Resonant Hamiltonian Systems Associated to the One-dimensional Nonlinear Schrödinger Equation with Harmonic Trapping

We study two resonant Hamiltonian systems on the phase space L2(R → C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that appears as the modified scattering limit of a specific NLS equation. 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, a robust wellposedness theory, including global wellposedness in L2, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.