**The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach**

### **Tongshuai Liu and Huanhe Dong \***

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao266590,China;Liutongshuaii@163.com

 **\*** Correspondence: mathsdong@126.com

Received: 4 January 2019; Accepted: 7 February 2019; Published: 13 February 2019

**Abstract:** In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential *q*(*<sup>x</sup>*, *t*) can be represented by the solution of this Riemann-Hilbert problem.

**Keywords:** prolongation structure; mNLS equation; Riemann-Hilbert problem; initial-boundary value problem
