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

Abstract

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.

Keywords:

• Second-order flow
• three-wave hierarchy
• prolongation technique
• Riemann–Hilbert method
• N-soliton solution

AMS Subject Classifications:

• 35Q51
• 37K40

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11971067 and 12071042, Beijing Natural Science Foundation under Grant Nos. 1202006 and 1182009, and the Fundamental Research Funds for the Central Universities under Grant No. 2020NTST22.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

All data included in this study are available upon request by contact with the corresponding author.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11971067] and Natural Science Foundation of Beijing Municipality [1202006 and 1182009].