Darboux Transformation, Soliton Solutions of a Generalized Variable Coefficients Hirota Equation

It is known that the variable coefficients Hirota equations have been widely studied in the amplification or absorption of propagating pulses, as well as in the generation of supercontinuum in inhomogeneous optical fibers. In this paper, a generalized variable coefficients Hirota equation is considered. Firstly, we constructed the classical and generalized Darboux transformations of the equation. Next, we obtained multisoliton solutions based on the classical Darboux transformation and rogue wave solutions using the generalized Darboux transformation. Finally, we discussed the evolutions of solitons.

DT is an effective method to obtain a new solution from the initial solution, and it can be repeated any number of times.The main idea of DT approach is to prove the canonical equivalence of the Lax pairs and to obtain soliton solutions through continuous iteration.To construct the explicit solutions, Gu and his collaborators constructed a classical DT in matrix form and provided purely algebraic algorithms for a group of isospectral integrable systems [15,27].To obtain the rogue waves of nonlinear Schrödinger equation, Guo et al. [28] derived a generalized DT through a limit procedure.These methods have also been extended to study the variable coefficients and nonlocal equations.
For the problem at hand, we focus on studying a generalized variable coefficients Hirota equation where i = √ −1 and u is a complex function with the variables (t, x), α = α(t), β = β(t), δ = δ(t) are real functions with variable t and the parameter γ is a nonzero constant.The significance of the study for this equation is that it is often associated with the amplification or absorption of propagating pulses and the generation of supercontinuum in inhomogeneous optical fibers [29][30][31].In optical fibers, α, β, γ , δ represent the group dispersion velocity, third order dispersion, self-steepening and the amplification or absorption coefficient respectively [32].For different value of α, β, γ , δ, the amplitude, intensity, width and period of the oscillation show different results.We constructed the classical and generalized DTs of Eq. ( 1) and obtained the multisolutions and rogue wave solutions.The evolutions of solutions are discussed.The propagation of solitons can be controlled by adjusting the values of relevant parameters.The results might be of potential applications in the design of optical communication systems.Some related works associated with (1) have been researched.The auto-Bäcklund transformation and a family of the analytic solutions has also been given, see [33].When α, β, δ are all constants, multisolitons, breathers and rogue waves have been derived, see [34][35][36][37].When α = δ = 0, β = β(t), γ = γ (t), the multisoliton solutions have been obtained, see [38][39][40][41].When β = 0, α = α(t), δ = δ(t), γ = γ (t), multisoliton solutions, rogue wave solutions, semi-rational solutions, breathers are obtained, please see [42][43][44][45][46] for details.
The paper is organized as follows.In section "Lax Pair and Darboux Transformation", we derive the Lax pair, classical DT and generalized DT of Eq. (1).In section "Multisoliton Solutions", we use the classical DT to obtain multisoliton solutions from the zero seed solution.In section "Rogue Wave Solutions", we use the generalized DT to obtain the rogue wave solutions from the non-zero seed solution.Finally, the main results are summeried.

Lax Pair and Darboux Transformation
In this section, we will derive the Lax pair and DTs of Eq. (1) which include classical DT and generalized DT.The Lax pair of soliton equation means that the equation can be written as a pair of linear problems.The DTs build the relationships between the seed solution u[0] and the new solution u[N ].

Theorem 1
The Lax pair for the generalized variable coefficients Hirota equation (1) can be expressed as follows where U , V are the matrices determined by u and u * with isospectral parameter λ (* denotes the complex conjugate), Proof According to the compatibility condition, i.e. ϕ xt = ϕ t x , of Eq. ( 2), we can obtain zero-curvature equation Substituting Eqs. ( 4) and (1) into Eq.( 5), we can verify the validity of Therefore, we can give the Lax pair for Eq.(1).

Darboux Transformation
Taking j = 1, 2, 3 . .., we assume u[ j] is j soliton solution of Eq. ( 1), ϕ[ j] be j solution of the Lax pair (2) at u[ j] and T [ j] is a gauge transformation between ϕ[ j − 1] and ϕ[ j] which represents the transformation relationship between two sets of solutions of Lax pair.The iteration process of the DT is described using a flowchart Theorem 2 The N-fold classical DT of the generalized variable coefficient Hirota equation Here the gauge transformation T Proof (1) Gauge transformation Assuming ϕ satisfies the Lax pair Eq. ( 2) and ϕ satisfies the Lax pair here U , V have the same forms with U , V except that u, u * in the matrices U , V are replaced with u , u * in the matrices U , V .If we set and call T a gauge transformation.We can obtain the gauge transformation T satisfies Substituting T = λI − S into Eqs.( 10) and ( 11), we have Setting ( f , g) T is a solution of the Lax pair (2) at λ = λ 0 , we see that satisfies Eqs. ( 12) and ( 13).Then we find the gauge transformation of the Lax pair (2), Denoting S = s 11 s 12 s 21 s 22 , and comparing the coefficients of λ in (12), the following relationship between two sets of solutions in Eq. (1) will be derived, Based on Eq. ( 14), we can see Substituting Eq. ( 17) into Eq.( 16), we obtain (2) One-fold classical DT . We can use Eqs.( 14) and ( 15) to obtain the gauge transformation ) According to Eqs. ( 15) and ( 18), we can obtain the one-fold classical DT where k = 0, 1.By means of Eqs. ( 14) and ( 15), the gauge transformation 20), we get and By Eq. ( 18), we derive Substituting Eq. ( 20) into Eq.( 24), we obtain the two-fold classical DT where k = 0, 1, 2. From Eqs. ( 14) and ( 15), the gauge transformation Utilizing Eq. ( 25), we get and By using Eq. ( 18), we observe Substituting Eq. ( 25) into Eq.( 29), we obtain the three-fold classical DT where k = 0, 1, . . ., N − 1. Continuing the above iteration process, we obtain The relationship between We obtain the N-fold classical DT Theorem 3 The N-fold generalized DT of the generalized variable coefficients Hirota equation (1) is Here the gauge transformation T 1 where .
Proof From Eqs. ( 14) and ( 15), we see that if λ = λ 0 , the gauge transformation It means that the same solution can not be reused in the iteration of DT.In the generalized DT, we consider the case of λ = λ 1 + ε, where ε is a small complex parameter.Assuming ϕ 1 (λ 1 + ε) is a solution of Lax pair (2) at λ = λ 1 + ε and u = u[0], and it can be expanded at ε = 0 as the following Taylor series where (1) One-fold generalized DT The one-fold generalized DT of Eq. ( 1) is the same as the one-fold classical DT.The gauge transformation ) and the one-fold generalized DT (2) Two-fold generalized DT From the classical DT (7), we see that at λ = λ 1 + ε and u = u [1], and so is , and can calculate that where By means of Eqs. ( 14) and ( 15), we get the gauge transformation (41) Combining with Eqs. ( 16) and ( 39), we find the two-fold generalized DT (3) Three-fold generalized DT Similarly, we see that (T 1 [2]T 1 [1])| λ=λ 1 +ε ϕ 1 (λ 1 +ε) is a solution of Lax pair (2) at λ = λ 1 +ε and u = u [2], and so is is a solution of Lax pair (2) at λ = λ 1 and u = u [2].We write , and can calculate that where Then we have the gauge transformation and the three-fold generalized DT (4) N-fold generalized DT Continuing the above process, we see that lim and can calculate that Then the gauge transformation and the N-fold generalized DT have been established.

Multisoliton Solutions
In order to obtain the multisoliton solutions for Eq. ( 1), we start from the seed solution u[0] = 0. Substituting u[0] = 0 into Lax pair (2), we get Through direct calculation, the solution ϕ From Eq. ( 20), if we take λ 1 = 1 + 2i, we find the one-soliton solution The solution Substituting Eqs. ( 22) and ( 52) into (25) and taking λ 2 = 2 + 3i, we get the two-soliton solution Next, we will discuss the evolutions of the soliton solutions and show the relationship between solitons and the group dispersion velocity α, third order dispersion β and the amplification or absorption coefficient δ.In  In Fig. 1, the soliton structure oscillates periodically because third order dispersion coefficient β is a trigonometric function.In Fig. 2, the image of the soliton solution has an upper convex shape and converges at x = 0 because third order dispersion coefficient β is an exponential function.In Fig. 3, the soliton with variable propagation velocities illustrates the non-travelling-wave characteristics because third order dispersion coefficient β  51) and ( 53) is of parabolic-type.In Figs.1c, 2c and 3c, the head-on interactions form a peak at each interaction region between the two solitons, respectively.In Figs. 4, 5 and 6, we can see that the image of the soliton solutions are related to the characteristics of the group dispersion velocity coefficient α.In Fig. 7, the soliton image is affected by several nonzero parameters including the group dispersion velocity coefficient α, third order dispersion coefficientβ, self-steepening coefficientγ , and amplification or absorption coefficient δ.

Rogue Wave Solutions
In this section, we will derive the rogue wave solutions for Eq.(1).we start with the seed solution u[0] = e i (α(t)γ +δ(t))dt .By combining it with Eqs. ( 39) and (42), we can describe one-rogue wave solution and two-rogue wave solution as follows here we take λ 1 = −i.
Then we will discuss the evolutions of the rogue wave solutions and show the relationship between rogue waves and the group dispersion velocity α, third order dispersion β and the amplification or absorption coefficient δ.In Figs. 8, 9 and 10, (a) and (b) represent threeand two-dimensional plots of one-rogue wave solution depicted by Eq. (54) respectively.(c) and (d) represent three-and two-dimensional plots of two-rogue wave solution depicted by Eq. (55) respectively.Figure 11 show the evolutions of one-rogue wave solution with different values of α, β.
In Figs. 8, 9 and 10, we take the group dispersion velocity coefficient α, third order dispersion coefficientβ, amplification or absorption coefficient δ as t respectively, and observe the evolutions of the rogue wave solutions.In Fig. 11a, b, the one-rogue wave increases in amplitude and rotates counterclockwise as the third-order dispersion coefficient β increases.In Fig. 11a, c, the shape and width of the one-rogue wave change as the group dispersion velocity coefficient α take different functions.

Conclusion
In summary, we have studied the generalized variable coefficients Hirota equation Eq. ( 1) and derived its Lax pair and DTs.Specifically, we have constructed a classical DT and got multisoliton solutions.Furthermore, rogue wave solutions also have been proposed explicitly by generalized DT.Finally, we have analyzed the dynamical features of these exact solutions.We believe that the results could be valuable in solving the inhomogeneity problems in optical fibers and plasma.