In recent years, nonlinear localized waves, including solitons,^[1–3] rogue waves,^[4–7] and breathers,^[8–10] have been one of the intense studies in the field of nonlinear science. Many attentions have been focused on the common solitons including bright and dark solitons. Owing to the instability of small amplitude perturbations, the breathers may develop and even grow in size to disastrous proportions. In much current literature, there are mainly two kinds of breathers such as Ma breathers (time-periodic breather solutions)^[11] and Akhmediev breathers (space-periodic breather solutions).^[4,12] While the rogue waves are modeled as transient wave packets localized in both time and space, they reveal a unique phenomenon that seems to appear from nowhere and disappear without a trace.^[13] Up to now, the authentic interpretations of the generation mechanism of the rogue waves are modulation instability and rational function solution.^[14] The rogue wave phenomenon appears in a class of fields, among them nonlinear optics,^[15,16] capillary flow,^[17] superfluidity,^[18] Bose–Einstein condensates,^[19] plasma physics,^[20] and even finance.^[21]

Recent studies have been extended to localized wave solutions including rogue waves and some other nonlinear wave in various nonlinear systems.^[22–24] In Ref. [25], the hybrid solutions consisted of rogue waves and cnoidal periodic waves in the focusing NLS equation were constructed by the Darboux transformation (DT) scheme. Using the Hirota bilinear method, the authors obtained the rogue wave triggered by the interaction between lump soliton and a pair of resonance kink stripe solitons.^[26,27] As one special type of interactional solutions, the semi-rational solution exhibits a range of abundant and appealing dynamics in nonlinear models,^[28] such as dark–bright–rogue wave pair,^[29,30] rogue wave interacting with solitons and breathers.^[31–33] In many cases, it shows that these semi-rational solutions appear as a mixture of polynomials with exponential functions. In Ref. [34], a new dark–antidark soliton pair solutions and some special semi-rational solutions of the coupled Sasa–Satsuma equations were discussed by DT. Utilizing the Hirota bilinear method, the authors constructed a new kind of semi-rational solution that the rogue waves interact with solitons and breathers at the same time in the Boussinesq equation.^[35] These results further uncover some striking dynamic structures in nonlinear models.

It is necessary to generate shorter (femtosecond, even attosecond) pulses with high frequency in fibre to meet the demand for high bit rates in optical communication. In the field of ultra-short pulses, where the width of optical pulse is in the order of femtosecond (10⁻¹⁵ s) and the spectrum of these ultrashort pulses is approximately of the order 10¹⁵ s⁻¹, the standard NLS equation is less accurate. This boosts greatly the study of higher order nonlinear effects in optics, which are modeled by various higher order NLS systems. In this paper, our aim is to investigate the localized waves of the following coupled cubic–quintic nonlinear Schrödinger (CCQNLS) equations^[36–39] 1 which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. For convenience, the independent variables z and t of the CCQNLS equations in Refs. [36]–[39] are replaced by variables t and x in the CCQNLS system (1), respectively. The two components q₁ and q₂ are the complex smooth envelop functions and they denote that two electromagnetic fields propagate along the coordinate z (or t) in the two cores of an optical waveguide, and t (or x) is the local time. Besides, each non-numeric subscripted variable stands for partial differentiation and the asterisk denotes complex conjugation. The parameters ρ₁ and ρ₂ are all real constants. When q₁ = u, q₂ = 0, and ρ₁ = 2β, the CCQNLS system (1) can turn into the integrable Kundu–Eckhaus equation.^[40] There have been many reports on the Kundu–Eckhaus equation,^[41] such as its Hamiltonian structure,^[42] solitons solutions,^[43] and rogue wave solutions.^[44,45] Recently, the multi-soliton solutions and the bound states of the solitons of the CCQNLS system (1) were discussed in Refs. [36] and [37]. In Ref. [38], the authors constructed the generalized DT of the CCQNLS system and obtained its rogue wave solutions. Besides, the DT of the multi-component CCQNLS system were obtained and its rogue waves were also constructed.^[39]

Here, we are interested in the hybrid solutions between rogue waves and some other nonlinear wave solutions in the CCQNLS system (1), for example, multi-dark, multi-bright soliton, and multi-breather. To the best of our knowledge, there are no reports on this kind of interactional solution of the CCQNLS system up to the present. Baronio et al.^[46] obtained some semi-rational solutions in the coupled NLS, which include a first-order rogue wave, a first-order rogue wave interacting with a bright soliton, a dark soliton, and a breather, respectively. However, the higher-order interactional solutions cannot be generated by Baronio’s method. Based on the generalized DT^[14,47] and the special vector solution of the Lax pair of the three-component coupled NLS system, we obtained its higher-order interactional solutions successfully.^[31] Starting from an appropriate periodic seed solution, a special vector solution of the Lax pair of the system (1) is elaborately constructed. Based on this kind of special vector solution, some abundant higher-order localized waves of the CCQNLS equations are generated by the generalized DT. In particular, the first- and second-order semi-rational localized wave solutions are discussed in detail. These semi-rational localized wave solutions are discussed in three cases: the first one is the first- and second-order vector rogue wave solutions; the second one is the hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; the last one is the interactional solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. These interesting and appealing solutions will further reveal some striking dynamic structures in the CCQNLS system.

Our paper is organized as follows. In Section 2, we construct the generalized DT to Eq. (1), then the first- to fourth-step generalized Darboux transformations are obtained by using the direct iterative rule. In Section 3, some nonlinear localized wave solutions of Eq. (1) are obtained. In particular, the first- and second-order localized waves are discussed in detail and some figures of these kinds of localized wave solutions are also exhibited. In Section 4, we give some conclusions.

In this section, we construct the generalized DT^[14,47] to the CCQNLS system (1). The Lax pair of the system (1) can be expressed as^[38,39] 2 where Ψ = (ψ(x,t),φ(x,t),χ(x,t))^T with T denoting the transpose of the vector, while U and V are all the 3 × 3 matrices and they can be given as where and λ is the spectral parameter. Besides, the CCQNLS system (1) can be derived from the compatibility condition U_t − V_x+[U,V] = 0 through symbolic computation, where the brackets represent the matrix commutator.

Let Ψ₁ = (ψ₁,φ₁,χ₁)^T be a special solution of the Lax pair (2) with q₁ = q₁[0], q₂ = q₂[0], and λ = λ₁, then based on the DT constructed in Refs. [36]–[38], we give the first-step elementary DT of the CCQNLS equations (1) 3 4 5 where the symbol † denotes the transpose and complex conjugation of the vector.

According to the above elementary DT expressions (3)–(5), we construct the generalized DT to the CCQNLS equations(1). Let Ψ₁ = (ψ₁,φ₁,χ₁)^T = Ψ₁(λ₁+δ) be a special vector solution of the Lax pair (2) with q₁ = q₁[0], q₂ = q₂[0], λ = λ₁+δ, and δ is a small parameter. Next, Ψ₁ can be expanded as the Taylor series at δ = 0 6 where

By means of the formulas (3)–(6), the first-step generalized DT of Eq. (1) can be directly constructed. In the following content of this section, we use these notations , , and , (j = 1,2,3,...,N).

(i) The first-step generalized DT 7 8 9 10 where Ψ₁[0] = = (ψ[0],ϕ₁[0],χ₁[0])^T, Γ1[1]=(eη1[1]e−η1[1]e−η1[1]),  Λ1=(λ1λ1*λ1*),H[0]=(ψ1[0]ϕ1[0]*χ1[0]*ϕ1[0]−ψ1[0]*0χ1[0]0−ψ1[0]*), Ω1=|ψ1[0]|2+|ϕ1[0]|2+|χ1[0]|2,η1[1]=∫(λ1−λ1*) Ω1−2Δ1 dx,Δ1ρ1[ (q1[0]ϕ1[0]ψ1[0]*+q1[0]*ϕ1[0]*ψ1[0])Ω1+2i(λ1*−λ1)|ψ1[0]|2|ϕ1[0]|2]+ρ2[ (q2[0]χ1[0]ψ1[0]*+q2[0]*χ1[0]*ψ1[0])Ω1+2i(λ1*−λ1)|ψ1[0]|2|χ1[0]|2].

(ii) The second-step generalized DT

It is shown that T[1]Ψ₁ is a vector solution of the Lax pair (2) with q₁ = q₁[1], q₂ = q₂[1], and λ = λ₁ + δ. Considering the following limit 11 a nontrivial solution of the Lax pair (2) with q₁ = q₁[1], q₂ = q₂[1], and λ = λ₁ can be obtained. Here, we have used the above generalized DT one time and also utilized the identity . Then the second-step generalized DT holds 12 13 14 15 where and

(iii) The third-step generalized DT.

In a similar way, by using the following limit 16 the special vector solution of the Lax pair (2) with q₁ = q₁[2], q₂ = q₂[2], λ = λ₁ can be derived. Besides, the following two identities have been applied in the above process. Then the third-step generalized DT can be obtained as 17 18 19 20 where Ψ₁[2] = (ψ₁[2],φ₁[2],χ₁[2])^T and T₁[2] = T₁[2]|_{λ = λ1} = λ₁I − H[1]Λ₁H[1]⁻¹,

(iv) The fourth-step generalized DT.

In addition, the special vector solution of the Lax pair (2) with q₁ = q₁[3], q₂ = q₂[3], and λ = λ₁ can be presented as follows: 21 by using the following three identities Continuing the above process, we can construct the fourth-step generalized DT as follows: 22 23 24 25 where Ψ₁[3] = (ψ₁[3],φ₁[3],χ₁[3])^T and T₁[3] = T₁[3]|_{λ = λ1} = λ₁I−H[2]Λ₁H[2]⁻¹,

Owing to the complexity and irregularity of the expressions of Ψ₁[j], such as Eqs. (16) and (21), the unified formula of the N-step generalized DT of the CCQNLS equations cannot be given easily. Combining the special vector solution of the Lax pair (2) and the higher-order generalized DT, we can obtain the corresponding higher-order localized wave solutions consisting of higher-order rogue wave, higher-order rogue wave interacting with multi-soliton or multi-breather.

In this section, some nonlinear localized wave solutions^[31] of the CCQNLS equations (1) are constructed through the above generalized DT. Here, the first- and second-order localized waves are discussed in detail and some figures of these kinds of localized wave solutions are also exhibited. Besides, some dynamic structures of these nonlinear waves are demonstrated.

3.1. The first-order localized wave solutions

In the following, we choose a nontrivial seed solution of the CCQNLS equations (1) 26 where , d₁ and d₂ are all real constants (d₁ ≠ d₂). For convenience, the above seed solutions are chosen periodically in time variable t without depending on space variable x. Choosing q₁ = q₁[0], q₂ = q₂[0], the special vector solutions of the Lax pair (2) can be elaborately expressed as 27 where here, m_k, n_k, and α are all real free parameters. Choosing the spectral parameter with a small real parameter ε, the Taylor expansion of the above vector function Ψ₁ at ε = 0 can be derived as 28 where and with

It can be found that the vector function is a solution of the Lax pair (2) at q₁ = q₁[0], q₂ = q₂[0], and . Through the first-step generalized DT (8)–(10), we can gain the first-order localized wave solutions of the CCQNLS system (1). Owing to the complicated integral operation existing in , we choose to obtain the expressions of the first-order semi-rational solutions of Eq. (1) after fixing the values of the free parameters. According to different values of the free parameters α, d₁, and d₂, we can obtain three types of the first-order nonlinear localized waves in the CCQNLS equations.^[31,32]

i) When α = 0, in order to gain the nontrivial solutions of Eq. (1), we set that d₁ and d₂ are all not zero, hereby, the components q₁ and q₂ are proportional to each other. Here, the first-order semi-rational solutions degenerate to the rational ones and we can obviously find they are the first-order rogue waves, see Fig. 1. Choosing d₁ = 1, d₂ = −2, ρ₁ = 1/10, and ρ₂ = 1/20, the expression of the first-order rogue wave solutions of Eq. (1) can be written as 29 30

Fig. 1.

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	Fig. 1. (color online) Evolution plot of the first-order rogue wave in the CCQNLS: (a) q1,(b)q2.

ii) Setting one of the two free parameters d₁ and d₂ is zero and α ≠ 0, we can obtain the first kind of the second-order semi-rational solutions that a first-order rogue wave interacting with a bright soliton and a dark soliton respectively in the two components q₁ and q₂. Choosing d₁ = 0, d₂ = 1, ρ₁ = 1/10, and ρ₂ = 1/20, the first kind of the first-order semi-rational solution of Eq. (1) can be expressed as 31 32 where

Here, the interesting interactional phenomenon can be demonstrated in Figs. 2 and 3. Moreover, by decreasing the absolute value of α, it can be shown that a first-order rogue wave and a bright (dark) soliton separate in Fig. 2. While increasing the absolute value of α, it demonstrates the first-order rogue wave merges with one soliton in Fig. 3. We notice that in Fig. 2(b), a dark soliton and a rogue wave emerge on the distribution of the spacial-temporal structure, and the amplitude of the rogue wave is about three times greater than the height of the background. However, when the bright soliton and the rogue wave divide in Fig. 2(a), we can find that the rogue wave cannot be easily identified. At this time, the amplitude of the plane wave background in q₁ component is zero and the amplitude of the rogue wave is dependent on this background, so the rogue wave is not observed easily.

Fig. 2.

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	Fig. 2. (color online) Evolution plot of the interactional solution between the first-order rogue wave and one-soliton in the CCQNLS equations with the parameters chosen by α = 1/10. (a) A first-order rogue wave and a bright soliton separate in q1 component; (b) a first-order rogue wave and a dark soliton separate in q2 component.

Fig. 3.

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	Fig. 3. (color online) Evolution plot of the interactional solution between the first-order rogue wave and one-soliton in the CCQNLS equations with the parameters chosen by α = 10. (a) A first-order rogue wave merges with a bright soliton in q1 component; (b) a first-order rogue wave merges with a dark soliton in q2 component.

iii) Setting d₁ ≠ 0, d₂ ≠ 0, and α ≠ 0, the second kind of the first-order semi-rational solutions consisting of one first-order rogue wave and one breather can be generated in both q₁ and q₂ components. Choosing d₁ = 1, d₂ = −2, ρ₁ = 1/10, and ρ₂ = 1/20, the second kind of the first-order semi-rational solution of Eq. (1) can be written as 33 34 where It is shown that the two components q₁ and q₂ are all the hybrid solutions between one breather and one first-order rogue wave in Figs. 4 and 5. Similarly, by decreasing the absolute value of α, we can observe that the first-order rogue wave and one breather separate in Fig. 4. Meanwhile through increasing the absolute value of α, the first-order rogue wave and one breather merge in Fig. 5.

Fig. 4.

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	Fig. 4. (color online) Evolution plot of the interactional solution between the first-order rogue wave and one-breather in the CCQNLS equations with the parameters chosen by α = 10−4. A first-order rogue wave and a breather separate in the two components: (a) q1 component, (b) q2 component.

Fig. 5.

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	Fig. 5. (color online) Evolution plot of the interactional solution between the first-order rogue wave and one-breather in the CCQNLS equations with the parameters chosen by α = 1/10. A first-order rogue wave merges with a breather in the two components: (a) q1 component, (b) q2 component.

3.2. The second-order localized wave solutions

In this subsection, we construct the second-order localized wave solutions of the CCQNLS equations (1), which include the second-order rogue wave, two dark (bright) solitons interacting with the second-order rogue wave, and two parallel breathers together with the second-order rogue wave.^[32,33]

Considering the following limit 35 where 36 a special vector solution Ψ₁[1] is generated through choosing q₁ = q₁[1], q₂ = q₂[1], and in the Lax pair (2).

By virtue of the above formulae (37) and (38), we can obtain concrete expressions of the modules for the second-order localized wave solutions. Here, we omit presenting the expressions since they are rather cumbersome to write down. According to different values of the free parameters α, d₁, and d₂, we can also obtain three kinds of the second-order nonlinear localized waves of the CCQNLS system (1).^[32,33]

(I) α = 0, d₁ ≠ 0, and d₂ ≠ 0.

Here, the second-order semi-rational solutions degenerate to the rational ones, thus, the second-order rogue wave of the CCQNLS equations can be obtained. When s₁ = 0, the second-order rogue wave is fundamental pattern, see Fig.6; while s₁ ≠ 0, the the second-order rogue wave is triangular pattern, see Fig. 7.

Fig. 6.

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	Fig. 6. (color online) Evolution plot of the second-order rogue wave of fundamental pattern in the CCQNLS equations with the parameters chosen by α = 0, ρ1 = 1/10, ρ2 = 1/20, d1 = 1, d2 = −2, m1 = 0, n1 = 0: (a) q1 component,(b) q2 component.

Fig. 7.

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	Fig. 7. (color online) Evolution plot of the second-order rogue wave of triangular pattern in the CCQNLS equations with the parameters chosen by α = 0, ρ1 = 1/10, ρ2 = 1/20, d1 = 1, d2 = −2, m1 = -100, n1 = 100: (a) q1 component,(b) q2 component.

(II) α ≠ 0, d₁ = 0, and d₂ ≠ 0.

At this point, the first kind of the second-order semi-rational solutions between two dark (bright) solitons and one second-order rogue wave can be given. Figure 8(a) shows two bright solitons together with a fundamental second-order rogue wave, besides, figure 8(b) demonstrates two dark solitons coexisting with a fundamental second-order rogue wave.

Fig. 8.

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Fig. 8. (color online) Evolution plot of the interactional solution between the second-order rogue wave of fundamental pattern and two-soliton in the CCQNLS equations with the parameters chosen by α = 1/200, ρ1 = 1/10, ρ2 = 1/20, d1 = 0, d2 = 1, m1 = 0, n1 = 0. (a) Two bright solitons together with a fundamental second-order rogue wave in q1 component; (b) Two dark solitons together with a fundamental second-order rogue wave in q2 component.

In Fig. 8(a), when the two bright solitons interact with the second-order rogue wave, the second-order rogue wave in q₁ component is not observed easily for the same reason as the first-order case. When setting s₁ ≠ 0, the fundamental second-order rogue wave can split into three first-order rogue waves, in addition, the second-order rogue wave of triangular pattern and two solitons separate in Fig. 9. By increasing the absolute value of α, the two dark (bright) solitons merge with the second-order rogue wave of triangular pattern, see Fig. 10.

Fig. 9.

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Fig. 9. (color online) Evolution plot of the interactional solution between the second-order rogue wave of triangular pattern and two-soliton in the CCQNLS equations with the parameters chosen by α = 1/200, ρ1 = 1/10, ρ2 = 1/20, d1 = 0, d2 = 1, m1 = 50, n1 = −50. (a) Two bright solitons and a second-order rogue wave of triangular pattern separate in q1 component; (b) Two dark solitons and a second-order rogue wave of triangular pattern separate in q2 component.

Fig. 10.

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Fig. 10. (color online) Evolution plot of the interactional solution between the second-order rogue wave of triangular pattern and two-soliton in the CCQNLS equations with the parameters chosen by α = 100, ρ1 = 1/10, ρ2 = 1/20, d1 = 0, d2 = 1, m1 = 50, n1 = −50. (a) Two bright solitons merge with a second-order rogue wave of triangular pattern in q1 component; (b) Two dark solitons merge with a second-order rogue wave of triangular pattern in q2 component.

(III) α ≠ 0, d₁ ≠ 0, and d₂ ≠ 0.

Hence, we arrive at the second kind of the second-order semi-rational solutions between two breathers and a second-order rogue wave in the two components q₁ and q₂, see Figs. 11–13. Meanwhile, setting s₁ ≠ 0, the fundamental second-order rogue can split into three first-order rogue waves, see Figs. 11 and 12. By decreasing the absolute value of α, it shows that the two parallel breathers and the second-order rogue of triangular pattern separate in Fig. 12. However, increasing the absolute value of α, it demonstrates that the two parallel breathers merge with the second-order rogue wave in Fig. 13.

Fig. 11.

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	Fig. 11. (color online) Evolution plot of the interactional solution between the second-order rogue wave of fundamental pattern and two-breather in the CCQNLS equations with the parameters chosen by α = 1/1000, ρ1 = 1/10, ρ2 = 1/20, d1 = 1, d2 = −1, m1 = 0, n1 = 0. Two parallel breathers and a fundamental second-order rogue wave exist in the two components: (a) q1 component, (b) q2 component.

Fig. 12.

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Fig. 12. (color online) Evolution plot of the interactional solution between the second-order rogue wave of triangular pattern and two-breather in the CCQNLS equations with the parameters chosen by α = 1/1000, ρ1 = 1/10, ρ2 = 1/20, d1 = 1, d2 = −1, m1 = 50, n1 = 73−50. Two parallel breathers and a fundamental second-order rogue wave separate in the two components: (a) q1 component, (b) q2 component.

Fig. 13.

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	Fig. 13. (color online) Evolution plot of the interactional solution between the second-order rogue wave of triangular pattern and two-breather in the CCQNLS equations with the parameters chosen by α=100, ρ1 = 1/10, ρ2 = 1/20, d1 = 1, d2 = −1, m1 = 50, n1 = −50. Two parallel breathers merge with a fundamental second-order rogue wave in the two components: (a) q1 component, (b) q2 component.

Through discussing the different values of the free parameters d₁, d₂, and α in the the first-order localized wave solutions, we can get various kinds of interactional solutions. Instead of considering various arrangements of the two potential functions q₁ and q₂, we consider the same combination as the same type solution. Hence we can get three types of the first-order localized wave solutions using our method: 1) One first-order rogue wave; 2) one dark or bright soliton together with one first-order rogue wave; 3) one breather interacting with one first-order rogue wave. However, the expressions of the second-order localized waves are very tedious and complicated, we do not give these expressions in the general form. The three types of hybrid solutions which are similar with the first-order case are only demonstrated. Whether the second-order localized waves own more types or not, we cannot draw a firm conclusion now.

In conclusion, choosing a periodic seed solution of Eq. (1), a peculiar vector solution of the Lax pair (2) is elaborately derived. Based on the special vector solution, we present some interesting and appealing nonlinear localized waves in the CCQNLS equations (1) through the generalized DT. The multi-parametric and semi-rational solutions of Eq. (1) are obtained, where some free parameters play an important role in controlling the dynamic properties of these localized nonlinear waves, such as α, d₁, d₂, and s_i (i = 1,2,...,N).^[32,33] The first- and second-order hybrid solutions of the CCQNLS equations are mainly discussed in three cases: I) when α = 0, d₁ ≠ 0, and d₂ ≠ 0, the semi-rational solutions degenerate to the rational ones, e.g., the first- and second-order rogue waves; II) when α ≠ 0, d₁ = 0, and d₂ ≠ 0, the first kind of the higher-order semi-rational solutions are presented, such as hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; III) when α ≠ 0, d₁ ≠ 0, and d₂ ≠ 0, the second kind of the higher-order semi-rational solutions are shown, such as hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers.

Baronio et al.^[46] obtained the first-order semi-rational solutions in the coupled NLS, however, the higher-order interactional solutions are not constructed. We construct the higher-order localized wave solutions of the CCQNLS system (1) through the generalized DT. For one thing, iterating the generalized DT process, we can generate more complicated localized wave solutions possessing more abundant striking dynamics. For another thing, there were many other interactional solutions in other nonlinear models.^[31–33] The rogue waves together with cnoidal periodic waves in the focusing NLS equation are obtained by DT method.^[25] Additionally, a study about rogue waves interacting with solitons and breathers at the same time was published. We will investigate the above two aspects in our future work. Furthermore, we hope that these kinds of nonlinear localized waves of the CCQNLS equations (1) will be verified in physical experiments in the future.