Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

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The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u(t), t ∈, [0, ∞), of the problem, one assigns a self-adjoint semi-infinite Jacobi matrix J(t) whose spectral measure dπ(λ; t) undergoes simple evolution in time t. The solution of the Cauchy problem goes as follows. One writes out the spectral measure dπ(λ; 0) for the initial value u(0) of the solution and the corresponding Jacobi matrix J(0) and then computes the time evolution dπ(λ; t) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix J(t) from dπ(λ; t) and hence finds the desired solution u(t). In the present paper, this approach is generalized to the case in which the role of J(t) is played by a block Jacobi matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure dπ(ζ; t) defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.