Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation with a One-Sided Non-Zero Boundary Condition

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrodinger (NLS) equation with one-sided non-zero boundary value as x → +∞ is presented. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that [q(x, t) − qr(t)ϑ(x)] ∈ L1,1(R) [here and in the following ϑ(x) denotes the Heaviside function] with respect to x ∈ R for all t ≥ 0, for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations and as a Riemann-Hilbert problem on a single sheet of the scattering variables $λ_r = sqrt{k^2 + A^2_r}$, where k is the usual complex scattering parameter in the IST. The direct and inverse problems are also formulated in terms of a suitable uniformization variable that maps the two-sheeted Riemann surface for k into a single copy of the complex plane. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with the same amplitude as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of physically relevant NLS solutions with nontrivial boundary conditions, either via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.