Optical soliton solutions and their nonlinear dynamics described by a novel time-varying spectral Hirota equation with variable coefficients

The celebrated Hirota equation has important applications in nonlinear optics and quantum physics. This paper proposes a new type of time-varying spectral Hirota (tvsH) equation with variable coefficients and constructs its soliton solutions by extending the Riemann–Hilbert (RH) method. Specifically, Lax representation of the tvsH equation is first provided, which consists of time-varying spectral matrices, and the corresponding RH problem is further established by analyzing the Lax pair. Then, by combining inverse scattering and the spatiotemporal evolution of matrix vector solutions, an explicit expression for n-soliton solution of the tvsH equation is constructed in the absence of reflection potential. In addition, the collision soliton dynamics of the tvsH equation under isospectral and non-isospectral conditions are analyzed by modulating the time-varying coefficients. Most importantly, through the dynamic analysis of single and double solitons in the tvsH equation, we discover some new waveforms that have never been reported before, such as oscillating parabolic solitons and amplitude parabolic variation solitons. These waveforms may have significant theoretical and practical implications.