Local and non-local conserved quantities for generalized non-linear Schrödinger equations

It is shown how to construct infinitely many conserved quantities for the classical non-linear Schrödinger equation associated with an arbitrary Hermitian symmetric space G/K. These quantities are non-local in general, but include a series of local quantities as a special case. Their Poisson bracket algebra is studied, and is found to be a realization of the "half" Kac-Moody algebra k_R ⊗ ℂ [λ], consisting of polynomials in positive powers of a complex parameter λ which have coefficients in the compact real form of k (the Lie algebra of K).