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 Schr¨odinger (NLS) equation with non-zero boundary values $q_{l/r} (t) ≡ A_{l/r} e−2i A^2_{l/r} t+iθ_{l/r} $as x →∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with $A_l \ne A_r$ and $θ_l \ne θ_r$ . The direct problem is shown to be well-defined for NLS solutions q(x, t) such that$q(x, t) − q_{l/r} (t)∈ L^{1,1}(R^{∓})$ 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 $λ_{l/r} =\sqrt{k^2 + A^2_{l/r}$ , 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 longtime 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.