On the continuous resonant equation for NLS. I. Deterministic analysis

Abstract

Abstract. We study the continuous resonant (CR) equation which was derived in [7] as the large-box limit of the cubic nonlinear Schrödinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schrödinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the striking structure of the equation. We study in particular the dynamics on a few invariant subspaces: eigenspaces of the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space. We focus on stationary waves and their stability.

Out-of-equilibrium dynamics and statistics of dispersive PDE

“…This is particularly the case in the special dimension d = 2 (or d = 1 with the quintic nonlinearity), in which the (NLS) nonlinearity that we start with is mass-critical. We mention some of those properties from [11,16,17]:…”

“…• (CR) enjoys many stationary solutions, including (up to a phase factor e iωt ) the Gaussian e The dynamics of equation (CR) in 2D were further analyzed by P. Germain, L. Thomann, and the author from the deterministic [16] and probabilistic [17] point of views. Finally, in a joint work with Laurent Thomann [28], we exhibit the dynamics of (CR) in a completely independent fashion, as an asymptotic system for NLS with partial harmonic trapping 4 :…”

Ground State of the Conformal Flow on 𝕊3

“…In particular, some intriguing algebraic properties of the spectrum may provide a hint in searching for a Lax pair for the conformal flow. The conformal flow system (1.1) is structurally similar to the Fourier representations of the cubic Szegő equation [5][6][7] and the lowest Landau level (LLL) equation [1,8,9]. These two equations also possess ground states that saturate inequalities analogous to (1.15).…”

“…These two equations also possess ground states that saturate inequalities analogous to (1.15). Their nonlinear orbital stabilities were established in [5] and [4,8], respectively, by compactness-type arguments. Such arguments are shorter compared to the Lyapunov approach; however, we prefer the latter because it gives a hands-on control of the perturbations and, more importantly, can also be applied to local constrained minimizers (or maximizers).…”

“…The ground state of the cubic Szegő equation is a maximizer of energy under two constraints [5] (rather than one, as in our case). On the other hand, the ground state of the LLL equation is the maximizer of energy under one constraint (as in our case), but the ground state can be continued with respect to parameter p by an action of the symmetry transformation (namely, the magnetic translation) [4,8]. We eliminate the drift of the ground state with respect to parameter p by using the conserved quantities E.˛/ and Z.˛/.…”