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

Abstract

We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L 2 -critical with respect to scaling.The first question we discuss is whether ensembles of orbits with L 2equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction M (q) = |q| 2 < 4π. We conjecture that this restriction is unnecessary.Further, we prove that the problem is globally well-posed for initial data in H 1/6 under the same restriction on M . Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.

Large-data equicontinuity for the derivative NLS

“…In addition to pinpointing Theorem 1.2 as an important question in the theory of (DNLS), the paper [20] also proved it for sets Q with sup{M (q) : q ∈ Q} < 4π. This is precisely the threshold delineated by the algebraic soliton that we discussed earlier.…”

“…Our goal in this paper is to prove that the flow map for (DNLS) preserves L 2equicontinuity. This question was posed in [20], where it was shown to have robust consequences both for a priori bounds and for the well-posedness problem. To formulate matters precisely, we need one preliminary definition:…”

“…Theorem 1.2 is phrased in terms of Schwartz-class solutions, not only because we know that such solutions exist, but also because they provide an effective way to address data with less regularity or decay. Consider, for example, the following consequence of Theorem 1.2 and the results in [20] regarding a priori bounds for (arbitrarily large) solutions:…”

$H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation

“…Before stating our main result, let us give a very brief review of what is known about the wellposedness of the DNLS equation. More detailed overviews can be found, for example, in the introductions of [2] and [14]. Local wellposedness in H s (R) for s ≥ 1 2 was proven by Takaoka [25], improving earlier work [22] by Ozawa.…”

“…There have also been some recent works below the aforementioned s = 1 2 threshold of uniform H s continuity with respect to initial data [3,26]. Klaus-Schippa [17] gave H s a priori estimates for 0 < s < 1 2 in the case of small mass, Killip-Ntekoume-Vis ¸an [14] improved the small mass assumption to 4π and furthermore proved a global wellposedness result in H s (R), 1 6 ≤ s < 1 2 , for initial data with mass less than 4π. Very recently, Harrop-Griffiths, Killip, and Vis ¸an [9] have removed the small mass assumption both from their H s a priori bounds, 0 < s < 1 2 , as well as from their global wellposedness result in H s (R) with 1 6 ≤ s < 1 2 .…”