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 $L^2$-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction $M(q)=\int |q{|}^2<4\pi$. 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.

References in corpus (6)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• $H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation
• Global well-posedness for the derivative nonlinear Schrödinger equation
• A priori estimates for the derivative nonlinear Schrödinger equation
• Global well-posedness for the fifth-order KdV equation in $H^{-1}(\mathbb{R})$
• Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

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• Well-posedness for the NLS hierarchy
• Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation
• Nowhere continuity of the flow map of an integrable derivative nonlinear Schrödinger system on the torus
• Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations