In the past several years, localized waves including dark or bright soliton, breather and rogue wave have been of great interests in nonlinear science. The dark and bright soliton are special cases of soliton. The breather is localized in time or space, such as Ma breather (time-periodic breather solution)^[1] and Akhmediev breather (space-periodic breather solution).^[2] While the rogue wave (also called freak wave, monster wave, killer wave, rabid-dog wave, and other names) is localized in both time and space, and seems to appear from nowhere and disappear without a trace.^[3–6] There have been many articles on rogue waves of single-component systems, such as the nonlinear Schrödinger (NLS) equation,^[7–9] the derivative NLS equation,^[10,11] the Kundu–Eckhaus equation,^[12–14] the Sasa–Satsuma equqtion,^[15] the higher-order dispersive NLS equation,^[16] and so on.

However, a variety of complex systems,^[17–19] such as Bose–Einstein condensates and nonlinear optical fibers, usually involve more than one component. So recent studies are extended to localized waves in multicomponent coupled systems, and many interesting and appealing results have been obtained. The bright–dark–rogue solution^[20,21] and other higher-order localized waves^[22] are all found in two-component coupled NLS equation. Some semi-rational, multi-parametric localized wave solutions are obtained in coupled Hirota equation.^[23–25] A four-petaled flower structure rogue wave is exhibited in three-component coupled NLS equation.^[26]

Motivated by the works of Baronio^[27] and Guo,^[9,28] we study the localized wave solutions of the three-component coupled NLS equation

Here we are interested in the interactional solutions between rogue waves and some nonlinear wave solutions in the Eq. (1), for example, dark, bright soliton and breather. To the best of our knowledge, this is not reported in other articles. By using Darboux-dressing transformation, Baronio et al.^[21] has obtained some semi-rational solutions in two-component coupled NLS equation, which include rogue wave, dark–bright–rogue wave and breather–rogue wave. But, Baronio’s method is very complicated and can not obtain higher-order localized waves. In order to overcome this difficulty, we construct a specifical vector solution of Lax pair for the vector NLS equation, which is firstly put forward.^[22,24] Combining the generalized Darboux transformation (DT) with the special vector solution, we have conveniently obtained several interesting higher-order localized waves.

The localized waves, such as second-order dark–bright–rogue wave and second-order breather–rogue wave, have been discussed in detail.^[22,24] N-component NLS equation has been solved to get multi-dark soliton.^[29] Meanwhile, performing Hirota bilinear method to Eq. (1), one can obtain two-bright-one-dark soliton and one-bright-two-dark soliton.^[30] Four-petaled flower structure rogue wave has also been found.^[26] However, three-component and two-component NLS equation are not exactly the same. Using our method, some meaningful results can be obtained. From the special seed solutions of Lax pair, we can get the basic solutions of Eq. (1) with several free parameters by generalized DT. Then, choosing the appropriate values of these free parameters, some interesting interactional solutions are exhibited.

In this section, we construct the generalized DT of Eq. (1). The system (1) admits the following Lax pair

Based on the DT of the Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem,^[28] the generalized DT of Eq. (1) can be also constructed. In Eq. (2), U and V are 4 × 4 matrixs, so it is more complicated than two-component NLS equation for getting a special vector solution of Lax pair. Let ϕ₁ = (ϕ₁,φ₁,χ₁,ψ₁)^T = ϕ₁(λ₁ + δ) be a solution of Eqs. (2) and (3) with q₁ = q₁[0], q₂ = q₂[0], q₃ = q₃[0] and λ = λ₁ + δ, then ϕ₁ can be expanded as the Taylor series at δ = 0,

Here, I is the 4 × 4 identity matrix. We can see that Eqs. (7)–(9) give rise to the N-order localized waves solutions of Eq. (1). If we iterative above procedures, some higher-order localized waves solutions may be obtained. Certainly, the determinant representation of the high-order localized wave solutions can be derived by Crum theorem.^[31] In order to avoid cumbersome calculation of determinant of high order matrix, we prefer to iterative the DT of degree one. Besides, it is very convenient to figure out these expressions through some computer softwares.^[32] By choosing appropriate eigenfunction ϕ₁, we can get some interesting localized waves solutions of Eq. (1) by the above formulas.

We begin with the nontrivial seed solution of Eq. (1)

Here . It is straightforward to calculate that the vector function is a solution of the Lax pair Eq. (1) at q₁ = q₁[0], q₂ = q₂[0], q₃ = q₃[0], and . Hence, by using Eqs. (7)–(9), we can arrive at

The validity of Eqs. (21)–(23) can be directly verified by putting them back into Eq. (1). At this point, we get first-order localized wave solutions of Eq. (1) with two free parameters α and β, which play important role in controlling the dynamics of these localized waves. Next, we discuss the dynamics of these solutions through three different cases.

Fig. 1.

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	Fig. 1. Evolution of the first-order rogue wave of the three-component coupled NLS equation by choosing d1 = 1, d2 = −1, and d3 = −0.5. Three component q1 (a), q2 (b) and q3 (c) have the similar structure.

Fig. 2.

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	Fig. 2. Evolution of the first-order dark–bright–rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 1/10, β = −1/10, d1 = 1, and d2 = d3 = 0.

Fig. 3.

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	Fig. 3. Evolution of the first-order breather–rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 1/100, β = −1/100, d1 = 1, d2 = 1, and d3 = −0.5.

Here, we give a classification about values of parameters α, β, and d_i (i = 1,2,3) corresponding to different types of first-order local wave solutions.

Case 1 When α ≠ 0 and β ≠ 0, the solutions q_i(i = 1,2,3) are all first-order rogue wave (RW).

Case 2 One of these two parameters α and β is 0, for convenience, we consider the case of α = 0, β ≠ 0. The classification is shown in Table 1.

Table 1.

Table 1.

Table 1. Classification of first-order local wave solutions generated by the first-step generalized DT. . di q1 q2 q3 d1 ≠ 0, d2 = d3 = 0 RW and one-dark soliton 0 RW and one-bright soliton d1 = 0,d2 ≠ 0, d3 =0 0 RW 0 d1 = 0,d2 = 0,d3 ≠ 0 RW and one-bright soliton 0 RW and one-dark soliton d1 ≠ 0, d2 ≠ 0, d3 = 0 RW and one-dark soliton RW and one-dark soliton RW and one-bright soliton d1 ≠ 0, d2 = 0,d3 ≠ 0 RW and one-breather 0 RW and one-breather d1 = 0,d2 ≠ 0,d3 ≠ 0 RW and one-bright soliton RW and one-dark soliton RW and one-dark soliton d1 ≠ 0,d2 ≠ 0,d3 ≠ 0 RW and one-dark soliton RW and one-breather RW and one-breather

Table 1. Classification of first-order local wave solutions generated by the first-step generalized DT. .

Case 3 When α ≠ 0 and β ≠ 0, the classification is shown in Table 2.

Table 2.

Table 2.

Table 2. Classification of first-order local wave solutions generated by the first-step generalized DT. . di q1 q2 q3 d1 ≠ 0, d2 = d3 = 0 RW and one-dark soliton RW and one-bright soliton RW and one-bright soliton d1 = 0,d2 ≠ 0, d3 = 0 RW and one-bright soliton RW and one-dark soliton RW and one-bright soliton d1 =0,d2 =0,d3 ≠ 0 RW and one-bright soliton 0 RW and one-dark soliton d1 ≠ 0, d2 ≠ 0, d3 = 0 RW and one-breather RW and one-breather RW and one-bright soliton d1 ≠ 0, d2 =0,d3 ≠ 0 RW and one-breather RW and one-bright soliton RW and one-breather d1 = 0,d2 ≠ 0,d3 ≠ 0 RW and one-bright soliton RW and one-dark soliton RW and one-dark soliton d1 ≠ 0,d2 ≠ 0,d3 ≠ 0 RW and one-breather RW and one-breather RW and one-breather

Table 2. Classification of first-order local wave solutions generated by the first-step generalized DT. .

Next, we consider the following limit:

Fig. 4.

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	Fig. 4. Evolution of the second-order rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 0, β = 0, d1 = 1, d2 = −1, d3 = 2, m1 = 100, and n1 = 0.

Fig. 5.

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	Fig. 5. Evolution of the second-order dark–bright–rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 1/1000, β = −1/1000, d1 = 1, d2 = 0, d3 = 0, m1 = 0, and n1 = 0.

Fig. 6.

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	Fig. 6. Evolution of the second-order dark–bright–rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 1/1000, β = −1/1000, d1 = 1, d2 = 0, d3 = 0, m1 = 100, and n1 = 0.

Fig. 7.

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	Fig. 7. Evolution of the second-order dark–bright–rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 10, β = −10, d1 = 1, d2 = 0, d3 = 0, m1 = 100, and n1 = 0.

Fig. 8.

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	Fig. 8. Evolution of the second-order breather–rogue wave of the three-component NLS equation (a) q1, (b) q2, (c) q3 by choosing α = 1/100, β = −1/100, d1 = 1, d2 = −1, d3 = −1, m1 = 100, and n1 = 0.

In Eq. (11), there are two important parameters α and β. In Ref. [22], except for the parameters d[1] and d[2], there is only one parameter α. Thus these two parameters, α and β here, determine greatly different structures of localized waves in three-component NLS equation.

We give some interesting localized waves of three-component NLS equation by the generalized Darboux transformation. With a fixed spectral parameter and a special vector solution of Lax pair of Eqs. (2) and (3), we apply the Taylor series expansion to Eq. (1), and give the generalized Darboux transformation.^[34] Applying the formula (7)–(9),^[35] the interactions between rogue wave and some nonlinear waves (dark, bright solitions and breather) are obtained. In the expressions of these solutions, some parameters play an important role in dynamic properties, such as α, β, d_i (i = 1,2,3), and s₁.

We mainly discuss the dynamics of these solutions through three different cases. (i) When α = 0 and β = 0, the first- and second-order rogue wave are given, which are similar to one-component and two-component NLS equation. (ii) When α ≠ 0, β ≠ 0, d₁ ≠ 0, d₂ = 0, and d₃ = 0, the first-order one-dark-rogue and one-bright-rogue wave can be gained. Meanwhile, the second-order one-dark-one-bright-rogue wave and two-bright-rogue wave are also presented. The parameter s₁ determines the shape of rogue wave, such as fundamental pattern and triangular pattern. (iii) When α ≠ 0, β ≠ 0, and d_i ≠ 0 (i = 1,2,3), the first- and second-order one-breather–rogue wave are observed.^[36] With increasing the absolute values of α and β, we can observe that rogue wave and those other nonlinear waves merge distinctly.

The localized waves of three-component coupled NLS equation are not absolutely identical with ones of two-component coupled NLS equation.^[22] Second-order one-dark-one-bright-rogue wave can be obtained in q[1] component, instead of second-order two-dark-rogue wave in the two-component case. Furthermore, we get second-order rogue wave which contains four fundamental ones and this type of rogue wave interacts with one-dark-one-bright soliton, which is different with the case of two-component. We can only get one-breather–rogue wave solution, which is not the two-breather–rogue wave ones in two-component NLS equation. Through considering both two-component and three-component NLS equation, we may well understand the localized waves of the multi-component NLS equation.^[37]