In recent decades, study of the coupled nonlinear Schrödinger equation of Manakov type 1 has attracted a lot of attention due to its use for modelling equations in diverse areas including deep water waves, nonlinear fibre optics, plasma physics and others (see, e.g.^[1–3] and references therein). This model was first proposed by Manakov to study intense electromagnetic pulse propagation in birefringent fibre. Subsequently, this system was derived as a key model for light-wave propagation in optical fibres.^[4] Much research on the coupled nonlinear Schrödinger equation (1) has been conducted. For example, multi-soliton solutions, collisions, algebro-geometric solutions, asymptotic behaviour of solutions, and other properties have been widely investigated for the coupled nonlinear Schrödinger equation of Manakov type.^[5–13]

The aim of this paper is to study the Cauchy problem of the nonlocal two-wave interaction system from the Manakov hierarchy^[14–17] 2 via the Riemann–Hilbert approach,^[18–22] where u = u(x, t) and v = v(x, t) are the wave envelopes, and the star denotes complex conjugation, by which the multi-soliton solutions for this system are constructed. We assume here the rapidly decreasing conditions u(x,t),v(x,t) → 0 as x → ±∞, and . System (2) can be used to describe the soliton wave phenomenon in nonlinear waves arising in a nonlinear dispersive medium.^[15] Furthermore, a similar prototypical system of evolution equations which models the nonlocal interaction of two waves in quadratic nonlinear media is investigated in Ref. [17]; the solvability of its initial value problem is shown and its conservation laws are explicitly presented. The related three-wave resonant interaction system is studied in Refs. [14] and [16], the bounded rational soliton solutions are obtained in Ref. [14] by means of the Darboux–Dressing method, and the multi-dark–dark–dark soliton solutions for this three-wave resonant interaction system are constructed through the generalized Darboux transformation method.^[16] Moreover, various types of other exact solutions have been obtained such as breathers, rogue waves and other solutions. For the latest literature in this area, we refer to Refs. [23]–[29].

The structure of this paper is as follows. In Section 2, we start with the spectral analysis of the Lax pair for the nonlocal two-wave interaction system (2). Then we formulate the corresponding Riemann–Hilbert problem for the nonlocal two-wave interaction system. In Section 3, we study the inverse scattering transformation of the nonlocal two-wave interaction system (2) and solve the Riemann–Hilbert problem. In Section 4, the spatial and temporal evolution of scattering data is discussed. In Section 5, N-soliton solutions of the nonlocal two-wave interaction system (2) are constructed.

In this section, we shall study the direct scattering problems for the nonlocal two-wave interaction system (2) and construct the corresponding Riemann–Hilbert problem. Let us consider the spectral and auxiliary problems associated with the nonlocal two-wave interaction system (2) 3 4 where λ is a spectral parameter, and Y(x,t,λ) is a matrix function. Assume that the potentials u and v decay to zero sufficiently fast as x → ±∞. It is convenient for us to introduce a new matrix spectral function 5 where 6 and E₁ is a solution of the above spectral equations at x → ±∞. Therefore, the spectral problem equations (3) and (4) can be rewritten as 7 8 where [σ_i,J] = σ_iJ − Jσ_i, i = 0, 1, 9 It is easy to see that both the matrices Q and Q₁ possess the “symmetry” conditions, i.e. 10 11 where the superscript † denotes the Hermitian of a matrix.

In the scattering process, the time t is fixed and can be treated as a dummy variable, hence it will be suppressed in our notation. We assume that the matrix Jost solutions J_±(x,λ) of the spectral problem (7) fulfil the asymptotic condition 12 where I denotes the 3 × 3 unit matrix. Noticing the identity 13 and tr(Q) = 0, we arrive at 14 for all (x,λ), where detJ denotes the determinant of matrix J and tr(·) represents the trace of a matrix. Besides, it is easy to verify that the matrix Jost solutions J_± solve the Volterra integral equations: 15 Hence J_±(x,λ) admit analytical continuations off the real axis λ ∈ ℝ if the above Volterra integrals converge. Let us split J_± into column vectors, i.e., J_± = ([J_±]₁,[J_±]₂,[J_±]₃). Then the column vectors [J₋]₁,[J₊]₂,[J₊]₃ can be analytically continued to the upper half plane λ ∈ ℂ₊, while the column vectors [J₊]₁,[J₋]₂,[J₋]₃ can be analytically continued to the lower half plane λ ∈ ℂ₋. Furthermore, if we denote E = e^iλσ₀x, it follows easily that both J₊E and J₋E are fundamental matrix solutions of the spectral problem (3), which indicates that they are not independent and are linearly related by a scattering matrix S(λ) 16 with S(λ) = (s_ij(λ))_3×3. In view of Eqs. (14) and (16), we have 17

In what follows, we shall construct the Riemann–Hilbert problem. To this end, we introduce the following matrix function 18 where 19 From the analytic property of J_±, we get that P⁺ is analytic in λ ∈ ℂ₊. In view of Eq. (15), it follows that 20

In order to obtain the analytic counterpart of P⁺ in ℂ₋, we start with the adjoint spectral equation of Eq. (7): 21 It is not difficult to verify that satisfies Eq. (21), where is partitioned into rows in the following form 22 In view of Eq. (16), we have 23 where R(λ) = S⁻¹(λ) = (r_kj)_3×3. Similar discussions lead to the conclusion that the row vectors , , are analytic in λ ∈ ℂ₊, while other row vectors , , are analytic for λ ∈ ℂ₋. Noticing the analytic property of J₋ and , it follows that s₁₁ admits analytic continuation to ℂ₊, and s₂₂, s₃₃, s₂₃ and s₃₂ can be analytically continued to ℂ₋, while s₁₃, s₃₁, s₁₂ and s₂₁ are only defined for λ ∈ ℝ. Similarly, r₂₂, r₂₃, r₃₂ and r₃₃ can be analytically continued to ℂ₊, r₁₁ allows analytical continuation to ℂ₋, while r₁₂, r₂₁, r₁₃ and r₃₁ are defined only for λ ∈ ℝ. Therefore, we obtain that the adjoint Jost solutions 24 are analytic for λ ∈ ℂ₋. In addition, we have that 25

By now, we have obtained two matrix functions P⁺ and P⁻, which are analytic in ℂ₊ and ℂ₋, respectively. On the real line, they are related by 26 where 27 where r₁₁s₁₁+r₁₂s₂₁+r₁₃s₃₁ = 1 is followed by the fact that R(λ)S(λ) = I. Equation (26) is just the matrix Riemann–Hilbert problem we needed, and the associated canonical condition is as follows: 28

In this section, we shall solve the Riemann–Hilbert problem (26) in both regular and non-reguluar cases.^[21] Before this, from Eqs. (14) and (18), it is easy to see that 29 30

We first consider the regular Riemann–Hilbert problem (26), i.e., det P⁺ = s₁₁ ≠ 0 and det P⁻ = r₁₁ ≠ 0 in their analytic domains. Equation (26) can be rewritten as 31 where . By Plemelj’s formula, we obtain that the solution to the regular Riemann–Hilbert problem (26) under condition (28) takes the form 32

Next, we consider the non-regular Riemann–Hilbert problem (26), that is, detP⁺(λ) and detP⁻(λ) possess simple zeros, which are denoted by λk ∈ ℂ₊ and , 1 ≤ k ≤ N, respectively, where N is the number of these zeros. Therefore λ_k and are also zeros of the scattering coefficients s₁₁(λ) and r₁₁(λ), where 33 Now we discuss the symmetry properties of the scattering matrix S(λ) and the Jost functions. Taking the Hermitian on both sides of the spectral problem (7), in view of Eq. (10), we have 34 Since fulfills Eq. (21), it follows that 35 from which, together with Eqs. (18) and (24), P^± satisfies the property 36 In view of Eq. (16), we have 37 Then we obtain by using Eq. (23) that S satisfies the property 38 from which we have the following symmetric relation 39 Since all the zeros are simple ones, then both kerP⁺(λ_k) and are one-dimensional and spanned by a single column vector μ_k and row vector , respectively. Hence we have 40 41 Taking the Hermitian of the first equation in Eq. (40) and utilizing the property (36), we obtain 42 Then, comparing it with the second equation in Eq. (40), we see the eigenvectors μ_k and satisfy the property 43

The solution to this non-regular Riemann–Hilbert problem (26) under the canonical normalization condition (28) is unique and is given by the following theorem. Theorem 1

The solution to the non-regular Riemann–Hilbert problem (26) with zeros determined by Eq. (40) under the canonical normalization condition (28) is 44 45 where 46 47 and M is an N × N matrix with its (j,k)th element given by 48 49 and is the unique solution to the following regular Riemann–Hilbert problem: 50 where are analytic in ℂ_±, respectively, and as λ → ∞.

In this section, we shall give N-soliton solutions of the nonlocal two-wave interaction system, which corresponds to the case G = I. Therefore the integral term in Eq. (73) vanishes, that is, 74 and 75 where , vectors μ_j are defined by Eq. (40), and M is given by Eq. (48).

Next, we shall study the properties of these N-soliton solutions. We first consider the simple case where N = 1, namely, single-soliton solutions. For simplicity, we set 76 Then the single-soliton solutions (75) read 77 Setting α₁ = β₁ = γ₁ = i and θ₁ = φ₁ = 1, we plot the graphs of the single-soliton solutions for the nonlocal two-wave interaction system (2) as shown in Fig. 1.

Fig. 1.

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	Fig. 1. (color online) Single-soliton solutions for the nonlocal two-wave interaction system (2).

For N ≥ 2, we introduce the following notations 78 In view of Eqs. (55) and (56), we have 79 Then the N-soliton solutions (75) read as 80 where the element of the matrix M = (M_jk)_N×N is defined by 81 Moreover, N-soliton solutions (75) can be written as 82 where two (N + 1) × (N + 1) matrices F₁ and F₂ are given by 83 84

In this paper, we discuss the N-soliton solutions for the nonlocal two-wave interaction system arisen from the Manakov hierarchy by means of the Riemann–Hilbert method. Our presentation is based on the Riemann–Hilbert formulation, which can be summarized as follows: we firstly construct a Riemann–Hilbert problem for the nonlocal two-wave interaction system (2) through the spectral analysis of the corresponding Lax pair. Secondly, we solve the Riemann–Hilbert problem for the nonlocal two-wave interaction system in both the regular and non-regular cases. Thirdly, by reconstructing the potentials and discussing the spatial and temporal evolution of the scattering data, we explicitly give expressions for the N-soliton solutions of the nonlocal two-wave interaction system (2). Finally, the dynamical behaviour of the single-soliton solution is displayed graphically.