Supercritical Geometric Optics for Nonlinear Schrödinger Equations

Abstract

We consider the small time semi-classical limit for nonlinear Schrödinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y. Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows. Support by the ANR project SCASEN is acknowledged.

Loss of regularity for supercritical nonlinear Schrödinger equations

“…However, the argument in [21] relies very strongly on the fact that the nonlinearity is defocusing, and cubic at the origin. In [1], we have proposed an approach that justifies WKB analysis in Sobolev spaces for (1.5) for any σ 2, in space dimension n 3 (higher dimensions could also be considered with the same proof, up to considering sufficiently large values of σ). In [18] and [32], WKB analysis was justified in spaces based on analytic regularity, in a periodical setting and for analytic manifolds, respectively.…”

“…In [18] and [32], WKB analysis was justified in spaces based on analytic regularity, in a periodical setting and for analytic manifolds, respectively. As noticed in [1], analytic regularity is necessary to justify WKB analysis with a focusing nonlinearity. The analytic regularity essentially allows to view the nonlinearity as a semilinear perturbation, and to construct an approximate solution that solves (1.5) up to a source term of order e −δ/ε for some δ > 0.…”

“…In general, the above integrations by parts do not make sense for all t ∈ [0, T ], since we consider weak solutions only. Note however that for σ 2 and n 3, the analysis in [1] shows that we can work with strong solutions, so the following analysis is not needed in this case (for σ = 1 and n 1, the same holds true, from [21]). Also, if one is just interested in proving Corollary 1.6 by contradiction, no further justification is needed for integrations by parts, and one can skip the end of this section.…”

“…It followed from a supercritical WKB analysis for the cubic nonlinear Schrödinger equation, which had been justified by E. Grenier [21]. For σ 2, adapting the results of [21] seems to be a much more delicate issue, and a rigorous analysis in this setting for n 3 has been given very recently [1]. An important remark, in the proof of Theorem 1.1 that we present here, is that it is not necessary to justify WKB analysis as precisely as in [21], or [1], to obtain this loss of regularity.…”

Geometric Optics and Boundary Layers for Nonlinear-Schrödinger Equations

“…Nevertheless, it does not give precise qualitative information on the solution of (1), for example, it does not allow to prove that the solution remains smooth on an interval of time independent of ε if the initial data are smooth or to justify WKB expansion up to arbitrary orders in smooth norms. In the work [2], the possibility of getting the same result as in [9] for pure power nonlinearities f (ρ) = ρ σ in the case Ω = R d was studied. It was first noticed that, thanks to the result of [15], the system    ∂ t a + ∇ϕ · ∇a + a 2 ∆ϕ = 0…”