Inverse Scattering Transform for the Focusing Nonlinear Schrodinger 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 q(r)(t) =A(r)e(-2iA tau 2t+i theta r), A(r) >= 0, 0 <= theta(r) < 2 pi, as x -> +infinity is presented. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that [q(x,t) - qr(t)v(x)] is an element of L-1,L-1(R) [here and in the following v(x) denotes the Heaviside function] with respect to x is an element of 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 variable lambda(r) = root k(2) + A(r)(2), where k is the usual complex scattering parameter in the 1ST. 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 -> +/-infinity, 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.