Instability of the periodic nonlinear Schrodinger equation

TL;DR

This paper investigates the instability and ill-posedness of periodic nonlinear Schrödinger equations with odd integer powers, revealing that solutions can become highly unstable even with small initial data in negative Sobolev spaces.

Contribution

It demonstrates that these equations are more unstable than previously known, showing discontinuity of the solution map and existence of solutions that diverge in distribution topology.

Findings

Solution map discontinuous for cubic case in distribution space

Existence of solutions close initially but diverging in distribution topology for higher powers

Ill-posedness extends to all negative Sobolev spaces

Abstract

We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L^2 mass. These equations are known to be illposed in H^s for all negative s, in the sense that the solution map fails to be uniformly continuous from H^s to itself, even for short times and small norms. Here we show that these equations are even more unstable. For the cubic equation, the solution map is discontinuous from H^s to even the space of distributions. For the quintic and higher order nonlinearities, there exist pairs of solutions which are uniformly bounded in H^s, are arbitrarily close in any C^N norm at time zero, and fail to be close in the distribution topology at an arbitrarily small positive time.