Self-similar solutions to the derivative nonlinear Schrödinger equation

Abstract

A class of self-similar solutions to the derivative nonlinear Schrödinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions.2000 Mathematics Subject Classification. 35Q55.

Global well-posedness for the derivative nonlinear Schrödinger equation in $L^2(\mathbb{R})$

“…Long-standing physical intuition dictates that dispersive equations will be illposed below the scaling-critical regularity. For the case of (DNLS), this is justified by the self-similar solutions constructed in [10,31]. Beyond proving that wellposedness fails in H s (R) with s < 0, these solutions even show that it fails in weak-L 2 , which is a scale-invariant space!…”

“…ii) Sextic paraproducts. If m ∈ S loc (6) then we have the estimates (8) then we have the estimates (10) then we have the estimates…”

“…where m ∈ S loc (10), to which we apply (6.12). Next, we use (3.48) to write √ κ g21 2+γ [7] (κ) + √ −κ g21 2+γ [7] (−κ) q ψ 24 µ dx…”

“…where m ∈ S loc (10), which can be bounded using (6.13). Further, from (3.46) we have √ κ g21 2+γ [3]…”

On the well-posedness problem for the derivative nonlinear Schrödinger equation

“…Moreover, we do not know of any results (in either geometry) that would preclude well-posedness all the way down to the scaling critical space L 2 . On the other hand, the self-similar solutions constructed in [8] (see also [26]) show that smooth solutions can break-down in a dramatic way if one permits mere weak-L 2 decay at spatial infinity.…”