The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary values ql/r(t)=Al/re2iAl/r2t+il/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl θ r. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that (q(x,t)-q_{l/r}(t) L1,1(R±) with respect to x for all t ≥ 0, and the corresponding 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 _{l/r}=\sqrt{k2+A2 t/r where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes 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 fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.