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

Abstract

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 time. The initial value problem solved, i.e. the author obtains 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'.