The solutions for nonlinear Schrodinger equations

We consider the nonlinear Schrodinger equation {iu(t) +Delta u +/- f(u) = 0, f(u) = vertical bar u vertical bar(2m)u(n), m, n is an element of N, (x, t) is an element of R-N x R u(0, x) = u(0)(x). (*) We give a formal solution of the Cauchy problem (*). By this formal solution, we obtain that there exists a constant B independent of p, N such that for any initial data parallel to u(0)parallel to H-s < B, 0 <= s < N/2, 2m + n = 1 + 4/(N - 2s), the Schrodinger equation (*) has a unique global solution u is an element of L-r (-infinity, infinity; B-p(s),(2)). Moreover, when m = 0, we give a unique (weak) solution of the Cauchy problem (*) in D' (distribution). (C) 2013 Elsevier Ltd. All rights reserved.