Rogue waves are giant single waves observed in the ocean. Their average height is at least twice the height of the surrounding waves. One encounters similar phenomena in propagation of pulses in optical fibres and waves in Bose-Einstein condensate, in superfluids, in optical cavities, in the atmosphere and even in finance. It is known that one-dimensional motion of the envelope of these type of waves is fairly adequately described by the focusing nonlinear Schrödinger equation. The authors try to find the oscillatory solutions of this equations in terms of rational functions by employing a modified Darboux procedure. This procedure leads to a hierarchical system of equations of which the authors succeed to give only the first two members. However, for \(t = 0\) they give recurrence relations to determine the sequence of rational functions on x-axis through which it is relatively easy to find the polynomials in the numerators and denominators. These rational functions have no poles on the real axis. However, they possess complex poles and these poles are employed to evaluate some integrals by using the residues.