The nonlinear Schrödinger equation with $t$-periodic data: I. Exact results

Abstract

We consider the nonlinear Schrödinger equation on the half-line with a given Dirichlet (Neumann) boundary datum which for large $t$ tends to the periodic function $g_0^b(t)$ ($g_1^b(t)$). Assuming that the unknown Neumann (Dirichlet) boundary value tends for large $t$ to a periodic function $g_1^b(t)$ ($g_0^b(t)$), we derive an easily verifiable condition that the functions $g_0^b(t)$ and $g_1^b(t)$ must satisfy. Furthermore, we introduce two different methods, one based on the formulation of a Riemann-Hilbert problem, and one based on a perturbative approach, for constructing $g_1^b(t)$ ($g_0^b(t)$) in terms of $g_0^b(t)$ ($g_1^b(t)$).

23 pages

References in corpus (2)

• The Unified Method: I Non-Linearizable Problems on the Half-Line
• The Unified Method: II NLS on the Half-Line with $t$-Periodic Boundary Conditions

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• The nonlinear Schrödinger equation with $t$-periodic data: II. Perturbative results
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• The Robin problem for the nonlinear Schrödinger equation on the half-line
• The Ablowitz-Ladik system on a finite set of integers
• Long-time behavior for the Cauchy problem of the 3-component Manakov system