Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping

Abstract

We study two resonant Hamiltonian systems on the phase space L 2 (R → C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Schrödinger equation with harmonic trapping, a robust wellposedness theory, including global wellposedness in L 2 and all higher L 2 Sobolev spaces, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.

Complex plane representations and stationary states in cubic and quintic resonant systems

“…Our results have direct implications for a number of equations of mathematical physics whose resonant systems fall in our class, in relation to Bose-Einstein condensates [13], the Schrödinger-Newton system in a harmonic potential [23], and relativistic wave equations in highly symmetric spacetimes [15,17]. Examples with quintic nonlinearities include the nonlinear Schrödinger equation in a one-dimensional harmonic trap, previously treated from a mathematical perspective in [33], and the conformally invariant quintic wave equation on a two-sphere, brought forth in a our present study. We have also presented in the appendix a few extra quintic resonant systems that benefit from the analytic stuctures we have formulated, which have been obtained by a generalization of explicitly known cubic resonant systems.…”

“…We now proceed filling in the details of step 1 and step 2 required to complete the proof. In handling (34) below, we shall suppress the factor e −iλt which is common to the entire expression (34) and already matches (33).…”

“…The first example is the quintic one-dimensional nonlinear Schrödinger equation in a harmonic trap. The resonant system for this equation has been previously studied in mathematical literature [33], while motivations to consider quintic nonlinearities from the standpoint of Bose-Einstein condensate physics are given in [32]. Our purpose is to clarify how this case fits in our framework.…”

“…Mathematically rigorous considerations of the resonant approximation to the quintic onedimensional nonlinear Schrödinger equation in a harmonic trap are given in [33]. In a nutshell, one starts with the quintic analog of the nonlinear Schrödinger equation (2) given by…”

“…This is literally possible in trapped Bose-Einstein condensates, where the cubic term can be switched off using Feshbach resonances, leading to the emergence of a quintic nonlinear Schrödinger equation [32]. Finally, in section 5, we present two concrete quintic PDEs of mathematical physics whose resonant systems fit into our setup: first, the quintic nonlinear Schrödinger equation in a one-dimensional harmonic trap (recently treated in [33]), and second, the quintic conformally invariant wave equation on a two sphere (which generalizes the cubic considerations on a three-sphere in [15]).…”