Continued-fraction solutions to the Riccati equation and integrable lattice systems

Abstract

Continued-fraction solutions to the matrix Riccati equation are discussed which are constructed by using the concept of form invariance. It is demonstrated that this technique is related to the AKNS method of deriving integrable nonlinear lattice systems. This gives an explanation why continued-fraction solutions related to the Toda lattice were obtained in a previous work. Continued fractions corresponding to Kac-Van Moerbeke, discrete nonlinear Schrodinger and discrete modified KdV lattice equations are constructed. A method for linearising the Kac-Van Moerbeke lattice equations is rederived and particular solutions are generated. The authors approach demonstrates the crucial role played by the boundary condition at the finite end of the lattice for the existence of this method. These results are extended to the other two lattice systems above in the semi-infinite case and corresponding particular solutions generated in terms of Bessel functions.