Riemann-Hilbert problem associated with an integrable generalized mixed nonlinear Schrödinger equation

ABSTRACT

The generalized mixed nonlinear Schrödinger (GMNLS) equation is a completely integrable model under certain parameter conditions, which has important applications in nonlinear optics, weak nonlinear dispersive water wave and quantum field theory. In this work, we use the unified transformation method to construct a Riemann-Hilbert problem and discuss the initial-boundary value problems of the GMNLS equation on the half-line. Furthermore, we also obtain that some spectral functions are not independent of each other, but meet a paramount global relation.

KEYWORDS:

• Lax pair
• Riemann-Hilbert problem
• generalized mixed nonlinear Schrödinger equation
• initial-boundary value problems
• unified transformation method

AMS SUBJECT CLASSIFICATIONS:

• 35G31
• 35Q15
• 37K10
• 45D05

Acknowledgments

The authors are grateful to thank Professor Ling Zhang and Doctor Pingping Fang for helpful discussions and thanks the referees for helpful suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (grant No. 12147115), the China Postdoctoral Science Foundation (grant No. 2022M712833), the Natural Science Foundation of Anhui Province (grant No. 2108085QA09), the Natural Science Foundation of Chuzhou University (grant No. 2022qd038), the University Natural Science Research Project of Anhui Province (Grant Nos. KJ2021A1094, KJ2021B03).