Bound-state solitons in three-wave resonant interactions

Three-wave resonant interactions are commonly observed in the dispersive and weakly nonlinear media in fluid mechanics, plasma physics, nonlinear optics, solid-state physics, Bose-Einstein condensates and acoustics. This paper investigates a three-wave resonant interaction system in the one-dimensional space-time form. For that system of the soliton-exchange case, a Lax pair is introduced. Then, we utilize the generalized Darboux transformation method to derive the semi-rational solutions depicting the bound-state dark-bright mixed solitons. The three wave packets can manifest themselves as the bound-state bright-dark-bright solitons on the zero-nonzero-zero background, and bound-state dark-bright-bright solitons on the nonzero-zero-zero background. The bound states between the bright/dark solitons exhibit periodic attractions and repulsions, while the bound states among the bright/dark solitons and multi-pole bright/dark solitons exhibit non-periodic attractions and repulsions. We explore how the phase shift of the single bright/dark soliton component affects the bound states among a single bright/dark soliton and the double-pole bright/dark solitons. All the results obtained for the one-dimensional space-time evolution can be reinterpreted in terms of the two-dimensional steady-state interaction. This work may provide explanations for the complex and variable natural mechanisms of three-wave resonant interactions in various physical contexts.