Reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures

We study finite-dimensional reductions of the dispersionless 2D Toda hierarchy showing that the consistency conditions for such reductions are given by a system of radial Loewner equations. We then construct their Hamiltonian structures, following an approach proposed by Ferapontov.

💡 Research Summary

The paper investigates finite‑dimensional reductions of the dispersionless two‑dimensional Toda hierarchy and constructs their Hamiltonian structures. The hierarchy is represented by two formal Laurent series λ(p;λ₁,…,λ_N) and \bar λ(p;λ₁,…,λ_N), which satisfy Lax equations generated by Poisson brackets {f,g}=p∂p f∂x g−p∂p g∂x f. A reduction is defined by requiring that λ and \bar λ depend only on a finite set of parameters λ_i (i=1,…,N) and that each λ_i obeys diagonal hydrodynamic‑type equations ∂{t_n}λ_i=v_i^{(n)}∂x λ_i, ∂{\bar t_n}λ_i=\bar v_i^{(n)}∂x λ_i. The authors prove that such a reduction is consistent if and only if the functions λ, \bar λ satisfy a system of radial Loewner equations ∂{λ_i}λ = (p λ)/(p−v_i) ∂{λ_i}u₀,  ∂{λ_i}\bar λ = (p \bar λ)/(p−v_i) ∂{λ_i}u₀, where v_i=v_i^{(1)} are the characteristic velocities of the first flow and u₀ is the constant term in λ’s expansion.

The compatibility of the Loewner system leads to the well‑known Gibbons‑Tsarev equations ∂{λ_j}v_i = (v_i v_j)/(v_j−v_i) ∂{λ_j}ϕ,   ∂{λ_i}∂{λ_j}ϕ = 2 v_i v_j/(v_i−v_j)² ∂{λ_i}ϕ ∂{λ_j}ϕ, with ϕ=log \bar u_{−1}. These equations guarantee that the reduced system is semi‑Hamiltonian: the characteristic velocities satisfy the semi‑Hamiltonian condition and the associated Riemann curvature tensor can be expressed as a quadratic form in a set of symmetry vectors w_i^α, R_{ijij}=∑_α ε_α w_i^α w_j^α.

Following Ferapontov’s approach, the authors first solve the linear system ∂{λ_j}ln√g{ii}=∂{λ_j}v_i/(v_j−v_i), i≠j, to obtain a diagonal metric g{ii}=φ_i(λ_i)^{-1}∂{λ_i}ϕ, where φ_i are arbitrary functions of a single variable. Choosing the potential metric g{ij}=δ_{ij}∂{λ_i}ϕ, they compute the rotation coefficients β{ij} and, using the Gibbons‑Tsarev relations, derive an explicit integral representation for the non‑vanishing curvature components: R_{ijij}=−(1/2πi)∮_C w_i(λ) w_j(λ) dλ, where the generating functions of the symmetries are w_i(λ)=v_i (p(λ)−v_i)^{-2} ∂_λ p(λ), and C is a closed contour encircling the critical points v_i in the λ‑plane.

The quadratic expansion of the curvature tensor yields a non‑local Hamiltonian operator of the form Π^{ij}=g^{ii}δ^{ij}∂_x+Γ^{ij}_k λ^k_x+∑_α ε_α w_i^α λ^i_x ∂_x^{-1} w_j^α λ^j_x, where the first term is local (differential), the second term involves Christoffel symbols of the metric, and the third term is a non‑local (integral) contribution. When φ_i≡1 the metric becomes purely potential and the Hamiltonian structure reduces to a purely non‑local operator, in agreement with previously known non‑local Poisson brackets for dispersionless hierarchies.

The paper concludes with an explicit N=2 example, constructing λ(p) and the corresponding characteristic velocities, verifying the Loewner system, the Gibbons‑Tsarev compatibility, and the full Hamiltonian operator. This concrete case illustrates how the abstract construction works in practice.

In summary, the work establishes that finite‑dimensional reductions of the dispersionless 2D Toda hierarchy are governed by radial Loewner equations, that these reductions are semi‑Hamiltonian via the Gibbons‑Tsarev system, and that Ferapontov’s curvature‑symmetry factorisation provides a systematic method to build both non‑local and purely non‑local Hamiltonian structures for these reductions, thereby extending the Hamiltonian theory of dispersionless integrable systems.