**1. Introduction**

In mathematics and physics, nonlinear partial differential equations play an important role due to their abundant mathematical structure and properties. Many works on nonlinear evolution equations have been studied, such as the Hamiltonian structure [1,2], the infinite conservation laws [3,4], the Bäcklund transformation [5,6] and so on [7–9]. Besides, the exact solution of these equations, which can be expressed in various forms by different methods, is also a significant subject of soliton research [10–22]. In recent years, with the development of soliton theory, more and more researchers pay attention to the Riemann-Hilbert approach. The Riemann-Hilbert approach was introduced by Fokas to analyze the initial-boundary values problem for linear and nonlinear partial differential equations [23,24]. In the past 20 years, many researchers have discussed a lot of nonlinear integrable equations for the initial-boundary values problem [25–32,32–41]. They have all made a grea<sup>t</sup> contribution to the development of this method. The core idea of this method is to construct the associated Riemann-Hilbert problem by the Lax pair of the integrable equation, and then in addition to the initial-boundary values problem, the long-time asymptotic behavior of the solution can be analyzed [42–46]. However, as we all know, it is difficult to determine whether a nonlinear evolution equation possesses a Lax pair or not. As far as we are concerned, the prolongation structure method is an efficient way to obtain the Lax pair, which was firstly proposed in 1975 by Wahlquist and Estabrook [47]. In recent years, a large number of scholars have improved this method, for example, Hermann deduced the prolongation structure method connection in 1976 [48], Deconinck applied the prolongation structure method to semi-discrete systems firstly [49], Wang used this approach to ge<sup>t</sup> the integrability of many nonlinear wave equation [50] and so on [51,52]. In this way, we can ge<sup>t</sup> the

Lax pair of the nonlinear evolution equation easily as long as it is integrable.

In this paper, we mainly talk about the modified nonlinear Schrödinger(mNLS) equation

$$iq\_t + q\_{xx} + i(|q|^2 q)\_x + 2\rho |q|^2 q = 0,\tag{1}$$

which is very important in plasma physics. Recently, many properties of this equation have been studied, such as the Hamiltonian structure [53], the Darboux transformation [54], the numerical solutions [55,56] and so on [57,58]. Actually, it can become the derivative NLS equation by certain gauge transformation [59]. In this paper, we mainly discuss the mNLS equation on the half line. For simplicity, we let *ρ* = 1. Supposing that the solution *q*(*<sup>x</sup>*, *t*) of the mNLS equation exists, and the initial-boundary values are defined as follows,

Initial values:

$$q\_0(\mathbf{x}) = q(\mathbf{x}, \mathbf{0}), \mathbf{0} < \mathbf{x} < \infty,\tag{2}$$

Boundary values:

$$q\_{\mathcal{S}}(t) = q(0, t), \mathcal{g}\_1(t) = q\_{\mathcal{X}}(0, t), 0 < t < T. \tag{3}$$

In order to formulate a Riemann-Hilbert problem, we need to reconstruct the Lax pair of Equation (1). Based on the initial-boundary values, the corresponding spectral functions can be defined. Eventually, the potential function *q*(*<sup>x</sup>*, *t*) can be expressed in terms of the solution of this Riemann-Hilbert problem.

This paper is divided into four sections. The construction of the prolongation structure for the mNLS equation is in Section 2 and then in Section 3, we reconstruct the Lax pair to formulate the Riemann-Hilbert problem and some conditions and relations are derived. In the last section, we define the spectral functions according to the initial-boundary values and the Riemann-Hilbert problem is investigated.

### **2. The Prolongation Structures of the mNLS Equation**

In order to obtain the Lax pair of the mNLS equation, we analyze the prolongation structure of this equation. This process mainly involves a fundamental theorem in Lie algebra [51].

**Theorem 1.** *Suppose X and Y are two elements of Lie algebra g* = *sl*(*n* + 1, *C*) *with* [*<sup>X</sup>*,*<sup>Y</sup>*] = *aY,* (*a* = 0) *and X* ∈ *range ad Y, it means that there exist Z* ∈ *g such that* [*<sup>Y</sup>*, *Z*] = *X, so we obtain Y* = *e*± *and X* = ±12 *ah, where e*± *are the nilpotent and h is the neutral elements of g.*

In the beginning, we introduce these variables

$$
\vec{u} = p, \boldsymbol{u}\_{\boldsymbol{x}} = \boldsymbol{v}, \vec{u}\_{\boldsymbol{x}} = p\_{\boldsymbol{x}} = q. \tag{4}
$$

Then Equation (1) is equivalent to this set of equations as follows

$$\begin{cases} u\_x - v = 0, \\ p\_x - q = 0, \\ iu\_t + v\_x + 2iu u\_x \vec{u} + iu^2 \vec{u}\_x + 2u^2 \vec{u} = 0, \\ ip\_t - q\_x + 2i\vec{u}\_x \vec{u} + i\vec{u}^2 u\_x - 2\vec{u}^2 \vec{u} = 0. \end{cases} \tag{5}$$

We define the set of two-forms *I* = {*<sup>α</sup>*1, *α*2, *α*3, *<sup>α</sup>*4},where

$$\begin{cases} \mathbf{a}\_1 = du \wedge dt + vdt \wedge dx, \\ \mathbf{a}\_2 = dp \wedge dt + qdt \wedge dx, \\ \mathbf{a}\_3 = idu \wedge d\mathbf{x} - dv \wedge dt + (2iuvp + iu^2q + 2u^2p)dt \wedge dx, \\ \mathbf{a}\_4 = idp \wedge d\mathbf{x} + dq \wedge dt + (ip^2v + 2ipqu - 2p^2u)dt \wedge dx. \end{cases} \tag{6}$$

It is easy to find that *I* is a closed ideal, actually, *dI* ⊂ *I*. After that, we define the differential one-forms

$$
\omega^i = dy^i - F^i(u, v, p, q; y^i)dx - G^i(u, v, p, q; y^i)dt. \tag{7}
$$

At the same time, we suppose *Fi* = *Fij yj*, *Gi* = *Fij yj*. According to the general theory of exterior differential systems, if ˜ *I* = *I* + *ω<sup>i</sup>* is a closed ideal, it must satisfy

$$d\omega^i = \sum\_{i=1}^4 (f^i\_{\slash} a^j) + n^i\_{\slash} \wedge \omega^j. \tag{8}$$

Combining (5)–(8), we obtain

$$\begin{cases} F\_v = F\_q = 0, \\ iG\_v + F\_u = 0, \\ iG\_q - F\_p = 0, \\ -G\_u v - G\_p q + (2iuvp + iqu^2 + 2u^2p)G\_v \\ -(2ipqu + ip^2v - 2p^2u)G\_q + [F, G] = 0. \end{cases} \tag{9}$$

where the bracket [, ] denotes the Lie bracket, namely [*F*, *G*] = *FG* − *GF*.

After a lengthy calculation, one solution of this set of equations can be derived

$$\begin{aligned} F &= \mathbf{x}\_0 + \mathbf{u}\mathbf{x}\_1 + p\mathbf{x}\_2, \\ G &= \mathbf{i}\mathbf{x}\_1\mathbf{v} - \mathbf{i}\mathbf{x}\_2q - \mathbf{u}^2p\mathbf{x}\_1 - p^2\mathbf{u}\mathbf{x}\_2 + \mathbf{i}\mathbf{u}\mathbf{x}\_3 - ip\mathbf{x}\_4 - ip\mathbf{u}\mathbf{x}\_5 + \mathbf{x}\_6. \end{aligned} \tag{10}$$

with the integrability conditions

$$\begin{aligned} 2i\mathbf{x}\_1 - \mathbf{x}\_3 - i[\mathbf{x}\_1, \mathbf{x}\_5] &= 0, 2i\mathbf{x}\_2 + \mathbf{x}\_4 + i[\mathbf{x}\_2, \mathbf{x}\_5] = 0, \\ i[\mathbf{x}\_0, \mathbf{x}\_3] + [\mathbf{x}\_1, \mathbf{x}\_6] &= 0, -i[\mathbf{x}\_0, \mathbf{x}\_4] + [\mathbf{x}\_2, \mathbf{x}\_6] = 0, \\ i[\mathbf{x}\_0, \mathbf{x}\_5] + [\mathbf{x}\_1, \mathbf{x}\_4] - [\mathbf{x}\_2, \mathbf{x}\_3] &= 0, [\mathbf{x}\_1, \mathbf{x}\_3] = 0, [\mathbf{x}\_2, \mathbf{x}\_4] = 0, [\mathbf{x}\_0, \mathbf{x}\_6] = 0. \end{aligned} \tag{11}$$

where all {*xi*}, *i* = {1, 2, ..., 6.} are pending matrices. Here {*<sup>x</sup>*1, *x*2, ..., *<sup>x</sup>*6} depend on an incomplete Lie algebra, called prolongation algebra.

The next step is to embed the prolongation algebra in *sl*(*n* +1, *<sup>C</sup>*). According to (11) and Theorem 1, we deduce that *x*1 and *x*2 is nilpotent and *x*5 is neutral element. So we have

$$\mathbf{x}\_1 = \begin{pmatrix} 0 & \xi^x \\ 0 & 0 \end{pmatrix}, \mathbf{x}\_2 = \begin{pmatrix} 0 & 0 \\ -\xi & 0 \end{pmatrix}, \mathbf{x}\_5 = \begin{pmatrix} -\xi^2 & 0 \\ 0 & \xi^2 \end{pmatrix}. \tag{12}$$

Bringing the above results into (11), we obtain

$$\begin{aligned} \mathbf{x}\_0 &= \begin{pmatrix} -i\xi^2 + i & 0 \\ 0 & i\xi^2 - i \end{pmatrix}, \mathbf{x}\_3 = \begin{pmatrix} 0 & -2i\xi^3 + 2i\xi^2 \\ 0 & 0 \end{pmatrix}, \\ \mathbf{x}\_4 &= \begin{pmatrix} 0 & 0 \\ -2i\xi^3 + 2i\xi & 0 \end{pmatrix}, \mathbf{x}\_6 = \begin{pmatrix} -2i\xi^4 - 2i + 4i\xi^2 & 0 \\ 0 & 2i\xi^4 + 2i + 4i\xi^2 \end{pmatrix}. \end{aligned} \tag{13}$$

where *ξ* is spectral parameter. Hence, the expressions of *F* and *G* can be presented eventually

$$\begin{aligned} F &= \begin{pmatrix} -i\xi^2 + i & \xi q \\ -\xi \bar{q} & i\xi^2 - i \end{pmatrix}, \\ G &= \begin{pmatrix} -2i\xi^4 - 2i + 4i\xi^2 + i\xi^2|q|^2 & 2\xi^3 q - 2\xi q - \xi|q|^2 q + i\xi q\_\mathbb{X} \\ -2\xi^3 \bar{q} + 2\xi \bar{q} + \xi|q|^2 \bar{q} + i\xi \bar{q}\_\mathbb{X} & 2i\xi^4 + 2i - 4i\xi^2 - i\xi^2|q|^2 \end{pmatrix}. \end{aligned} \tag{14}$$

So, the mNLS equation admits Lax pair

$$
\psi\_{\mathbf{x}} = F\psi, \psi\_{\mathbf{t}} = G\psi,\tag{15}
$$

where *ψ* = (*<sup>v</sup>*1, *<sup>v</sup>*2)*<sup>T</sup>*.
