A solution of the initial value problem for half-infinite integrable lattice systems

In previous studies solutions of a number of half-infinite nonlinear lattice systems were constructed from continued fraction solutions to corresponding Riccati equations. A method for linearizing the Kac-Van Moerbeke lattice equations was reconstructed and extended to the discrete nonlinear Schrodinger equation, relativistic Toda lattice equations as well as other examples. This approach demonstrated the important role played by the boundary condition at the finite end and solutions were obtained for given behaviour of this end with time. Here the initial value problem will be solved, i.e. we will obtain solutions of these half-infinite lattice equations corresponding to prescribed values at t = 0. Such solutions were obtained for the Kac-Van Moerbeke lattice through studying the time behaviour of continued fractions related to Jacobi matrices and the corresponding 'hamburger moment problem'. A similar approach is used here and we find for the discrete nonlinear Schrodinger equation that the continued fractions which arise are related to the 'trigonometric moment problem'. We also consider the discrete modified KdV equation, relativistic Toda lattice and discrete-time Toda lattices and in these cases T-fractions, which are related to the 'strong Stieltjes moment problem', are used to solve the initial value problem.