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This paper extends the second-order three-wave flow of Fordy and Kulish to propose a generalized second-order flow of the three-wave hierarchy. Then the Lax pair of the generalized second-order three-wave flow is constructed by the standard prolongation technique, and the infinite conservation laws for this flow are derived. Finally, the Riemann–Hilbert approach is applied to the initial value problem of this flow, and the  N -soliton solutions to the second-order three-wave flow are derived explicitly. Collision behaviors of the two and three solitons are demonstrated graphically, which shows that our results have potential applications to the resonant three-wave interactions in nonlinear media.

The Riemann–Hilbert approach to the generalized second-order flow of three-wave hierarchy