Review of continued fractions solutions of riccati equations and integrable lattice equations

الملخص EN

Continued fraction solutions to the matrix Riccati equation are discussed.

The relation between the elements of the continued fraction and the coefficients of the matrix Riccati equation are used to generate a class of well known non-linear lattice equations.

It is demonstrated that the relation between the elements of the continued fraction and the power series was used to construct a new approach for linearising these equations.

This method of linearisation is extended to other lattice equations solvable in the framework of the discrete A.K.N.S.

system if a boundary condition is imposed at the finite end.

A connection with this discrete A.K.N.S.

system is found.

Other nonlinear lattice equations obtained as the compatibility condition between two linear systems are linearised and solutions of these lattice equations are obtained.

This new approach of linearisation illustrates the important role played by the boundary condition at the finite end of the lattice