Binary Symmetry Constraints of N - wave Interaction Equations in 1 + 1 and 2 + 1 Dimensions

Binary symmetry constraints of the N -wave interaction equations in 1 + 1 and 2 + 1 dimensions are proposed to reduce the N -wave interaction equations into finite-dimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of the constrained flows of the N -wave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the N -wave interaction equations, and thus the integrability by quadratures are shown for the N -wave interaction equations by the constrained flows. Running title: Symmetry Constraints of N -wave Equations