Abstract:

Soliton resonance and soliton molecules play a significant role in the study of optics, physics, and fluid mechanics. This paper systematically investigates the soliton resonance and soliton molecule properties of the Hirota equation in the context of nonlinear optical systems. First, based on the Lax pair, the $N$-fold Darboux transformation of the Hirota equation is constructed, from which the exact analytical expressions of the $N$-soliton solutions are derived. Subsequently, by adjusting the spectral parameters and initial phase parameters, novel soliton resonance and soliton molecule solutions of the Hirota equation are constructed, and their dynamical characteristics are analyzed. The research results indicate that localized soliton resonance can evolve into soliton molecules, with soliton molecules being the limiting form of localized soliton resonance. Furthermore, the study reveals that the energy carried by solitons exhibits a growth trend during the formation of soliton resonance and soliton molecules. It also demonstrates that the spatiotemporal evolution patterns of soliton molecules become increasingly complex as the soliton order increases. Meanwhile, the wave density of soliton molecules is positively correlated with the coefficients of higher-order dispersion and higher-order nonlinear terms. These findings provide new theoretical insights into the soliton dynamics of the Hirota equation and contribute to a better understanding of soliton interaction mechanisms in nonlinear optical systems.

Key words: Hirota equation, Darboux transformation, soliton resonances, soliton molecules