Riemann-Hilbert approach and $N$-soliton solutions for a new four-component nonlinear Schr\"odinger equation

A new four-component nonlinear Schr\"{o}dinger equation is first proposed in this work and studied by Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a $5\times5$ matrix spectral problem for the four-component nonlinear Schr\"{o}dinger equation. Then based on the Lax pair, we analyze the spectral problem and the analytical properties of the Jost functions, from which the Riemann-Hilbert problem of the equation is successfully established. Moreover, we obtain the $N$-soliton solutions of the equation by solving the Riemann-Hilbert problem without reflection. Finally, we derive two special cases of the solutions to the equation for $N=1$ and $N=2$, and the local structure and dynamic behavior of the one-and two-soliton solutions are analyzed graphically.


Introduction
The nonlinear Schrödinger equation (NLS) is an important integrable model. It is closely related to many nonlinear problems in theoretical physics such as nonlinear optics and ion acoustic waves of plasmas. Some higher-order coupled NLS equations are proposed, to describe more deep physical effects, including self-deepening, selffrequency shifting, and cubic-quintic nonlinearity. Among the different solutions of these models, soliton solutions play a crucial role in explaining some related complex nonlinear phenomena. With the development of nonlinear science, there are many ways to find solutions for nonlinear integrable models, including inverse scattering transform [1], Darboux transform [2], Hirota bilinear method [3], Lie group method [4], etc. Among them, inverse scattering transform method is one of the most effective tools for solving the initial value problem of nonlinear integrable systems to get the soliton solutions. For second-order spectral problems, inverse scattering theory is equivalent to Riemann-Hilbert (RH) approach, but for higher-order spectral problems the development of inverse scattering theory is not perfect, part of the inverse scattering problem needs to be transformed into RH problem. RH approach is developed by Zakharov et al [5], applied to integrable systems [6]- [31] as a more general method than inverse scattering method. This method has been successfully used to study the integrable system with single component. However, to the best of authors' knowledge, there are very few studies on the multi-component problems. The well-known general two-component coupled nonlinear Schrödinger equation of the form [32]        ip t + p xx + 2(a|p| 2 + c|q| 2 + bpq * + b * qp * )p = 0, iq t + q xx + 2(a|p| 2 + c|q| 2 + bpq * + b * qp * )q = 0, (1.1) where a and c are real constants, b is a complex constant, and " * " denotes complex conjugation. In physics, a and c describe the SPM and XPM effects, and b and b * describe the four-wave mixing effects.
2) has the following one-soliton solutions where n 1 and m 1 are arbitrary real numbers, α 1 , β 1 , τ 1 , ζ 1 and γ 1 are arbitrary imaginary numbers, and e −ξ 1 can be obtained from (4.19). The structure of this work is as follows. In the second part, we derive a Lax pair associated with a 5×5 matrix spectral problem for the FCNLS equation (1.2). Then based on the Lax pair with a 5 × 5 matrix, we analyze the spectral problem and the analytical properties of the Jost functions. In the third part, we establish the RH problem based on the previous conclusions. Next, we give the symmetry of the scattering matrix, and study the temporal and spatial evolution of the scattering data. In the fourth part, by solving the RH problem, we obtain the N-soliton solutions of the FCNLS equation (1.2), and analyze the propagation behaviors of one-soliton solutions and two-soliton solutions. Finally, some conclusions are presented in the last section.

The Lax Pair and eigenfunction
We first derive the Lax pair of the FCNLS equation (1.2) via the following theorem.
where Φ is a column vector function, and matrices U and V are written as here λ being the spectral parameter and p 1 = a 11 q * 1 +a 21 q * 2 +a 31 q * 3 +a 41 q * Proof. The compatibility condition of the two equations in Eqs.(2.1) Then we obtain that then we can get the equivalent Lax pair where [Λ, µ] = Λµ − µΛ is the commutator. We can get the following full differential where e λΛ µ = e λΛ µe −λΛ .

Asymptotic analysis
To formulate an RH problem, we seek solutions of the spectral problem with the 5 × 5 unit matrix as λ → ∞. Let us consider the solution of Eq.(2.10) as follows where µ (0) , µ (1) and µ (2) are independent of λ. Substituting Eq.(2.11) into Eqs.(2.9), and comparing the same order of frequency for λ, we obtain (2.14) with the asymptotic conditions here each [µ + ] l (l = 1, 2, 3, 4, 5) denotes the l-th column of the matrices [µ ± ], respectively. The symbol I is the 5 × 5 unit matrix, and the two solutions [µ ± ] are uniquely determined by the Volterra integral equations for λ ∈ R Then we analysis the Eqs.(2.16), (2.17) To find the analytic area of each column, we just consider Re[2iλ(x − y)] < 0 and 5 are analytic in the lower half-plane C − . Now we investigate the properties of µ ± . Since tr(P) = 0 and Liouville's formula, we know that the determinants of µ ± are independent of the variable x. Therefore we obtain from Eq.(2.15) that Since µ ± E are both matrix solutions of the spectral problem Eqs. (2.9), where E = e −iλΛx . Therefore, these two solutions are interdependent, and they must be related by a scattering matrix S (λ) = (s k j ) 5×5 To formulate an RH problem for the FCNLS equations (1.2), we consider the inverse matrices of µ ± as where each [µ −1 ± ] l , (l = 1, 2, 3, 4, 5) denotes the l-th row of µ −1 ± , respectively.   Proof. Using Eq.(2.9) it is easy to verify that µ −1 ± satisfy the equation of K According to (2.19), it's easy to find according to the analytic property of µ −1 + and µ − , we can proof the theorem. The matrix R(λ) can be analyzed in the same way.

Riemann-Hilbert problem
In this part, an RH problem is formulated by using the properties of µ ± . We construct matrix function P 1 = P 1 (x, λ) and P 2 = P 2 (x, λ). The function P 1 = P 1 (x, λ) is analytic in C + , and the function P 2 = P 2 (x, λ) is analytic in C − . Let with P 1 → I, as λ → +∞, At present, we restrict P 1 to the left-hand side of the real λ-axis as P + , and the restrict P 2 to the right-hand side of the real λ-axis as P − . On the real line, they are meet According to Eq.(3.3), we obtain the canonical normalization conditions as follows P 1 → I, as λ → +∞, To solve the RH problem, we consider the following theorem.
As we can see matrix P has the symmetry relation symbol " †" represents the Hermitian of a matrix, and (3.10) According to Eq.(2.9) and Eq.(3.9), µ ± meet the following relation the scattering matrix S (λ) satisfies the equation (3.13) Proof. According to Eq.(3.6), we have From Eq.(3.7), Eq.(3.8) and Eq.(3.13), we see det P 1 (λ) = (det P 2 (λ * )) * , if det P 1 have a zero λ, det P 2 have a zero λ * . So we suppose that det P 1 has N simple zeros {λ j } N 1 in C + , and det P 2 has N simple zeros {λ * j } N 1 in C − . These zeros with the nonzero vectors v j andv j , set up of the full generic discrete data, which satisfy the equations where v j is column vector, andv j is the row vector. From Eq.(3.14), Eqs.(3.16) one obtains that the eigenvectors admit the following relation.
Then we analyze the time-spatial revolution with v j . We take the derivative of the first equation of Eqs.(3.16) with respect to x, apply the same method to t.
On the basis of and the Lax pair Eqs.(2.9), we have

(3.19)
Applying the same method to P 1,t , we get where v j,0 are complex constant vectors. From Eq.(3.17), we havê

Multi-soliton solutions
Now, we are going to expand P 1 (λ) at large-λ as From Eq.(4.2), we can generate where (P (1) 1 ) i j is the (i, j)-entry of matrix P (1) 1 . To obtain soliton solutions, we set G = I in (3.4). The solutions for this special RH problem (3.4) can be given as Then setting nonzero vectors v k,0 = (α k , β k , τ k , ζ k , γ k ) T and θ k = −i(λ k x + 2λ 2 k t), we generate (4.8) Obviously It should be noted that the parameter b i j (i ≤ 5, j ≤ 4) do not work on the construction solutions, so the specific expression is not given for convenience. As a consequence, general N-soliton solution for the FCNLS equation (1.2) can be derived as follows where (4.12) To make the expression (4.11) simpler, we define the following matrix F, G, H and K.
From the expressions of Υ 1 , Υ 2 , Υ 3 and Υ 4 , ̟ 1 , ̟ 2 , ̟ 3 , ̟ 4 we can know that all of them rely on both the real part n 1 and the imaginary part m 1 of the eigenvalue λ 1 . Figure 1, Figure 2, Figure 3 and Figure 4 represent the localized structures and dynamic behaviors of the single-soliton solution. All the analysis of q 1 be the same with q 2 , q 3 and q 4 .
(a) (b) (c)   In the case of N = 2, the two-soliton solutions can be obtained as where

Conclusions and discussions
In this work, we have proposed a FCNLS equation (1.2) associated with a 5 × 5 Lax pair, which was investigated via the RH approach. Based on the Lax pair with a 5 × 5 matrix, we start with the analyze of the spectral problem and the analytical properties of the Jost functions, from which the RH problem of the equation is established. Then, we obtain the N-soliton solutions of the FCNLS equation (1.2), by solving the RH problem without reflection. Finally, we derive two special cases of the solutions to the equation for N = 1 and N = 2, and the local structure and dynamic behavior of the one-and two-soliton solutions are analyzed graphically.