Imaginary eigenvalues of Zakharov–Shabat problems with non-zero background

Highlights: • The focusing Zakharov–Shabat problem with non-zero boundary conditions is studied. • Sufficient conditions are identified which ensure that the scattering problem admits purely imaginary eigenvalues. • The results are illustrated with several examples. - Abstract: The focusing Zakharov–Shabat scattering problem on the infinite line with non-zero boundary conditions for the potential is studied, and sufficient conditions on the potential are identified to ensure that the problem admits only purely imaginary discrete eigenvalues. The results, which generalize previous work by Klaus and Shaw, are applicable to the study of solutions of the focusing nonlinear Schrödinger equation with non-zero background.