The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data

Abstract

We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an inverse scattering transform formalism for solutions satisfying such step-like boundary conditions. Using this formalism, we prove that there exists a global solution of the corresponding Cauchy problem and establish a representation for this solution in terms of the solution of a Riemann-Hilbert problem. By performing a steepest descent analysis of this Riemann-Hilbert problem, we identify three asymptotic sectors in the half-plane $t \geq 0$ of the $xt$-plane and derive asymptotic formulas for the solution in each of these sectors. Finally, by restricting the constructed solutions to the half-line $x \geq 0$, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem.

Figures (9)

• Figure 1: The left, middle, and right asymptotic sectors in the $xt$-plane.
• Figure 2: Illustration of the asymptotic sectors $\mathcal{L}$, $\mathcal{M}$, and $\mathcal{R}$ in the case when $4\beta - 2\alpha < 0 < 4\beta+4\alpha$.
• Figure 3: Illustration of the asymptotic sectors $\mathcal{L}$, $\mathcal{M}$, and $\mathcal{R}$ in the case when $0 < 4\beta - 2\alpha < 4\beta+4\alpha$.
• Figure 4: Illustration of the asymptotic sectors $\mathcal{L}$, $\mathcal{M}$, and $\mathcal{R}$ in the case when $4\beta - 2\alpha < 4\beta+4\alpha < 0$.
• Figure 5: The zero set, branch cut (thick line between $E_1$ and $k_0$) and signature table of $\text{\upshape Im\,} g$; the vertical component of the zero set asymptotes to the line $\text{\upshape Re\,} k = -\xi/4$.

Theorems & Definitions (58)

• Theorem 2.1: Solution of the direct problem
• proof
• Remark
• Definition
• Theorem 2.2: Solution of the Cauchy problem with step-like initial data
• proof
• Theorem 2.3: Asymptotics in the middle sector
• proof
• Theorem 2.4: Asymptotics in the left sector
• proof