Localized wave solutions to a variable-coefficient coupled Hirota equation in inhomogeneous optical fiber

The first- and second-order localized waves for a variable-coefficient coupled Hirota equation describe the vector optical pulses in inhomogeneous optical fiber and are investigated via generalized Darboux transformation in this work. Based on the equation’s Lax pair and seed solutions, the localized wave solutions are calculated, and the dynamics of the obtained localized waves are shown and analyzed through numerical simulation. A series of novel dynamical evolution plots illustrating the interaction between the rogue waves and dark-bright solitons or breathers are provided. It is found that functions have an influence on the propagation of shape, period, and velocity of the localized waves. The presented results contribute to enriching the dynamics of localized waves in inhomogeneous optical fiber.

Motivated by the above considerations, a variablecoefficient coupled Hirota equation is studied in this work [33]: where q j ( j = 1, 2) is the complex envelope in the electric field, x is the evolution time, t is the propagation distance, and * denotes a complex conjugate. Additionally, α(t), β(t), δ(t), β(t) α(t) , and 1 are the coefficients of group velocity dispersion (GVD), the nonlinear terms referring to self-phase modulation (SPM) and cross-phase modulation (XPM), the third-order dispersion (TOD), nonlinear terms related to self-steepening and delayed nonlinear response, and the gain or absorption modulus, respectively.
Previous research on Eq. (1) has been carried out, in which two types of N th-order rogue wave solutions with different dynamic structures were considered [34]. Optical vector breather solutions were obtained via DT as a symbolic iteration technique [35]. Shi et al. obtained the polynomial wave solutions and the rational wave solutions via a unified method [36]. Yang et al. derived one-and two-fold soliton solutions and oneand two-fold breather solutions [37]. Further, Yang et al. presented one-and two-fold breather-to-soliton conversion conditions [38].
Studies demonstrate that multi-wave interaction enriches the research results of nonlinear evolution equations and produces complementary effects in some coupled or vector systems [39][40][41]. Thus, unlike the above existing research results on Eq. (1), the dynamical characteristics of first-and second-order localized waves will be investigated in this paper. In the present work, the generalized DT is derived, and the dynamics of first-and second-order localized wave solutions are discussed by combining the classical DT and limit methods.
The remainder of this paper is organized as follows. In Sect. 2, the generalized DT is derived, and the higher-order localized wave solutions are obtained. Based on numerical simulation, the evolution plots of the first-and second-order localized waves are given in Sect. 3, and their dynamical characteristics are discussed. Finally, Sect. 4 provides several conclusions.

Generalized Darboux transformation
The generalized DT is an effective method for solving nonlinear evolution equations. In this section, the generalized DT is derived, and the iterative formula of the N th-order solutions is obtained for Eq. (1).
The following Lax pair of Eq. (1) is considered [38]: where where λ is the spectral parameter, = (ϕ, χ, φ) T is the vector solution of Eq. (2), and T denotes the transpose for a vector. It is easy to verify that U and V satisfy the compatibility condition The Darboux matrix T is constructed as follows [42]: where (2) corresponding to the spectral parameter λ=λ 1 , the seed solutions q 1 = q 1 [0] and q 2 = q 2 [0], and I is the identity matrix. Thus, the classical DT is defined as: The generalized DT of Eq. (1) is constructed based on the above classical DT. Assuming 1 = 1 (λ 1 , η) is a solution of Eq. (2) and η is a small parameter, the following Taylor expansion of η=0 is obtained: where It can be easily confirmed that of Eq. (2). Therefore, the generalized DT is defined as follows:

Dynamics of localized waves
In this section, the first-and second-order localized wave solutions of Eq. (1) are calculated, and the dynamics of localized waves are analyzed according to the evolution plots.
Assuming the plane waves q (t) are seed solutions of the localized waves, where and a 1 and a 2 are arbitrary real constants. The corresponding basic vector solution at λ= − 2i 3 a 2 1 + a 2 2 1 + η 2 is calculated as: where v 1 = 3λ − 9λ 2 + 4(a 2 1 + a 2 2 ) 9λ 2 + 4(a 2 1 + a 2 2 ) , v 2 = 3λ + 9λ 2 + 4(a 2 1 + a 2 2 ) Fig. 1 The first-order localized waves with a 1 = 1, a 2 = 0, = 1 10 and (a, d) in which , m j , and n j are arbitrary real constants. Let γ = a 2 1 + a 2 2 and expand function 1 (η) as a Taylor series at η = 0, where As the expression [ 1 T is complicated, its specific form is omitted. The dynamical characteristics of the first-and second-order localized waves are discussed subsequently. Obviously The evolution plots of the first-order localized waves are obtained by altering the values of the free parameters. The dynamics of the first-order localized waves are then discussed. Figure 1 depicts the interactions between first-order rogue waves and dark-bright solitons. When α(t), β(t), and δ(t) are constants, the first-order rogue wave in the component q 1 [1] interacts with one dark soliton and the velocity of the dark soliton remains constant during propagation, as shown in Fig. 1a. When α(t) and β(t) are constants and δ(t) = cos(t) 20 , the first-order rogue wave will interact with the periodic dark soliton, as shown in Fig. 1b. When α(t) and β(t) are variable coefficients and δ(t) = cos(t) 20 , unlike the previous plots, the rogue wave changes into an S-shape, as shown in Fig. 1c. Figure 1d-f shows that the rogue wave in the component q 2 [1] is not easily observed in a background of zero amplitude. Figure 2 shows the collision between a first-order rogue wave and one breather. Figure 2a and d displays the first-order rogue wave coexisting with a parabolic breather when α(t) and β(t) are constants, and δ(t) is a linear function. Figure 2b and e shows the evolution plots of the first-order rogue wave interacting with the periodic breather when α(t) and β(t) are the same as the former and δ(t) is a trigonometric function. Figure 2c and f illustrates the presence of periodic rogue waves, which are observed when α(t) and β(t) are trigonometric functions and δ(t) = t 100 . Moreover, Fig. 2 illustrates that the amplitude of q 1 [1] is greater than the amplitude of q 2 [1], which are both influenced by a 1 and a 2 .
Based on the following limit formula Fig. 3 The second-order localized waves with a 1 = 1, and Eqs. (8) and (9), the second-order localized wave solutions can be obtained as: Similarly, the dynamical characteristics of the secondorder localized wave solutions are analyzed by altering the values of the free parameters in the following cases. Figure 3a and d exhibits the second-order rogue wave interacting with two parabolic dark-bright solitons and displays the parabolic dark-bright solitons' accelerating and decelerating motions when α(t) and β(t) are constants, δ(t) = t 100 , m 1 = 0, and n 1 = 0. When δ(t) = t 2 100 , m 1 = 0, and n 1 = 0, the dark solitons change from parabolic to cubic, as shown in Fig. 3b. Unlike the former, in Fig. 3c, the second-order rogue wave is separated into three first-order rogue waves when m 1 = 30 and n 1 = 30. This result indicates that the amplitude of the separated second-order rogue wave becomes smaller and its energy is lower than that of the second-order one without separation. Furthermore, the rogue wave in the component q 2 [2] is difficult to observe in a background of zero amplitude, as shown in Fig. 3d-f. Figure 4 shows the dynamics of the second-order rogue wave and breathers when α(t) and β(t) are constants and δ(t) is a trigonometric function. Figure 4a and d displays the second-order rogue wave interacting with two periodic breathers when δ(t) = cos(t) 50 , m 1 = 0, and n 1 = 0. Based on the above parameters, the period of the two breathers decreases, and their propagation velocity becomes faster when δ(t) = cos(3t) 50 , as illustrated in Fig. 4b and e. Compared with Fig. 4a and d, b and e indicate that δ(t) has no effect on the amplitude of the breathers and changes the propagation velocity of the breathers. In addition, Fig. 4c and f shows that separation phenomenon occurs in the second-order rogue waves when changing the values of parameters m 1 and n 1 . Figure 5a demonstrates that the second-order rogue wave coexists with the two periodic parabolic dark solitons when α(t) and β(t) are constants and δ(t) = t+cos(5t) 30 . Furthermore, it is found that δ(t) determines the type of dark soliton. The second-order rogue wave and the two dark solitons are periodic when α(t), β(t), and δ(t) are trigonometric functions, and the height of the rogue waves' peak decreases along the positive and negative directions of the x axis (as shown in Fig. 5b). It is hard to observe the second-order rogue when the second-order rogue wave is together with bright solitons, as illustrated in Fig. 5c and d.
When α(t) and β(t) are variable coefficients, δ(t) is a constant, m 1 = 0, and n 1 = 0, the second-order rogue wave and K -shape dark-bright solitons are generated together, as shown in Fig. 6a and c. Figure 6b illustrates that the second-order rogue wave is separated into three first-order rogue waves when the values of m 1 and n 1 are changed. It is also found that the rogue wave is difficult to identify when a 2 = 0 in the component q 2 [2].

Conclusions
This work studied a variable-coefficient coupled Hirota equation by constructing generalized DT on the basis of classical DT and Taylor expansion. The equation was then used to obtain first-and second-order localized wave solutions, whereby localized wave evolution plots were obtained via numerical simulation. It was found that the parameters had an important effect on the dynamics of the localized waves. The parameters a 1 and a 2 had a significant influence on the type of localized waves. If a 1 = 0 and a 2 = 0, the rogue waves coexisted with dark-bright solitons; if a 1 and a 2 were not equal to 0, the rogue waves interacted with breathers. The parameters m j and n j ( j = 1, 2, · · · , N − 1) determined the separation of the rogue waves. When the parameters m 1 and n 1 were not equal to 0, the second-order rogue waves separated into three first-order rogue waves. Moreover, it was observed that the functionsα(t), β(t), and δ(t) influ-enced the propagation shape of localized waves. When α(t), β(t), and δ(t) were constants, common localized waves occurred. When α(t) and β(t) were constants and δ(t) was a linear, quadratic, or trigonometric function, the rogue waves interacted with the parabolic, cubic, or periodic dark-bright solitons and breathers. When α(t), β(t), and δ(t) were trigonometric functions, the localized waves had periodicity all along the propagation direction. These results contribute to the understanding of localized wave propagation in inhomogeneous optical fibers. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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