Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by $$N \gg 1$$ N ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $$\beta ^{-1}$$ β - 1 . Given a fixed $${1\le m \ll N}$$ 1 ≤ m ≪ N , we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $$\beta $$ β , for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

Quantum relativistic Toda chain

Investigated is the quantum relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl’sqwe are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six-vertexR-matrix. Both independent Baxter’sQ-operators are constructed explicitly as seria over local Weyl generators. The operator-valued Wronskian ofR-matrix. Both independent Baxter’sQsis also calculated.

Lecture on SUSY Gauge Theories and Integrable Systems

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.