Introduction to integrable systems: open Toda lattice, KP-, and KdV-hierarchies

The goal of this crash course is to make a brief introduction into the beautiful world of integrable hierarchies. We do not intend to give a general survey of integrable system theory but rather want to use few known examples to introduce notions and initial circle of ideas specific for integrable equations. We used [A] as a reference for the first section, [P] for the second section, and [MJD] and [K] for the third section. 1. Classical mechanics, Poisson structures, Hamiltonian formalism The classical Newton’s equations for a particle with coordinates q(t) = (q1(t), . . . , qn(t)) in a potential force field F have a form q̈ = F, F = −gradP, where P = P (q) is a potential energy. We rewrite these equations as { q̇ = p ṗ = −gradP, where p = (p1, . . . , pn) is the momentum. The space R = {(p, q)} is the phase space of the system. Definition 1.1. A Poisson structure on a manifold M is a bilinear bracket {·, ·} on the space of functions C(M)× C(M) → C(M) satisfying (1) skew-symmetry {F,G} = −{G,F} (2) Leibnitz rule {F,G ·H} = {F,G}+ {F,H} (3) Jacobi identity {{F,G}H}+ {{G,H}F}+ {{H,F}G} = 0 Remark 1.2. Leibnitz rule is equivalent to the following formula: in any local coordinate system {xi} on M one can express the Poisson bracket as {F,G} = ∑ i ∂F ∂xi {xi, G}.