Local and non-local conserved quantities for generalized non-linear Schr�dinger equations

Abstract

Abstract. It is shown how to construct infinitely many conserved quantities for the classical non-linear Schrόdinger 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 4 R (x) (C [A], consisting of polynomials in positive powers of a complex parameter λ which have coefficients in the compact real form of A (the Lie algebra of K).

Affine Lie algebraic origin of constrained KP hierarchies

“…It follows now from the above Lemma that H n = −2 Tr (EO n+1 (1)), which coincides with the result of Wilson [12] (see also [13]). The Z-S equations can be obtained from the Hamiltonians defined above and the first bracket structure given by…”

“…Consider now the Hamiltonian density H 1 = n i=1 q i r i = Tr (A 0 ) 2 and impose the basic relation H n = ∂ −1 ∂ n H 1 for higher Hamiltonians. It follows now from the above Lemma that H n = −2 Tr (EO n+1 (1)), which coincides with the result of Wilson [12] (see also [13]). The Z-S equations can be obtained from the Hamiltonians defined above and the first bracket structure given by…”

Kac-Moody symmetry of generalized non-linear Schr�dinger equations

“…This is a continuation of the work presented in [1], in which it was shown how to construct conserved quantities for the generalized non-linear Schrδdinger (GNLS) equation of Fordy and Kulish [2]: iί = 4ίΛV%-δ (1.1) (summation is implied over repeated indices) which is associated with a Lie algebrâ = A@m. q(x, t) is a matrix field in 1 + 1 dimensions whose components lie in m, and A is the centralizer of a special Cartan subalgebra element E satisfying the property [E,e β ]=-fe β (1.2) for all e a e-m (where α is positive). This means that the algebra ^ is "symmetric", i.e.…”

“…Each such A t is associated, via (1.5), with an evolution operator d t . It was shown in [1] that when A t is a polynomial in positive powers only, the collection of evolution operators can be labelled d Nik9 where ksi and AT is a positive integer, and that they have the commutation relation M+NfU>k] VM,iV^O;V/,fce/.…”