Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.

💡 Research Summary

The paper addresses the problem of classifying integrable Hamiltonian systems of hydrodynamic type in three‑dimensional space‑time (two spatial dimensions plus time). The authors restrict themselves to systems whose Poisson brackets are constant, i.e. the Poisson tensor Ω is a constant antisymmetric matrix satisfying the Jacobi identity. In this setting the evolution equations take the compact form

  uⁱ_t = Ωⁱʲ ∂_j H(u), i = 1,…,n,

where u = (u¹,…,uⁿ) are the field variables and H(u) is a scalar Hamiltonian density. The central question is: for which Hamiltonians H does the system admit a dispersionless Lax pair and an infinite hierarchy of hydrodynamic reductions, thereby guaranteeing integrability in the sense of the method of hydrodynamic reductions?

Methodology
The authors employ two complementary criteria. First, they look for a dispersionless Lax pair (ψ_t = A(ψ,λ) ψ_x, ψ_y = B(ψ,λ) ψ_x) with a spectral parameter λ. Existence of such a pair is equivalent to the system being a compatibility condition of two first‑order quasilinear equations, a hallmark of integrable dispersionless models. Second, they require that the system possess infinitely many reductions to systems of Riemann invariants r^α(x,t,y) satisfying

  r^α_t = v^α(r) r^α_x, r^α_y = w^α(r) r^α_x,

for arbitrary numbers of components m. The presence of an infinite family of such reductions is the modern definition of integrability for multidimensional quasilinear systems.

Two‑component case (n = 2)
With Ω fixed to the canonical 2×2 symplectic matrix, the Hamiltonian can be written as

  H(u¹,u²) = f(u¹) + g(u²) + h(u¹,u²).

The authors systematically analyse the functional freedom in h. By imposing the Lax‑pair condition and demanding an infinite set of reductions, they find that h must belong to a finite list of algebraic forms. Up to trivial equivalences (linear changes of variables, addition of total derivatives), the admissible Hamiltonians are:

1. Linear cross term: h = α u¹ u².
2. Quadratic‑linear mix: h = β (u¹)² u².
3. Linear‑quadratic mix: h = γ u¹ (u²)².
4. Logarithmic interaction: h = δ ln(u¹ + u²).
5. Square‑root interaction: h = ε √(u¹ u²).

For each case the authors explicitly construct the functions A(ψ,λ) and B(ψ,λ) of the Lax pair and present the corresponding reduction equations. The logarithmic and square‑root families illustrate genuinely nonlinear interactions yet still admit a dispersionless Lax representation.

Three‑component case (n = 3)
Here Ω is a constant 3×3 antisymmetric matrix, which can be parametrised by a constant vector a = (a₁,a₂,a₃) via Ω_{ij}=ε_{ijk} a_k. The Hamiltonian is expanded as

  H = F(u¹) + G(u²) + K(u³) + L(u¹,u²,u³).

The mixed term L encodes all possible three‑field interactions. By repeating the Lax‑pair and reduction analysis, the authors obtain a complete list of admissible L. Up to equivalence, the seven families are:

1. Linear cross terms: L = α u¹ u² + β u² u³ + γ u³ u¹.
2. Mixed quadratic‑linear terms: L = δ (u¹)² u² + ε u² (u³)².
3. Pure cubic interaction: L = ζ u¹ u² u³.
4. Square‑root of the product: L = η √(u¹ u² u³).
5. Logarithmic sum: L = θ ln(u¹ + u² + u³).
6. Separate quadratic terms: L = ι