A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations

Abstract

We study the asymptotic behavior of large data solutions to Schrödinger equations $i{u}_t+\Delta u=F(u)$ in ${\mathbb{R}}^d$, assuming globally bounded $H_x^1({\mathbb{R}}^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d\ge 5$ 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 Schrödinger equation, and a remainder which converges in $H_x^1({\mathbb{R}}^d)$ to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in $H_x^1({\mathbb{R}}^d)$. 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.

60 pages. A confusing typo (pointed out by Tristan Roy) regarding two parameters (mu_3 and mu_4) mistakenly being swapped with each other for a large portion of the paper has now been fixed

References in corpus (2)

• Asymptotic stability of N-soliton states of NLS
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions

Cited by in corpus (4)

• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation
• Existence and stability of infinite time bubble towers in the energy critical heat equation
• On solutions with compact spectrum to nonlinear Klein--Gordon and Schroedinger equations