Geodesic
The boundary value problem for the nonlinear Schr\"odinger equation
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 8, Tome 169 (1988), pp. 151-165

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A boundary value problem on the half-line for the nonlinear Schrödinger equation (NLS) and its generalisation is studied by the inverse scattering transform method. The connection between conservation laws and boundary conditions for integrable boundary value problems for higher NLS equations is established. It's shown that the generalized boundary value problem requires a joint analysis of regular and singular solutions for the NLS equations in the repulsive case. 
@article{ZNSL_1988_169_a15,
     author = {V. O. Tarasov},
     title = {The boundary value problem for the nonlinear {Schr\"odinger} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {151--165},
     publisher = {mathdoc},
     volume = {169},
     year = {1988},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1988_169_a15/}
}
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V. O. Tarasov. The boundary value problem for the nonlinear Schr\"odinger equation. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 8, Tome 169 (1988), pp. 151-165. http://geodesic.mathdoc.fr/item/ZNSL_1988_169_a15/