Riemann–Hilbert problems and soliton solutions for the reverse space-time nonlocal Sasa–Satsuma equation

The main work of this paper is to study the soliton solutions and asymptotic behavior of the integrable reverse space-time nonlocal Sasa–Satsuma equation, which is derived from the coupled two-component Sasa–Satsuma system with a specific constraint. The soliton solutions of the nonlocal Sasa–Satsuma equation are constructed through solving the inverse scattering problems by Riemann–Hilbert method. Compared with local systems, discrete eigenvalues and eigenvectors of the reverse space-time nonlocal Sasa–Satsuma equation have novel symmetries and constraints. On the basis of these symmetry relations of eigenvalues and eigenvectors, the one-soliton and two-soliton solutions are obtained and the dynamic properties of these solitons are shown graphically. Furthermore, the asymptotic behaviors of two-soliton solutions are analyzed. All these results about physical features and mathematical properties may be helpful to comprehend nonlocal nonlinear system better.


Introduction
Since Ablowitz and Musslimani publish relevant researches of the nonlocal nonlinear Schrödinger (NLS) equation [1,2], nonlocal systems have aroused widespread concern. The nonlocal nonlinear equation is quite different from the local one in that it has terms at the points (−x, t), (x, −t) or (−x, −t) not just at (x, t). Hence there is the evolution of solutions on a wider range of space and time. It means that the solutions have coupling property between x/t and −x/−t, which makes us remind the quantum entanglement between pairs of particles [3]. The terms of these three points (−x, t), (x, −t) and (−x, −t) correspond to reverse space nonlocal equation, reverse time nonlocal equation and reverse space-time nonlocal equation, respectively. It is worth mentioning that nonlocal systems could have some new features in solving the soliton solutions, and these solitons solutions have some distinct dynamic properties and phenomena which are different from the local ones. Hence there have been a number of works concerned with nonlocal systems. The nonlocal NLS equation has a wide range of applications in physics, for example it presents the propagation of optical solitons in optical fibers. The general soliton solutions of nonlocal NLS equation can be constructed through a series of methods, such as Darboux transformation [4][5][6][7], Hirota bilinear method [8][9][10], inverse scattering transform [11][12][13][14], Dbar-approach [15] and Bäcklund transformation [16]. The nonlocal modified Korteweg-de Vries (mKdV) equation is applied to show the dynamic properties in plasma physics, fluid dynamics and other physical fields, which was investigated via the Hirota bilinear method [17,18], Darboux transformation [19,20], inverse scattering transform [21,22] and∂-dressing method [23,24]. Some general soliton solutions of the nonlocal Fokas-Lenells (FL) equation can also be derived through Darboux transformation [25] and Hirota bilinear method [26]. Peng and Chen [27] analyzed the nonlocal Hirota equation with nonzero boundary conditions by Riemann-Hilbert method and PINN algorithm. Inspired by the novel physical phenomena of nonlocal problems, many scholars have paid extensive attention to nonlocal systems.
It is well known that inverse scattering transform is one of the most significant methods to construct the soliton solutions of nonlinear differential equations, which was first proposed by Gardner, Greene, Kruskal, and Miura in 1967 [28]. With further research, the Riemann-Hilbert problem as a simplified method is applied in integrable systems. By constructing associated Riemann-Hilbert problems, some nonlocal nonlinear equations are analyzed with inverse scattering transforms. In [29,30], a nonlocal NLS equation is constructed from the Manakov system, and its solitons and multisolitons are obtained. The N-soliton solutions of three nonlocal NLS equations which have different symmetry constraints are derived in [3]. The nonlocal reverse space-time mKdV hierarchies are investigated based on nonlocal symmetry reductions of matrix spectral problems [31,32]. Zhang and Yan [33] analyze both focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions. In [34,35], the dynamics of soliton solutions for the nonlocal Kundunonlinear Schrödinger equation is investigated. For nonlocal Sasa-Satsuma equations, the inverse scattering transform is also applicable [36]. Not only that, the reverse space-time Sasa-Satsuma equation could also be investigated by binary Darboux transformation [37]. All those nonlocal equations were derived from singlecomponent local Sasa-Satsuma equation, but the nonlocal equations from multi-component Sasa-Satsuma systems have been not studied yet. In this paper, the coupled two-component Sasa-Satsuma system is considered where u = u(x, t) and v = v(x, t) are complex-valued functions with the independent spatial variable x and temporal variable t, and the subscript x (or t) denotes partial derivative with respect to space x (or time t) of functions u and v. Then, we add the solution constraint as follows It is easy to verify that the two equations in the coupled Sasa-Satsuma system are consistent under the constraint, and the reverse space-time nonlocal Sasa-Satsuma equation of u is derived as which is a higher-order nonlocal NLS equation, and describe the soliton propagation in optical fibers. It is easy to see that the higher order nonlinear terms are nonlocal which affect the phase modulation, width and the velocity of the wave in nonlinear waveguides [38]. The reverse space-time nonlocal Sasa-Satsuma equation is parity-time (PT) symmetric, i.e. the solutions are invariant when x → −x, t → −t and i → −i.
The reverse space-time nonlocal Sasa-Satsuma equation (3) is integrable, whose Lax pair can be obtained by the following Lax pair of the coupled Sasa-Satsuma system where Y = Y (x, t, λ) is the matrix eigenfunction of the complex spectral parameter λ, and and = diag(−1, −1, −1, −1, 1).
Under the solution constraint (2), the Lax pair of reverse space-time nonlocal Sasa-Satsuma equation (3) is deduced by replacing v(x, t) and v * (x, t) with u(−x, −t) and u * (−x, −t) respectively in the above Lax pair (4). To our best knowledge, there has not been any work on solutions of such integrable nonlocal Sasa-Satsuma equation. We will establish Riemann-Hilbert problems of Eq. (3). Then, one soliton, two soliton and their asymptotic analysis will be presented.
The rest of this paper is shown as follows. In Sect. 2, the inverse scattering transform is carried out to solve the coupled Sasa-Satsuma system by constructing the Riemann-Hilbert problem. Then we investigate the symmetry relations of discrete scattering data for the reverse space-time nonlocal Sasa-Satsuma equation, such as symmetry constraints of scattering eigenvalues and eigenvectors. Based on the symmetry constraints, the one-soliton and two-soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation are derived from the N-soliton solutions of the coupled Sasa-Satsuma system with the above solution's restricted condition. And the figures of soliton solutions are presented, in the meantime the dynamics of solutions is analyzed. In Sect. 3, we make the asymptotic analysis for two-soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation. Finally, the conclusions of this paper are stated in Sect. 4.

Soliton and multisolitons solutions of the reverse space-time nonlocal Sasa-Satsuma equation
In order to derive the soliton and multisolitons solutions of the reverse space-time nonlocal Sasa-Satsuma equation (3), the coupled Sasa-Satsuma system (1) is analyzed firstly. It is obvious that the solutions of the reverse space-time nonlocal Sasa-Satsuma equation (3) could be obtained from the coupled Sasa-Satsuma system with the constraint (2). And it is worth mentioning that the key to obtaining the soliton and multisolitons solutions of this nonlocal equation (3) is the symmetry relations of its discrete scattering data in the process. In this section, we start from utilizing inverse scattering transform method to analyze the general soliton solutions of the coupled Sasa-Satsuma system (1) in the Riemann-Hilbert formulation. And then the symmetry constraints of discrete scattering data are analyzed in the process of solving the soltion and multisoltions solutions of reverse space-time nonlocal Sasa-Satsuma equation (3). Finally, the dynamic properties of these solutions for the reverse space-time nonlocal Sasa-Satsuma equation (3) are analyzed and illustrated by the solutions' figures.
2.1 Inverse scattering and N-soliton solutions for the coupled Sasa-Satsuma system The N-soliton solutions of the coupled Sasa-Satsuma system (1) in the Riemann-Hilbert formulation will be analyzed through inverse scattering transform method. Both solutions u and v are assumed to decay to zero rapidly as x → ±∞. From the Lax pair (4), Y can be solved as where the matrix function J is (x, t)-independent at infinity. Substituting (9) into (4), the partial derivatives of matrix function J with respect to x and t are derived as where In the scattering problem, the matrix Jost solutions J ± (x, t, λ) of Eq. (10) are introduced with the asymptotic property as x → ±∞ where I is the 5×5 unit matrix, and the subscripts ± denote to which end of the x-axis the boundary conditions are set. The notations are also introduced as follows where are five dimensional column vectors. In addition, some notations of the inverse Jost solutions are given as whereφ i ,ψ i (i = 1, 2 . . . 5) are five dimensional row vectors.
Since and are both solutions of the scattering problem (4), the linear relations are derived by a scattering matrix S(t, λ) as follows which indicates S(t, λ) is a x-independent 5×5 scattering matrix for real λ, and In addition, the matrix Jost solutions J ± (x, λ) can be obviously derived by the Volterra integral equations that are accessible Then, the Jost solution P + is derived as which is analytic in λ ∈ C + . And the inverse Jost solution P − is given as which is analytic in λ ∈ C − . In the above expressions (22) and (23), the H 1 and H 2 are shown as

Proposition 2.2 Based on the definition of P ± (x, t, λ), the matrix Riemann-Hilbert problem satisfies
(i) the relation between P + and P − From the definitions of P ± in (22) and (23) as well as the scattering relation (17), the determinants of P ± are derived as We suppose that λ k ∈ C + andλ k ∈ C − (1 ≤ k ≤ N ) are simple zeros of the s 55 (t, λ) and theŝ 55 (t, λ), respectively. In this case, the kernels of P + (x, t, λ k ) and P − (x, t,λ k ) contain only a single column vector ω k (x, t) and row vectorω k (x, t), respectively, Notice that the matrix Q satisfies the following property where the superscript † represents the Hermitian (i.e., conjugate transpose) of a matrix. Then the symmetry constraints of λ and ω are given as Since P + satisfies the Eqs. (10) and (11), ω k satisfies the following differential equations Under the symmetry constraints in (30), the above Eq. (31) is solved as where ω k0 is a constant column vector. Then, as λ approaches ∞, expansion of P + is given as Substituting the expansion (33) into Eq. (10) and comparing terms of the order O(1), the potential Q is shown as Hence u and v can be constructed as follows In general, the matrix Riemann-Hilbert problem obtains analytical solutions. If P − P + = I , then the solution P + can be explicit as where The normative column eigenvectors ω k0 and ω † k0 are obtained as Denoting the N-soliton solutions of the coupled Sasa-Satsuma system are derived as where and the expressions of b k and d k are given from equation (41) as follows In the above expressions, λ k (1 ≤ k ≤ N ) are complex numbers in the upper half plane, namely, λ k ∈ C + . And a j and c j are arbitrary complex constants.

2.2
The symmetry relations of discrete scattering data in the reverse space-time nonlocal Sasa-Satsuma equation It could be observed from the above analysis that the reverse space-time nonlocal Sasa-Satsuma equation (3) is derived from the coupled Sasa-Satsuma system (1). Under the constraint (2), the expression (7) of potential Q(x, t) is rewritten as and it is easy to see that the potential Q satisfies the symmetry constraint where (3), if λ k is a discrete eigenvalue, then so is −λ * k . It means that the discrete eigenvalues are either purely imaginary or in the form of (λ k ,−λ * k ) pairs. Proof In spatial matrix spectral problem Y x = (−iλ + Q)Y , we make x → −x and t → −t. Then a new spatial matrix scattering problem is derived as

Theorem 2.1 For the reverse space-time nonlocal Sasa-Satsuma equation
Hence, if λ k is a discrete eigenvalue, then −λ * k is a discrete eigenvalue for the reverse space-time nonlocal Sasa-Satsuma equation (3). Both λ k and −λ * k are in the upper half plane C + . This proves the symmetry constraint of eigenvalue λ k in Theorem2.1. (i) For one eigenvalue λ 1 = iη ∈ C + , i.e. η ∈ R and η > 0, its eigenvector is derived as follows where γ ∈ R.
(ii) For two eigenvalues λ k = iη k ∈ C + (k = 1, 2), i.e. η k ∈ R and η k > 0, their two eigenvectors ω 10 and ω 20 are given as follows where and in the above expressions, h = η 1 +η 2 η 2 −η 1 . (iii) For two nonimaginary eigenvalues (λ 1 , λ 2 ) ∈ C + , satisfying λ 2 = −λ * 1 , two corresponding eigenvectors ω 10 and ω 20 are (a 1 , b 1 , c 1 , d 1 , 1) , and they are related as where H is Proof In order to prove these symmetry relations of discrete eigenvectors in Theorem 2.2, we start with analyzing the relationship between u and v in the solutions (41) of the coupled Sasa-Satsuma system based on the solution constraint (2). And the symmetry relations combined with the symmetry constraint of eigenvalues in Theorem 2.1 are discussed separately for the one-soliton and two-soliton solutions.
For the first case of one-soliton solutions, we consider a single purely imaginary eigenvalue λ 1 = iη which is in the upper half plane C + , i.e. η > 0 and substitute λ 1 = iη into Eq. (40). It is easy to get Thus, the one-soliton solutions of the coupled Sasa-Satsuma system from Eq. (41) have the following expressions where b 1 = a * 1 and d 1 = c * 1 . Further with the solution constraint v(x, t) = u(−x, −t), the relations between a 1 and c 1 are formulated Hence, From the above equations, it could be observed that |a 1 | 2 = |c 1 | 2 = 1 4 . Hence the eigenvector for the onesoliton solution is given as where γ ∈ R.
For the second case of two-soliton solutions, two complex eigenvalues λ 1 , λ 2 ∈ C + are considered. From expression of solutions for the coupled Sasa-Satsuma system in (41), the two-soliton solutions u(x, t) and v(x, t) are given as where and the expressions of coefficients A k , B k , (k = 1, 2, 3, 4), and C j ( j = 1, 2, 3, 4, 5) are derived as follows (77) For the above two-soliton solutions, we make two different assumptions about discrete eigenvalues λ 1 and λ 2 .
The first assumption is that the eigenvalues λ 1 and λ 2 are purely imaginary, and λ k = iη k (k = 1, 2). Both λ 1 and λ 2 are in the upper half plane C + , i.e. η 1 > 0 and η 2 > 0. Then In this case, making x → −x, t → −t, there are Combined with the solution constraint v(x, t) = u(−x, −t), the coefficients A k , B k , C k (k = 1, 2, 3, 4) satisfy the conditions as follows From the above Eqs. (81)-(82), the parameters of eigenvectors ω 10 and ω 20 for two-soliton solutions of the reverse space-time Sasa-Satsuma equation (3) satisfy the conditions in Theorem 2.2.
The second assumption is that the eigenvalues λ 1 and λ 2 are not purely imaginary, and λ 2 = −λ * 1 ∈ C + . Then and In this case, letting x → −x, t → −t, there are Combined with the solution constraint v(x, t) = u(−x, −t), the coefficients A k , B k , (k = 1, 2, 3, 4), and C j ( j = 1, 2, 5) satisfy the conditions as follows Then, from the above conditions, we know 1 which can be rewritten as Eq. (55) in Theorem 2.2. This proves the symmetry constraints of discrete eigenvectors for one-soliton and two-soliton solutions in Theorem 2.2.

The one-soliton and two-soliton solutions of the nonlocal equation and their dynamics
The symmetry relations of discrete eigenvalues and eigenvectors are concluded in Theorems 2.1 and 2.2.
And then, the one-soliton and two-soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation (3) could be given under these symmetry constraints, and their figures are drawn by choosing different parameters. For one-soliton solutions, the single purely imaginary eigenvalue λ 1 = iη ∈ C + (η > 0) has its discrete eigenvector ω 10 = ( 1 2 e iγ , 1 2 e −iγ , 1 2 e iγ , 1 2 e −iγ , 1) as in the Theorem 2.2. Then, the one-soliton solution of the reverse space-time nonlocal Sasa-Satsuma equation (3) is derived as where γ is an arbitrary real constant.
Next, there will be some figures to describe the nonlocal one-soliton solution Eq. (90) (see in Fig. 1). Based on the expression of one-soliton solutions (90) of the reverse space-time nonlocal Sasa-Satsuma equation (3), we could observe that the coefficients of x and t are real numbers in the hyperbolic functions, and there is a fixed constant before the hyperbolic functions. So the figures of soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation have invariant amplitude, i.e. the peaks of the soliton solution are unchanged as wave travels. In Fig. 1, the figure of (a) presents a single-hump solution, and the shape of solution is not parallel with neither x axis nor t axis. The figure of (b) shows breather-type solution, and its realizations are periodic.
For two-soliton solutions, there are two cases. In the first case, we suppose that the discrete eigenvalues λ 1 and λ 2 are purely imaginary, and λ 1 = iη 1 , λ 2 = iη 2 . The eigenvalues are in the upper half plane C + , i.e. η k > 0 (k = 1, 2). Some definitions are needed here Since q = q * , q is real constant. And the other definitions are q = r 0 , a 1 = r 1 e iγ 1 , a 2 = r 2 e iγ 2 .
From Eq. (51), the expression of p is derived as follows In the above expression, the quantity under the square root is always positive, because h is real and |h| > 1. Hence, it is obvious that the relations between a 1 , a 2 and c 1 , c 2 are linear from Eqs. (52)-(53).
Then, substituting Eqs. (52)-(53) into Eq. (50), and taking f = r 2 /r 1 , the simplified equation of f is derived as where After the value of f is calculated, substituting the Eq. (52) into the expression p = 2|a 1 | 2 + 2|c 1 | 2 , the r 1 is certain where Then, we may get the value of r 2 from the equation r 2 = r 1 f . By now, the values of a 1 and a 2 have been determined by choosing free constants γ 1 and γ 2 , and the values of c 1 and c 2 can be calculated from a 1 , a 2 , p and q through Eqs. (52)-(53). The existence of the two-soliton solutions is determined by the existence of solutions for Eq. (94). Subsequent analysis will base on the discriminant for quadratic equation (94) We consider a special condition about r 0 = 1, thus p = 1, a = c = −h and b = 2h cos(γ 1 − γ 2 ). And there exists a single root of the quadratic equation (94) if and only if cos(γ 1 − γ 2 ) = 1. The discrete eigenvectors of two-soliton solutions are given as follows where cos(γ 1 − γ 2 ) = 1. In this case, the two-soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation (3) could be expressed by Eq. (62) according to Theorem 2.2. Through calculating, the two-soliton solutions with two purely imaginary eigenvalues could be given as follows where By choosing different parameters, some figures of two-soliton solutions with two purely imaginary eigenvalues are shown in Fig. 2. The figure (a) presents the elastic collisions between a single-hump soliton and a breather of the reverse space-time nonlocal Sasa-Satsuma equation. It is not difficult to find that the amplitudes of waves change after interaction, but the shapes of waves keep unchanged after collision. The figure (b) shows oblique elastic collision between two single-hump solutions. It could be observed that both the amplitudes and shapes of two single-hump solutions remain unchanged after interaction, but there exhibit little meandering and slight position shifts.
In the second case, the discrete eigenvalues λ 1 and λ 2 are assumed not purely imaginary, and λ 2 = −λ * 1 . Both λ 1 and λ 2 are in the upper half plane C + . The two-soliton solutions with discrete eigenvectors b With parameters: λ 1 = −λ * 2 = 0.6 + 1.2i. c With parameters: 10 and ω 20 satisfying Eq. (55) could be obtained from Eq. (62) in Theorem 2.2, which are given as follows where By choosing different parameters, some figures of twosoliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation (3) in this case are shown in Fig. 3. It could be observed that the process of collision and separation of two parallel breather-type waves is shown in (a)-(c). The two-soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation with not purely imaginary eigenvalues are periodic waves. And their shapes and amplitudes remain unchanged after interaction. The figures (a), (b) and (c) have same parameters of (a 1 , b 1 , c 1 ,

Asymptotic analysis of two-soliton solution in the reverse space-time nonlocal Sasa-Satsuma equation
Multiple solitons collisions may have effect on the shape of soliton, either changed or recovered. Thus, in order to investigate the elastic and inelastic interactions between the bound solitons and the regular one soliton, we analyze the asymptotic analysis of two-soliton solution in the reverse space-time nonlocal Sasa-Satsuma equation (3). Firstly, we make an assumption as Then the Re(θ 1 ) and Re(θ 2 ) can be derived as follows For convenience, we suppose v k = 12ξ 2 k − 4η 2 k , k = 1, 2.
According to the above two-soliton solution (62) of the reverse space-time nonlocal Sasa-Satsuma equa-tion (3) under the conditions in Theorem 2.2, without loss of generality, we make an assumption that η 1 , η 2 > 0. For fixed Re(θ 1 ), suppose that v 1 > v 2 .
Comparing with the single-soliton solution (90), we can see the asymptotic expressions u 1− , u 1+ , u 2− , u 2+ are single-soliton solutions. After the two singlesoliton collision, each soliton does not change its shape and velocity, but the position and phase shift.

Conclusions
In this paper, our research revolves around the reverse space-time nonlocal Sasa-Satsuma equation (3) which is constructed from the coupled Sasa-Satsuma system (1) under the solution constraint (2). The reverse spacetime nonlocal Sasa-Satsuma equation is integrable, and it is obvious that its Lax pair could be derived from the corresponding one of the coupled Sasa-Satsuma system. The coupled Sasa-Satsuma system are first investigated through the inverse scattering transform, and the N-soliton solutions are derived via constructing corresponding Riemann-Hilbert problem. Considering the solution constraint between the reverse spacetime nonlocal Sasa-Satsuma equation and the coupled Sasa-Satsuma system, the symmetry constraints of scattering eigenvalues and eigenvectors for the reverse space-time nonlocal Sasa-Satsuma equation are analyzed in detail. According to the different selections of scattering eigenvalues, the symmetry constraints of scattering eigenvectors are different. Then the onesoliton and two-soliton solutions of the reverse spacetime nonlocal Sasa-Satsuma equation are presented graphically through choosing different parameters, and their dynamics exhibit interesting wave patterns. The asymptotic analysis of two-soliton solutions of the reverse space-time nonlocal Sasa-Satsuma equation is studied in the last part. These results show that an integrable nonlocal equation has interesting physical meanings and some unusual mathematical properties, such as soliton's meandering feature and intricate symmetry relations of discrete scattering data. In the further, it might be useful to comprehend some physical phenomena and inspire some novel physical applications on other nonlocal nonlinear systems.