On the Continuous Limit of Integrable Lattices II. Volterra Systems and SP(N) Theories

A connection is suggested between the zero-spacing limit of a generalized N-fields Volterra (V N ) lattice and the KdV-type theory which is associated, in the Drinfeld–Sokolov classification, to the simple Lie algebra sp(N). As a preliminary step, the results of the previous paper [1] are suitably reformulated and identified as the realization for N=1 of the general scheme proposed here. Subsequently, the case N=2 is analyzed in full detail; the infinitely many commuting vector fields of the V 2 system (with their Hamiltonian structure and Lax formulation) are shown to give in the continuous limit the homologous sp(2) KdV objects, through conveniently specified operations of field rescaling and recombination. Finally, the case of arbitrary N is attacked, showing how to obtain the sp(N) Lax operator from the continuous limit of the V N system.