Symmetric matrix ensemble and integrable hydrodynamic chains

The partition function of the Symmetric Matrix Ensemble is identified with the $$\tau $$ τ -function of a particular solution of the Pfaff Lattice. We show that, in the case of even power interactions, in the thermodynamic limit, the $$\tau $$ τ -function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

On a class of integrable Hamiltonian equations in 2+1 dimensions

We classify integrable Hamiltonian equations of the form u t = ∂ x ( δ H δ u ) , H = ∫ h ( u , w ) d x d y , where the Hamiltonian density h ( u , w ) is a function of two variables: dependent variable u and the non-locality w = ∂ x − 1 ∂ y u . Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h ). We show that the generic integrable density is expressed in terms of the Weierstrass σ -function: h ( u , w ) = σ ( u ) e w . Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

Hamiltonian formalism of two-dimensional Vlasov kinetic equation

In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo–Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo–Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.