The derivative nonlinear Schrödinger equation on the half line

Abstract

We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on R and prove smoothing estimates by combining the restricted norm method with a normal form transformation.

Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation

“…where G(u), p i and q i are defined in ( 8)-( 10). To bound the first component of Φ, we use (11) to obtain…”

“…Nevertheless, it is not a priori clear if different extensions of initial data produce the same solution on R + . We state an extension argument in [11]:…”

“…Furthermore, we estimate the integral equation by Fourier restriction norm method (also known as X s,b method) to close the fixed point argument. The tools have been applied to different dispersive equations on the half line, such as the cubic NLS, the derivative NLS, the KdV equation and the "good" Boussinesq equation, see [2,8,11,12]. We are ready to state our main theorem of this paper.…”

Unconditional uniqueness for the Benjamin-Ono equation

“…Since then the unconditional well-posedness for NLS was further improved, see [16,23,24,36,41] and studied for various other nonlinear dispersive PDEs, see e.g. [3,63] for KdV, [40,45,47,41] for the modified KdV equation, [13,50] for the derivative NLS equation, and [35] for the periodic modified Benjamin-Ono equation.…”

Admissible Boundary Values for the Gerdjikov-Ivanov Equation with Asymptotically Time-Periodic Boundary Data

“…The initial boundary value problem for (1.5) in the quarter plane (x, t) ∈ R 2 | x ≥ 0, t ≥ 0 is overdetermined in the sense that the Dirichlet and Neumann boundary values at x = 0 cannot both be independently prescribed for a well-posed problem. Indeed, in [9] it was shown that the Dirichlet initial boundary value problem for (1.5) is locally well-posed in H s ([0, ∞)) for any s ∈ 1 2 , 5 2 , s = 3 2 , with given initial data q(x, 0) = g(x) and Dirichlet boundary data q(0, t) = h(t). In particular, for any g ∈ H s ([0, ∞)) and h ∈ H…”