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| We present the inverse scattering transform (IST) for the focusing nonlinear Schr\"odinger equation: $iq_t=q_{xx}+ 2|q|^2q$, with non-zero boundary conditions $q(x,t)\sim q_{l/r}(t)=A_{/r}e^{i\theta_{l/r}(t)}$ as $x\rightarrow \mp \infty$ in the fully asymmetric case.
The direct problem is shown to be well-posed for NLS solutions $q(x,t)$ such that $q(x,t)-q_{l/r}(t)\in L^{1,1}(\mathbb{R^\mp})$ with respect to $x$ for all $t\ge 0$, 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 $\lambda_{l/r}=\sqrt{k^2+A^2_{l/r}}$, 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 the same amplitude as $x\rightarrow \pm \infty$, here both reflection and transmission coefficients have a nontrivial time dependence. |
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