. A P ] 1 3 N ov 2 00 3 INSTABILITY OF THE PERIODIC NONLINEAR SCHRÖDINGER EQUATION

We study the periodic non-linear Schrodinger equation −iu t +u xx = ±|u| p−1 u for initial data which are assumed to be small in some negative order Sobolev space H s (T) (s < 0), but which may have large L 2 mass. In [6], [7] these equations were shown to be ill-posed in H s (T), in the sense that the solution map was not uniformly continuous from H s (T) to itself even for short times and small norms. Here we show that these equations are even more unstable, in different ways for different p. For instance, for the cubic equation (p = 3) we show that the solution map is not continuous as a map from H s (T) to even the space of distributions (C ∞ (T)) *. For the quintic equation, the solution map fails to be uniformly continuous from C ∞ to C −∞ in the sense that there exist pairs of solutions which are uniformly bounded in H s , are arbitrarily close in the C ∞ topology at time 0, and fail to be close in the distribution topology at an arbitrarily small time t > 0. Like unhappy families, every unfortunate scientific idea is unfortunate in its own way. 1. Two unstable Cauchy problems for non-linear Schrödinger equations This is the third in a series of papers [7],[8] investigating the failure of nonlinear evolution equations (in particular, defocusing non-linear Schrödinger equations) to be locally well-posed in certain Sobolev spaces H s. Informally, we say that a Cauchy problem is locally well-posed in H s if for any choice of initial data u 0 in H s , there exists a time T = T (u 0 H s) > 0 depending only on the norm of the initial data, for which a solution exists on the time interval [0, T ], is unique, and depends continuously on the initial data as a map from H s x to C 0 t H s x on the time interval [0, T ], and we say the Cauchy problem is ill-posed if it is not well-posed. At first glance, this seems like a simple enough clas-sificiation of Cauchy problems into two distinct classes. However, it appears that there are in fact many different types of ill-posedness, caused by different mechanisms of the underlying equation (blowup, instability of soliton-type solutions, energy transfer from …