A remark on normal forms and the "upside-down" I-method for periodic NLS: Growth of higher Sobolev norms

Abstract

We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish parallel to u(t)parallel to H-s less than or similar to(1+vertical bar t vertical bar)(alpha(s-1)+) with alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with alpha = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with alpha = 4/9. These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain's idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.