The eigenvalue problem for the focusing nonlinear Schrodinger equation: new solvable cases

In this paper, we study the semi-classical limit of the Zakharov-Shabat eig envalue problem for the focusing of NLS with some specific initial data. In all these cases, the eigenvalue problem is reduced to connection problems for the hypergeometric equation and for other classical equations. The spec ial initial data [Suppl. Frog. Theor. Phys, 55 (1974) 284] is contained in our family of initial data, parameterized by a real parameter mu, as a part icular case mu = 0. We find that beyond a certain value of the parameter mu , the pure-point spectrum becomes empty and all the scattering information is contained in the reflection coefficient. (C) 2000 Published by Elsevier Science B.V.