Separation of variables for soliton equations via their binary constrainedflows

Binary constrained flows of soliton equations admitting 2x2 Lax matrices ha ve 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax m atrices directly give rise to first N pairs of canonical separated variable s for their separation of variables. We propose a new method to introduce t he other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are e stablished. Finally, the factorization of soliton equations by two commutin g binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hier archies. (C) 1999 American Institute of Physics. [S0022-2488(99)02112-X].