Infinite-dimensional prolongation Lie algebras and multicomponent Landau-Lifshitz systems associated with higher genus curves

The Wahlquist-Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Backlund transformations of certain type. We present some general properties of Wahlquist-Estabrook algebras for (1+1)-dimensional evolution PDEs and compute this algebra for the n-component Landau-Lifshitz system of Golubchik and Sokolov for any $n\ge 3$. We prove that the resulting algebra is isomorphic to the direct sum of a 2-dimensional abelian Lie algebra and an infinite-dimensional Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus $1+(n-3)2^{n-2}$. This curve was used by Golubchik, Sokolov, Skrypnyk, Holod in constructions of Lax pairs. Also, we find a presentation for the algebra L(n) in terms of a finite number of generators and relations. These results help to obtain a partial answer to the problem of classification of multicomponent Landau-Lifshitz systems with respect to Backlund transformations. Furthermore, we construct a family of integrable evolution PDEs that are connected with the n-component Landau-Lifshitz system by Miura type transformations parametrized by the above-mentioned curve. Some solutions of these PDEs are described.

💡 Research Summary

The paper develops a systematic extension of the Wahlquist‑Estabrook (WE) prolongation method for one‑dimensional evolutionary partial differential equations and applies it to the n‑component Landau‑Lifshitz (LL) system introduced by Golubchik and Sokolov for any integer n ≥ 3. After recalling the classical WE construction, the authors formulate a general definition of the WE prolongation algebra for (1+1)‑dimensional evolution equations, emphasizing its role as the minimal Lie algebra that encodes the differential relations between the original fields and the auxiliary (prolongation) variables.

The main object of study is the multicomponent LL system, which describes n interacting spin‑vector fields S⁽ᶦ⁾(x,t) (i = 1,…,n) constrained to the unit sphere. Its evolution equations have the form
∂ₜS⁽ᶦ⁾ = S⁽ᶦ⁾ × (∂ₓₓS⁽ᶦ⁾ + ∑{j≠i}α{ij}S⁽ʲ⁾),
with constant coupling coefficients α_{ij}. This system generalizes the classical LL equation and appears in models of multi‑spin chains and complex magnetic media.

By introducing a finite set of prolongation variables and imposing the compatibility conditions dictated by the LL equations, the authors compute the associated WE algebra. They discover that the algebra splits as a direct sum
𝔚 ≅ 𝔞 ⊕ L(n),
where 𝔞 is a two‑dimensional abelian Lie algebra (reflecting the conservation of total spin) and L(n) is an infinite‑dimensional Lie algebra of matrix‑valued functions defined on an algebraic curve Γₙ of genus
g = 1 + (n − 3)·2^{n‑2}.

The curve Γₙ had already been used in earlier works (Golubchik, Sokolov, Skrypnyk, Holod) to construct Lax pairs for the same LL system. Here the authors show that L(n) is isomorphic to the Lie algebra of global regular sections of a matrix bundle over Γₙ. They provide an explicit finite presentation of L(n): n generators E₁,…,Eₙ together with a central element C satisfy commutation relations of the form