Integration of Some Differential-Difference Nonlinear Equations Using the Spectral Theory of Normal Block Jacobi Matrices

The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution $u(t)$, $t\in[0,\infty)$, of the problem, one assigns a self-adjoint semi-infinite Jacobi matrix $J(t)$ whose spectral measure $d\rho(\lambda;t)$ undergoes simple evolution in time $t$. The solution of the Cauchy problem goes as follows. One writes out the spectral measure $d\rho(\lambda;0)$ for the initial value $u(0)$ of the solution and the corresponding Jacobi matrix $J(0)$ and then computes the time evolution $d\rho(\lambda;t)$ of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix $J(t)$ from $d\rho(\lambda;t)$ 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 $d\rho(\zeta;t)$ 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.