Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space

Abstract

In this paper, we prove that there exists some small ε * > 0, such that the derivative nonlinear Schrödinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data u 0 ∈ H 1 (R) satisfies u 0 L 2 < √ 2π +ε * . This result shows us that there are no blow up solutions whose masses slightly exceed 2π, even if their energies are negative. This phenomenon is much different from the behavior of nonlinear Schrödinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on half-line R + , we show the blow-up for the solution with negative energy. 1 2 u(α 2 t, αx), α > 0. It has the same scaling invariance as the quintic nonlinear Schrödinger equation, i∂ t u + ∂ 2 x u + µ|u| 4 u = 0, t ∈ R, x ∈ R, and the quintic generalized Korteweg-de Vries equation, ∂ t u + ∂ 3 x u + µ∂ x (u 5 ) = 0, t ∈ R, x ∈ R.

Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition inH12

“…We also refer to [5] for a further numerical investigation of the structure of the singular profile near blowup times. Here, we also point out that negative energy solutions to DNLS on a bounded interval or on the half-line, with Dirichlet boundary conditions, blow up in finite time (see [42], [50]). …”

“…Thus, the question of whether the same is true for DNLS appears naturally (see also [15]). However, in a recent series of articles, Wu [50,51], and Guo and Wu [13] obtained global existence for DNLS above the mass threshold 2π. They showed how to incorporate the momentum P [u] in controlling theḢ 1 -norm of u.…”

Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

“…2. Prior to [15], in [40,41], Wu first obtained L 2 -norm threshold improvements for energy-space initial data.…”

Self-similar solutions to the derivative nonlinear Schrödinger equation

“…It is well known (see [7]) that DNLS (1) has a global solution in H 1 (R), provided that mass is less than 2π. The constant 2π is improved to 4π in [12,13]. A simple gauge transform u → v(t, x) = u(t, x) exp i 2…”

“…If one regards the nonlinearity of (7) as a perturbation, the decay rate of solutions may be kept but the phase of asymptotics may be modified. Therefore, from (12) and (13), the deformation from A NLS to B NLS arises naturally.…”