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In this paper, a new integrable nonlinear Schrödinger-type (NLST) equation is investigated by prolongation structures theory and Riemann-Hilbert (R-H) approach. Via prolongation structures theory, the Lax pair of the NLST equation, a

In recent decades, nonlinear partial differential equations (PDEs) play a significant role in mathematics and theoretical physics, which have attracted much attentions in soliton theory and integrable system [

We find that construction of the Lax pair is critical for solving the R-H problem. However, there is no unified way to build Lax pair so far. In 1975, Wahlquist and Estabrook [

In this paper, we get Lax pair of NLST equation [

This paper is structured as follows. In Section

In this part, we study prolongation structures of equation

Firstly, we introduce an important proposition in the representation theory of Lie algebra.

Let

Secondly, a new series of independent variables for (

Before we formulate a R-H problem, we provide a definition of Riemann-Hilbert problem.

Let the contour

The Lax pair of the NLST equation (

In this part, the inverse scattering transforms of the NLST equation (

We suppose that the potential

Let us now consider formulating a correlated R-H problem with variable

This implementation of the inverse scattering transform for the NLST is very similar to that for the cubic NLS [

Notice that

Moreover, by analyzing the properties of

The R-H problem requires (

In order to formulate the R-H problem of (

In this part, we derive

Let us assume that

A key step for solving soliton solutions is to calculate the potential matrix

Furthermore, from the scattering relationship (

Notice that zeros

Finally, from (

We set

Solution (

Modulus of single-soliton

One can see that the single-soliton solution (

We set

We choose

In this case, when

Collision modulus of two-soliton

We choose

In this case, we assume

Bound state modulus of two-soliton

In summary, we obtained the N-soliton solutions (

In this work, a new integrable NLST equation is investigated via prolongation structures theory and R-H approach. We apply the prolongation structures theory to the NLST equation; by discussing the continuation algebra of the NLST equation, a

Recently, we notice that there are many other approaches to obtain exact solutions in the field of integrable systems, like Hirota’s bilinear method [

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Shandong Provincial Natural Science Foundation (Grant No. ZR2019QD018), National Natural Science Foundation of China (Grant No. 61602188), and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant Nos. 2017RCJJ068 and 2017RCJJ069).