Том 50, № 4 (2016)

We solve the classification problem for integrable lattices of the form u,_t = f(u₋₂,..., u₂) under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains five equations, including three new ones. Difference Miura-type substitutions are found, which relate these equations to known polynomial lattices. We also present some classification results for generic lattices.

It is proved that if an entire function f: ℂ → ℂ satisfies an equation of the form α₁(x)β₁(y) + α₂(x)β₂(y) + α₃(x)β₃(y), x,y ∈ C, for some α_j, β_j: ℂ → ℂ and there exist no \({\widetilde \alpha _j}\) and ˜\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az² + Bz + C) ∙ σ_Γ(z - z₁) ∙ σ_Γ(z - z₂), where Γ is a lattice in ℂ; σ_Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z₁, z₂ ∈ ℂ; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).

Let (X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points {π_i} of the fiber π⁻¹(q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P¹ with a double pole at each p_i, then these solutions are doubly periodic solutions of the matrix KdV equation U_t = 1/4(3UU_x + 3U_xU + U_xxx). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points {p_i}; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.

We consider a self-adjoint matrix elliptic operator A_{ε, ε} > 0, on L₂(R^d;Cⁿ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA_ε^1/2), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA_ε^1/2) converges to cos(τ(A⁰)^1/2) in the norm of operators acting from the Sobolev space H^s(R^d;Cⁿ) (with a suitable s) to L₂(R^d;Cⁿ). Here A₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ∂_τ²u_ε(x, τ) = −A_εu_ε(x, τ).