**5. Conclusions**

In this paper, we mainly studied the initial-boundary values problem for the mNLS equation on the half line. Before we did this, with the help of prolongation structure theory, the Lax pair of this equation was derived. Then we reconstructed the Lax pair to obtain a Riemann-Hilbert problem, and therefore, the potential function has been represented by its solution. In future work, the long time asymptotic behavior for the solutions will be analyzed.

**Author Contributions:** All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

**Funding:** This work was supported by Nature Science Foundation of Shandong Province of China (No.ZR2018MA017), China Postdoctoral Science Foundation funded project (No.2017M610436), Open Fund of the Key Laboratory of Meteorological Disaster of Ministry of Education (Nanjing University of Information Science and Technology) (No.KLME201801).

**Conflicts of Interest:** The authors declare no conflict of interest.
