The dynamic analysis of the complex negative-order Ablowitz–Kaup–Newell–Segur equation using the localised wave structures

A complex negative-order Ablowitz–Kaup–Newell–Segur (AKNS) equation, which has potential utility in characterising the propagation of femtosecond pulses through erbium-doped fibre, is studied in this paper. The complex negative order AKNS equation’s \((n, N-n)\)-fold extended Darboux transformation (DT) is established by considering its recognised \(2\times 2\) matrix Lax pair. The equation’s soliton and breather solutions can be obtained by utilising the resulting DT. By asymptotic analysis, we are able to illustrate and discuss the soliton’s elastic interaction graphically and to analyse some significant dynamic features, such as wave velocity, amplitude and energy. Meanwhile we also study the rogue wave and mixed breather-rogue wave interaction solutions. Finally, the numerical simulations of one- and two-soliton solutions are performed to analyse their dynamical structures. The implications and characteristics discussed in this work can be useful in understanding the propagation of optical pulses.