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.

We construct realizations for the 2-toroidal Lie algebra associated with the Lie algebra A1 using vertex operators based on bosonic fields. In particular our construction realizes higher-level representations of the 2-toroidal algebra for any given pair of levels (k0, k1) with k0≠0. We also construct a smaller module of level (k0, 0) for the toroidal algebra from the Fock space using certain screening vertex operator, and this later representation generalizes the higher-level construction of the affine Lie algebra sl2.

We give a de nition for a new class of Lie algebras by generators and relations which simultaneously generalize the Borcherds Lie algebras and the Slodowy G.I.M. Lie algebras. After proving these algebras are always subalgebras of Borcherds Lie algebras, as well as some other basic properties, we give a vertex operator representation for a factor of them. We need to develop a highly non-trivial generalization of the square length two cuto theorem of Goddard and Olive to do this.

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation: φt=iαφxx+βφ2¯φx+γ|φ|2φx + ig(|φ|2)φ. Under certain conditions on the initial data and the function g(s), we study the existence and nonexistence of global solution for this equation. The blow-up solution and the blow-up time are also investigated.

In this paper we study the Cauchy problem for the generalized equation of finite-depth fluids ∂tu- G(∂2xu)-∂x (up/p)=0, where G(·) is a singular integral, and p is an integer larger than 1. We obtain the long time behavior of the fundamental solution of linear problem, and prove that the solutions of the nonlinear problem with small initial data for p>5/2+√21/2 are decay in time and freely asymptotic to solutions of the linear problem. In addition we also study some properties of the singular integral G(·) in Lq(R) with q>1.

We present-a general vertex operator construction based on the Fock space for an affine Lie algebras of type A. This construction allows us to give a unified treatment for both the homogeneous and rinciple realizations of the affine Lie algebras glN as well as for some extended affine Lie algebras coordinatized by certain quantum tori.