Multi-soliton solutions of the N-component nonlinear Schrödinger equations via Riemann–Hilbert approach

In this paper, we utilize the Riemann–Hilbert approach to discuss multi-soliton solutions of the N-component nonlinear Schrödinger equations. Firstly, by transformed Lax pair, we construct the matrix-valued functions P1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1,2}$$\end{document} that satisfy the analyticity and normalization and the corresponding jump matrix can be determined. Then, in the reflectionless case, we get the multi-soliton solutions ql\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{l}$$\end{document}(l=1,…,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(l=1,\ldots ,N)$$\end{document} of the N-component nonlinear Schrödinger equations, which are related to the spectral parameter η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}. Particularly, the 2-soliton solutions q1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{1}$$\end{document}, q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{2}$$\end{document}, and q3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{3}$$\end{document} of the three-component nonlinear Schrödinger equations are given and the corresponding 2-soliton diagrams are drawn.


Introduction
Riemann-Hilbert (RH) problem is the 21st question that Hilbert mentioned at the International Congress of Mathematicians in Paris [1]. It belongs to the scope of boundary value problem of matrix-valued functions on the complex plane. Usually, the RH problem is defined as follows [2]. Assume that is a directed path on the complex plane C, 0 = \{self-intersection of }. Suppose that there is a smooth map on 0 G(z) : 0 → M L(n, C).
In fact, the RH problem is a boundary value problem of matrix value function on complex plane, and we can convert it into integral equations. Because RH problem is a problem on complex plane, the biggest advantage of RH approach is to transform the problem solvable on complex plane. For example, some definite integrals are difficult to integrate in the real domain, but can be solved when treated as complex integrals.
It is known that the methods by which the solutions of the integral equations can be obtained include inverse scattering transformation [3], Darboux transformation [4], symmetry reduction method [5], Hirota bilinear method [6], Lax pair nonlinear method [7], Wronskian technique [8], and so on. RH approach is a direct and simple method to solve the soliton equations. The problem can be solved by Lax pair of integrable systems and analysis of the spectral function. One can construct the RH problem similar to the above description, and then get the soliton solutions of the original equations. Now RH approach has been developed into a powerful analytical tool to solve problems in a large class of pure and applied mathematics, which can be widely applied to initial boundary value problem [9][10][11][12][13][14][15][16], asymptotic of orthogonal polynomials [17], Bäcklund transformation [18,19], and long-time asymptotics [20][21][22]. Afterward, It is found that the RH approach can be used to obtain the solutions of integral equations by inverse scattering theory [23][24][25][26][27][28][29][30]. In recent years, Wazwaz solved multiple soliton solutions of the equations [31][32][33][34]. Then, RH approach has been widely used to get multi-soliton solutions of multidimensional equations [35][36][37][38][39].
In this paper, we mainly discuss multi-soliton solutions of the N -component nonlinear Schrödinger (NLS) equations by RH approach. The N -component NLS equations [40] take the form where q l = q l (t, x) (l = 1, . . . , N ) are functions and the subscripts mean the partial derivatives. The organization of this paper is given as follows. In Sect. 2, we transform Lax pair to construct the RH problem for the N -component NLS equations. In Sect. 3, the multi-soliton solutions for the N -component NLS equations are obtained, which are relevant to the spectral parameter. Then, the 2-soliton solutions of the three-component NLS equations are given and the corresponding 2-soliton graphs are drawn. In Sect. 4, we give the conclusion.

Riemann-Hilbert problem
Based on Eq. (1.1), we have the Lax pair η is the spectral parameter. Then, we get the Jost solution of Lax pair (2.1) with asymptotic form by In order to facilitate calculation, we define a matrix function = (x, t; η). Letting The Lax pair (2.1) can be rewritten as . Then, the Volterra integral equations can be expressed as By calculation, we can know (2.8) . Then, it can be obtained are analytic in C − . We can rewrite 1,2 as follows Based on the properties of 1,2 and tr Q=0, we can know that det 1,2 are independent for all x. By the asymptotic conditions 1,2 → I as |x| → ∞, we know that det 1,2 = 1.
(2.11) Therefore, 1,2 are linearly related by a spectral matrix S(η) = (s k j (η)) (N +1)×(N +1) , which can be expressed as Taking the inverse of both sides of Eq. (2.12), we can obtain −1 (2.14) Applying Eq. (2.14) and the analytic properties of column vectors of 1,2 , we can get the analytic properties of −1 1,2 , that is In order to construct a matrix RH problem, we need to determine two matrix functions Then, P 1,2 can be rewritten as . We can study the asymptotic expansion of P 1 Submitting (2.20) into the first equation of (2.6) and comparing the corresponding coefficients of η, we obtain

Conclusion
In general, we investigate the multi-soliton solutions of the N -component NLS equations via RH approach. By the Volterra equations, the corresponding analytical properties can be obtained. Then, we define P 1 and P 2 to construct the RH problem. In reflectionless case, making full use of the symmetric relation of the potential matrix and giving the zero point relation of the determinant of two analytic matrix functions in the problem, one can construct the multi-soliton solutions. For the multi-component NLS equations, it is more complicated than standard NLS equations. The multisoliton solutions of N-dimensional NLS equations via the RH approach have not been well studied before, and therefore we take the factor of multi-component into consideration.
RH approach can be used not only to solve the initial boundary value problem of integrable systems, but also to analyze the solution of long-time behavior, quantum field theory and statistical model, orthogonal polynomial theory, and random matrix theory. In addition, it can be used in plasma physics, ocean engineering, atmospheric sciences, Bose-Einstein condensate, nonlinear optics and so on. The Riemann-Hilbert approach can be applied to many equations of similar types. In future, we can innovate and improve the method and apply it to many more complex equations.