We have shown that it is possible to formulate a Bethe-Ansatz-type solution for a quantum mechanical nonlinear Schrödinger equation in an external electric field.
The authors continue their analysis of the extended derivative nonlinear Schrodinger equation with the help of the Bethe-ansatz technique. In the previous analysis by the QISM approach it was not possible to consider the reduction of the extended system to the derivative NLSE, as the theory then becomes non-ultralocal. Therefore, they have applied the approach of the Bethe ansatz to both the extended DNLSE and the usual DNLSE, and show that it is possible to construct multiparticle quantum states in both the cases.
We have formulated the quantum inverse problem for the unstable nonlinear Schr\"odinger equation that forms the basis of nonlinear optics, following the modified approach of Sklyanin. Due to the nonexistence of the vacuum for the corresponding Lax L operator, the algebraic Bethe ansatz cannot be formulated. This difficulty is removed by recourse to the functional Bethe ansatz. The whole analysis is performed by considering different boundary conditions at the two ends of the interval on the x axis. Both signs of the nonlinearity are treated, corresponding to ``bright'' and ``dark'' solitons.