Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication

Under investigation in this paper is a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication. Lax pair, one/N-fold (N=2,3,…) binary Darboux transformations and the limit forms of the one-fold binary Darboux transformation for such an equation are obtained. Based on one/N-fold binary Darboux transformations and the limit forms of the one-fold binary Darboux transformation, one- and N-dark soliton solutions are obtained. Soliton amplitude is not affected by the coefficients of the nonlinear-Schrödinger operator, Hirota operator, Lakshmanan-Porsezian-Daniel operator and quintic operator, but soliton velocity is linearly related to them. Overtaking interaction between the two solitons is illustrated. Parallel solitons are formed: soliton width and interval between the parallel solitons are decreased with the increasing value of each and every coefficient of the nonlinear-Schrödinger operator, Hirota operator, Lakshmanan-Porsezian-Daniel operator and quintic operator respectively. For the interactions among the three solitons, e.g. interaction among three overtaking solitons, and interaction between the parallel solitons and a single soliton are displayed. We find that the interactions between the two solitons and among the three solitons are elastic.