Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

The rogue wave solutions are discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Using the Darboux matrix, the generalized Darboux transformation is constructed and a recursive formula is derived. Based on the transformation, the first-order to the third-order rogue wave solutions are obtained. Then, the nonlinear dynamics of the first-order to the third-order rogue waves are studied on the basis of some free parameters. Several new structures of the rogue waves are found using numerical simulation.The conclusions will be a supportive tool to study the rogue waves better.


Introduction
For some decades, some researchers have focused on the rogue waves which were firstly used by Draper [1].Rogue waves are also called killer waves, freak waves, giant waves, monster waves, or extreme waves, which appeared in nautical mythology, entered the ocean waves [2][3][4], and gradually moved into other fields, such as optics [5][6][7], matter waves [8], superfluids [9], plasmas [10], atmosphere [11], and even finance [12].The common characteristics and differences in various fields of physics are under discussion [13].The phenomenon of the rogue waves is not fully understood up to now.Therefore, continuous studies on the rogue waves will enrich the concept and bring a full understanding of the mysterious phenomenon.
The first model describing the rogue waves was the focusing nonlinear Schrödinger (NLS) equation which is an integrable system and plays an important role in the study of the rogue waves.As is well known, there are many research results on the rogue waves of (1).The first-order rational solution was derived by Peregrine [14] and the second-order one was obtained using the modified Darboux transformation (DT) [15].The DT is an effective tool for constructing the solution of the integrable systems [16], which came from the work of Darboux on the Sturm-Liouville equation.Recently, the generalized DT [17] was proposed, which is a new and simpler method to construct the higher-order rational solution of the rogue waves of (1).By making use of the generalized DT, Zhaqilao [18] investigated the first-order and the second-order solutions of the rogue waves for the generalized nonlinear Schrödinger equation, Song et al. [19] obtained the higher-order solutions of the rogue waves for the fourth-order dispersive nonlinear Schrödinger equation, and Lv and Lin [20] solved the three coupled higher-order nonlinear Schrödinger equations with the achievement of -soliton formula and derived the lump solutions [21].Nonlinear phenomena can be described mathematically on the basis of the corresponding nonlinear evolution equations, through which we can study the potential nonlinear dynamics.The Heisenberg models of ferromagnetics with 2 Journal of Function Spaces different magnetic interactions in the semiclassical and continuum limits are considered as a class of the nonlinear evolution equations and had a connection with nonlinear spin excitations and the nonlinear Schrödinger equation [22,23].In the Heisenberg ferromagnetism, the nonlinear Schrödinger equation is used to describe the dynamics of nonlinear spin excitation [24].When the ferromagnetic spin chain is site-dependent, the nonlinear Schrödinger equation is modified as the inhomogeneous nonlinear Schrödinger one [25].
In the paper, we will consider an inhomogeneous fifthorder nonlinear Schrödinger equation as follows [26,27]: where (, ) is a complex function and  and  denote the scaled time and spatial coordinate, respectively. and  represent the variation of the bilinear and biquadratic exchange interactions at different sites along the spin chain and where   and   ( = 1, 2) are real constants. is a perturbation parameter and the asterisk is the complex conjugation.Equation ( 2) is generated from the deformation of the inhomogeneous Heisenberg ferromagnetic system with the prolongation structure, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain [28].It is an integrable nonlinear equation.Wang et al. [26] constructed the Darboux transformation and represented oneand two-soliton solutions of (2).
For the second-order to fourth-order nonlinear Schrödinger equations, a lot of research has been done, including the rogue waves [18,19], soliton solutions [29][30][31], modulation instability, integrability [32][33][34], and rational solutions [35].However, to the best of our knowledge, the rogue wave solution for the fifth-order nonlinear Schrödinger equation has never been reported in the literature.
The paper is to obtain th-order rogue waves solutions for the inhomogeneous fifth-order nonlinear Schrödinger equation using the generalized DT.On the basis of the DT matrix, the generalized DT is constructed and the formula is derived which will generate th-order solution of the rogue waves.The nonlinear dynamics of the first-order to the third-order rogue waves are discussed with the influence of some parameters and the interesting structures are showed.

Generalized Darboux Transformation
The Lax pair will ensure the integrability of the nonlinear equations, which is effective in constructing the solutions by using the Darboux transformation.The Lax pair is given as follows [26]: where  and  are 2 × 2 matrices where Φ = (, ) T is an eigenfunction of the Lax pair (4a) and (4b),  is a potential function,  is a constant spectral parameter, and the asterisk denotes complex conjugate.Using the Maple, (2) can be deduced from the compatibility condition, namely, the zero-curvature equation The Darboux transformation consists of the eigenfunction transformation and potential transformation, which is a gauge transformation of the solution for the Lax pair (4a) and (4b). is a Darboux matrix, which will transform the Lax pair (4a) and (4b) into a new one; that is to say, It is known that  [1] and  [1] have the same form as  and  when  and  * in the matrices  and  are substituted by  [1] and  [1]  * in the matrices  [1] and  [1] . [1] and  [1]  are given by  [1] = (  + )  −1 ,  [1] = (  + )  −1 .

(15c)
The initial value is On the basis of the former elementary DT, a generalized DT is derived for (2).We assume that is a special solution of the Lax pair (4a) and (4b) and  is a small parameter.Using the Taylor series, Ψ can be expanded at  = 0 where . ., ).From the above assumption, it is easily seen that Φ [0]  1 = Φ 1 [0] is a particular solution of the Lax pair (4a) and (4b) with  = [0] and  =  1 .Therefore, an th-step generalized DT is constructed by combining all the Darboux matrices where Equations ( 18), (19a), (19b), and (19c) are the recursive formulae of the th-step generalized DT for (2), which are simple to construct the higher-order rogue waves solutions and can be transformed into the 2 × 2 determinant representation making use of the Crum theorem [16].On the basis of the rogue waves solutions of (2), the figures of the first-to the third-order rogue waves will be discussed in the next section.

Rogue Waves Solutions
In order to calculate easily, the rogue waves solutions are derived when the inhomogeneous parameters are independent of ; that is to say,  1 =  1 = 0.The parameters are chosen as  2 = 0.5 and  2 = −25.
It is known that the higher-order solutions of the rogue waves will be obtained with the seed solution and initial eigenfunction.To begin with, the seed solution is equal to [0] =   .The corresponding eigenfunction Φ 1 () for the linear spectral problem at  = ℎ is where and  is a small parameter and Ω() is the separating function, which contains 2 free parameters   ,   ( = 1, . . ., ) and   ,   are real constants.Let ℎ = 1 +  2 and expanding the vector function Φ 1 () at  = 0, we have where 1 Φ [1]  1 = (  [1]   1 ) , Φ [2]  1 = (  [2]   1 and 1 is a solution of the Lax pair (4a) and (4b) at [0] =   and  = .Hence, substituting 1 , [0] =   , and  =  into (12), the first-order rogue wave solution of equation ( 2) is obtained where There is only a free parameter  in the first-order solution, which corresponds to the fifth-order nonlinear effects of (2).If different parameter values are taken, there are different nonlinear phenomena in (2).The picture of the solution is depicted, as shown in Figure 1. Figure 1(a) displays change of the amplitude and Figure 1(b) is the density plot.In Figures 1(a) and 1(b), we have  = 0.In Figures 1(c) and 1(d), there is  = 0.2.Fixing  = 0, the changes of | [1](0, )| in the direction of  axes with the parameters  = 0 are displayed in Figure 1(e).It is observed from Figure 1(e) that the maximum amplitude of | [1](0, )| is 3, which occurs at  = 0.
Then, considering the following limit lim where  1 [1] and Φ 1 [1] can be calculated by Maple.We neglect Φ 1 [1] since its expression is rather cumbersome to write it down.Then, we obtain the second-order rogue wave solution of ( 2) where and  1 ,  1 ,  2 , and  2 are given in Appendix B. In (27), there are three free parameters ,  1 , and  1 .Different values will be taken to demonstrate different secondorder nonlinear rogue waves.The corresponding graph of the rogue wave is displayed in Figure 2. In Figures 2(a) and 2(b), we have  = 0 and  1 =  1 = 0.It is a basic form which reaches a single maximum at the center in Figure 2(a).Figure 2(b) looks like "" type.By slightly adjusting the parameter  = 0.2 and other parameters being the same, the rogue waves are depicted, as displayed in Figures 2(c) and 2(d).In this case, two small peaks are located on both sides and the other two peaks are located near high peak.Nearly all power of the rogue wave is focused on the high peak.It is observed from Figure 2(a) that the maximum amplitude of | [1](0, )| is 5, which occurs at  = 0.
Then, changing the parameters  1 and  1 in Ω() and letting  = 0, the picture of the rogue waves is exhibited in Figure 3.In Figures 3(a) and 3(b), we have  1 = 10 and  1 = 0.In Figures 3(c) and 3(d), there are  1 = 0 and  1 = 20.In Figures 3(a)-3(d), the solution  [2] splits into three first-order rogue waves, which are not separated completely.In Figures 3(a)-3(d), three first-order rogue waves have a structure of an equilateral triangle.Furthermore, it is found that the temporal-interval in which the rogue waves appear is enlarged.The solution  [2] splits into three first-order rogue wave separated completely as the increase of  1 and  1 .When  = 0.1 and  1 =  1 = 40, the plot of the rogue waves is depicted in Figures 3(e) and 3(f).The numerical results indicate that the maximum amplitude of solution  [2] varies with the change of the parameters ,  1 , and  1 .
Likewise, we study the third-order solution  [3] of (2).Considering the limitation below lim )) Φ [1]   1 where  1 [2] and Φ 1 [2] are calculated by the Maple.We neglect  1 [2] and Φ 1 [2] since the expressions are too boring to write them down in the paper.Then, the third-order rogue wave solution of ( 2) is obtained There are five parameters ,  1 ,  1 ,  2 , and  2 in the solution  [3].The corresponding pictures of  [3] are described in Figure 4.In Figures 4(a) and 4(b), we have  = 0,  1 =  1 = 0, and  2 =  2 = 0.It is a fundamental pattern and there is a single maximum at the center where the maximum amplitude of  [3] is 7. Increasing the parameter  = 0.5 while the other parameters are unchanged, the solution  [3] is shown in Figures 4(c) and 4(d).
When we select the parameters as  = 0.02 and  1 = 500, the pictures of the solution  [3] are illustrated in Figures 5(a) and 5(b).Changing the parameter  as  = 0.25, the corresponding plots of the solution  [3] are depicted in Figures 6(a) and 6(b).Comparing with Figures 5 and 6, it is seen that the third-order rogue waves are composed of six first-order rogue waves and six waves form a triangle.However, it is found that six first-order rogue waves form an equilateral triangle in Figure 5(b) while the waves form an isosceles triangle in Figure 6(b).
When the parameters  = 0.02 and  2 = 1500, the plots of the solution  [3] are demonstrated in Figure 7.It can be observed from Figure 7 that the shape is a pentagon, where one wave is located in the center, and the others are the vertices of the pentagon.
If the parameters  = 0.02,  1 = 500, and  2 = 10000, the pictures of the solution  [3] are illustrated in Figure 8.It is found that Figure 8(a) also consists of the six first-order rogue waves.Interestingly, one wave is on the left side and the other four waves form a rectangle, and the last lies in the center.It is known from Figure 8(b) that six first-order rogue waves form a triangle on the left side and a rectangle on the right side.
With the increase of the order for the rogue waves solutions [], there are more free parameters, which will  exhibit more novel and interesting structures.It is supposed that the new structures of the higher-order rogue waves will be polygon of different shape.

Conclusions
In the paper, the th-order solutions of the rogue waves are constructed by using the generalized Darboux transformation (DT) for the inhomogeneous fifth-order nonlinear Schrödinger equation.Firstly, a short introduction about the DT is given.Secondly, the generalized DT is discussed making use of the Taylor expansion and a limit procedure.The first-, second-, and third-order rogue waves solutions are derived.Changing the free parameters in the solutions, some new structures of the rogue waves are obtained.By analyzing the nonlinear dynamics of the rogue waves, we can study the rogue waves in optical fibers.Furthermore, if the iteration process of the generalized DT is carried out continually, there will be more complicated and interesting structures of the rogue waves.It is hoped that the theoretical results will be confirmed by experiments in the future.