Other integrability courses [Aru07; Bei14; Zab25; Иса19]. Immediate importance of some of them for us can be very indirect. Broaden your perspective!

References