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Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies. (English) Zbl 1163.37022

The paper is devoted to the applications of block Toeplitz determinants and their asymptotics to the study of integrable hierarchies. Given a function \(\gamma (z)\) on the circle and define the Toeplitz matrix \(T_{N}(\gamma )\) with symbol \(\gamma \),
\[ T_{N}(\gamma )= \begin{pmatrix} \gamma^{(0)} & \cdots & \cdots & \gamma^{(-N)} \\ \gamma^{(1)} & \cdots & \cdots & \gamma^{(-N+1)} \\ \cdots & \cdots & \cdots & \cdots \\ \gamma^{(N)} & \cdots & \cdots & \gamma^{(0)} \end{pmatrix}, \]
where \(\gamma ^{(k)}\) are the Fourier coefficients \(\gamma (z)=\sum_{k}\gamma ^{(k)}z^k\). The term block Toeplitz is used for the case of matrix-valued symbol \(\gamma (z)\). In that case the entries \(\gamma ^{i-j}\) of the above matrix are \(n\times n\) matrices themselves. Denote \(D_N(\gamma )=det \;T_{N}(\gamma )\), and use the notation \(T(\gamma )\) for the \(\mathbb N\times \mathbb N\) matrix obtained letting \(N\to\infty \). The main theory of Toeplitz determinants is connected with the calculation of \(D_N(\gamma )\) as \(N\) tends to infinity. Another goal of the theory is to find expressions for \(D_N(\gamma )\) as well as for its limit in terms of Fredholm determinants. The above stated block Toeplitz determinants are applied to the computation of the \(\tau \) function of an (almost) arbitrary solution of the Gelfand-Dickey hierarchy
\[ {\partial L\over\partial t_j}=[(L^{{j\over n}})_{+},L], \]
where \(L\) is a differential operator of order \(n,j\neq nk\). The author proposes a new method for computing any Gelfand-Dickey \(\tau \) function defined on a Segal-Wilson Grassmannian manifold as the limit of block Toeplitz determinants associated to a certain class of symbols \(\mathcal W\)\((t;z)\). Truncated block Toeplitz determinants associated to the same symbols are shown to be \(\tau \) functions for rational reductions of the KP. An application of the theory for the Riemann-Hilbert problems is shown as well. The theory is illustrated with some interesting examples.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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