On the growth of high Sobolev norms for certain one-dimensional Hamiltonian PDEs (1506.04181v2)

Abstract: This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{\"o}dinger equation on the torus :$$i \partial_t u = |D|^\alpha u+|u|^2 u, \quad u(0, \cdot)=u_0,$$where $\alpha$ is a real parameter. We show that, apart from the case $\alpha = 1$, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.