Eigenvalues of the Zakharov-Shabat scattering problem for real symmetric pulses

The classical problem of determining the solitons generated from symmetric real initial conditions in the nonlinear Schrödinger equation is revisited. The corresponding Zakharov-Shabat scattering problem is solved for real and symmetric double-humped rectangular initial pulse forms. It is found that such real symmetric pulses may generate eigenvalues with nonzero real parts corresponding to separating soliton pulse pairs. Moreover, it is found that the classical formula relating the number of eigenvalues to the area of the pulse is not always correct.