A (concentration-)compact attractor for high-dimensional non-linear Schr\"odinger equations

We study the asymptotic behavior of large data solutions to Schr\"odinger equations \(i u_t + \Delta u = F(u)\) in \(\R^d\), assuming globally bounded \(H^1_x(\R^d)\) norm (i.e. no blowup in the energy space), in high dimensions \(d \geq 5\) and with nonlinearity which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as \(t \to +\infty\), these solutions split into a radiation term that evolves according to the linear Schr\"odinger equation, and a remainder which converges in \(H^1_x(\R^d)\) to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in \(H^1_x(\R^d)\). This is despite the total lack of any dissipation in the equation. This statement can be viewed as weak form of the "soliton resolution conjecture". We also obtain a more complicated analogue of this result for the non-spherically-symmetric case. As a corollary we obtain the "petite conjecture" of Soffer in the high dimensional non-critical case.