Nonlinear differential-difference equations related to the second Painleve equation

As a result of classification of second order ordinary differential equations without movable branch points, f''=F(z,f,f'), f=f(z), '=d/dz, where F is rational in f, algebraic in f' and analytic in z, a number of the so-called Painleve equations was obtained. Among them, six irreducible equations are best known. They led to the recognition of new functions, called the Painleve transcendents. The Painleve equations have numerous applications in modern mathematics and mathematical physics. They can be obtained by similarity reductions from certain integrable partial differential equations (e.g., KdV, mKdV and others). They possess a number of other remarkable properties (e.g., Backlund transformations, classical solutions, the Hamiltonian structure). Via the Hamiltonian structure the Painleve equations are related to their associated equations, the so-called sigma-equations. In this paper we derive Backlund transformations for two \sigma-forms of the second Painleve equation (with respect to two different Hamiltonians) and use these transformations to obtain nonlinear differential-difference and difference equations for the solutions.