Abstract
In this work, we study a generalised high-order nonlinear Schrödinger equation with time-dependent coefficients, embracing a wide range of physical influences. By employing the Darboux transformation, we construct explicit breather and rogue wave solutions, illustrating how the spectral parameter governs waveform transitions. In these dynamics, dispersion determines stability and symmetry, nonlinearity influences the peak amplitude and width, and third-order dispersion introduces asymmetry and drift in the wave profile. We have demonstrated that stabilization, destabilization and shifting of the centre of the localization, or drifting towards the soliton in space or even temporal directions, can be possible by manoeuvring the spectral parameter relating dispersion and nonlinearity in optical fibre. Manoeuvring the spectral parameter relates the dispersion a1(t) and nonlinearity from 100 t to 0.1 t leads to the stabilization of the soliton by a notable decrease in the amplitude for two hundred folds. The results reveal that the inclusion of higher-order term functions as a control mechanism for managing instability and localisation in nonlinear optical fibre systems, offering promising prospects for future developments in nonlinear optics.