Let A=C[x±1 1, x±1 2 be the ring of Laurant polynomials and B the set of skew derivations of A. Set L=A⊕B. In this paper, we study the automorphism group, derivations and universal central extension of the derived Lie subalgebra of L.

In this paper we discuss the mixed nonlinear Schrodinger Equation Ut=iαUxx+βU2Ux+γ|U|2Ux+ig(|U|2) U, U (x, 0)=Uo(x), where x, β,γ are constants with α≠0. If the real function g(s)εC2(R+) satisfies some conditions, and the initial data U0(x)εH1 (R) are such that the norm |U0|L2(R) is sufficiently small. then there is a unique weak soultion U (x, 1) of (*) such that U (x, 1)εCw(R; H1 (R)).

In this paper we study the representation theory for certain "half lattice vertex algebras." In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.

In the present paper, we study the nonzero level Harish-Chandra modules for the Virasoro-like algebra. We prove that a nonzero level Harish-Chandra module of the Virasoro-like algebra is a generalized highest weight (GHW for short) module. Then we prove that a GHW module of the Virasoro-like algebra is induced from an irreducible module of a Heisenberg subalgebra.