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

Authors: N. Song; W. Zhang; P. Wang; Y. K. Xue
Journal: Journal of Function Spaces
 (2017)

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## Abstract

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.


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## Body

## 1. 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 [2–4], and gradually moved into other fields, such as optics [5–7], 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(1)iqt+12qxx+q2q=0,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 n-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 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 fifth-order nonlinear Schrödinger equation as follows [26, 27]: (2)iqt-iεqxxxxx-10iεqxxx-20iεqxq∗qxx-30iεq4qx-10iεqx2qx+fqxx+2qfq2+∫-∞xfx′x′qx′,t2dx′-igqx=0,where q(x,t) is a complex function and t and x denote the scaled time and spatial coordinate, respectively. f and g represent the variation of the bilinear and biquadratic exchange interactions at different sites along the spin chain and(3)f=f1x+f2,g=g1x+g2,where fj and gjj=1,2 are real constants. ε 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 one- and 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 [29–31], modulation instability, integrability [32–34], 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 obtainNth-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 Nth-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.


## 2. 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]:(4a)Φx=UΦ=λU0+U1Φ,(4b)Φt=VΦ=AB-B∗-AΦ,where U and V are 2×2 matrices(5)U0=-i100-1,U1=0q-q∗0,A=-16iλ5ε+8iλ3εq2+4λ2εqqx∗-qxq∗-2ifλ2-2iλεqqxx∗+q∗qxx-qx2+3q4-ihλ+εq∗qxxx-qqxxx∗+qxqxx∗-qxxqx∗+6q2q∗qx-6q2qx∗q+ifq2+ifq2+∫-∞xfx′x′qx′,t2dx′,B=16λ4εq+8iλ3εqx-4λ2εqxx+2q2q-2iλεqxxx+6q2qx+2fλq+εqxxxx+8q2qxx+2q2qxx∗+4qx2q+6qx2q∗+6q4q+ifqx+hq,where Φ=(φ,ϕ)T is an eigenfunction of the Lax pair (4a) and (4b), q is a potential function, λ 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(6)Ut-Vx+U,V=0.The Darboux transformation consists of the eigenfunction transformation and potential transformation, which is a gauge transformation(7)Φ1=TΦof the solution for the Lax pair (4a) and (4b). T is a Darboux matrix, which will transform the Lax pair (4a) and (4b) into a new one; that is to say,(8)Φ1x=U1Φ1,Φ1t=V1Φ1.It is known thatU1 and V1 have the same form as U and V when q and q∗ in the matrices U and V are substituted by q1 and q1∗ in the matrices U1 and V1. U1 and V1 are given by (9)U1=Tx+TUT-1,V1=Tt+TVT-1.According to (9), a relationship between the new potential function q1 and the initial potential function q will be established. Thus, the Darboux matrix T is defined as [16](10)T=λI-S,S=HΛH-1,where(11)I=1001,H=φ1ϕ1∗ϕ1-φ1∗,Λ=λ100λ1∗.LetΦ1=φ1,ϕ1T be an eigenfunction of the Lax pair (4a) and (4b) with a seeding solution q=q[0] and λ=λ1. It is seen that (ϕ1∗,-φ1∗)T can also satisfy equation (4a) and (4b) with λ=λ1∗. Selecting different eigenfunctions Φk=(φk,ϕk)T at λk, respectively, the above Darboux transformation procedure can be easily iterated.Therefore, on the basis of the Darboux matrixT in (9), the initial Darboux transformation of equation (2) is given as(12)Φ10=T0Φ1,q1=q0-2iλ1-λ1∗φ10ϕ1∗0φ102+ϕ102,where(13a)T1=λ2I-H0Λ1H0-1,(13b)T0=I,H0=φ10ϕ1∗0ϕ10-φ1∗0,Λ1=λ100λ1∗,(13c)Φ10=φ10,ϕ10T=φ1,ϕ1T=Φ1,q0=q.IfN different basic solutions Φk=(φk,ϕk)T(k=1,2,…,N) of the Lax pair (4a) and (4b) at λ=λk(k=1,2,…,N) are given, the initial Darboux transformation can be iterated N times. Then the Nth-step DT for (2) is(14a)ΦNN-1=TN-1TN-2⋯T1T0ΦN,(14b)qN=q0-2i∑k=1Nλ1-λ1∗φkk-1ϕk∗k-1φkk-12+ϕkk-12,where(15a)Tk=λk+1I-Hk-1ΛkHk-1-1,(15b)Hk-1=φkk-1ϕk∗k-1ϕkk-1-φk∗k-1,Λk=λk00λk∗,(15c)Φkk-1=Tk-1Tk-2⋯T1T0λ=λkΦk,k=1,2,…,N.The initial value isΦ10=φ10,ϕ10T=φ1,ϕ1T=Φ1.On the basis of the former elementary DT, a generalized DT is derived for (2). We assume that(16)Ψ=Φ1λ1,δis a special solution of the Lax pair (4a) and (4b) and δ is a small parameter.Using the Taylor series,Ψ can be expanded at δ=0(17)Ψ=Φ10+Φ11δ+Φ12δ2+⋯+Φ1mδm+oδm,where Φ1j=1/j!∂j/∂λjΦ1λλ=λ1j=1,2,…,m.From the above assumption, it is easily seen thatΦ10=Φ10 is a particular solution of the Lax pair (4a) and (4b) with q=q0 and λ=λ1. Therefore, an Nth-step generalized DT is constructed by combining all the Darboux matrices(18)Φ1N-1=Φ10+∑l=1N-1T1lΦ11+∑l=1N-1∑k>lN-1T1kT1lΦ12+⋯+T1N-1T1N-2⋯T11Φ1N-1,qN=qN-1-2iλ1-λ1∗φ1N-1ϕ1N-1∗φ1N-12+ϕ1N-12,where(19a)T1k=λ1I-H1k-1Λ1H1k-1-1,(19b)H1k-1=φkk-1ϕk∗k-1ϕkk-1-φk∗k-1,(19c)Φ1k-1=φkk-1ϕkk-1,k=1,2,…,N.Equations (18), (19a), (19b), and (19c) are the recursive formulae of the Nth-step generalized DT for (2), which are simple to construct the higher-order rogue waves solutions and can be transformed into the 2n×2n 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.


## 3. Rogue Waves Solutions

In order to calculate easily, the rogue waves solutions are derived when the inhomogeneous parameters are independent ofx; that is to say, f1=g1=0. The parameters are chosen as f2=0.5 and g2=-25ε.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 toq0=eit. The corresponding eigenfunction Φ1(λ) for the linear spectral problem at λ=ih is (20)Φ1λ=C1eA+C2e-Aeit/2C2eA+C1e-Ae-it/2,where(21a)C1=h+h2-1h2-1,C2=-h-h2-1h2-1,A=h2-1x+ωt+Ωδ,(21b)ω=16ih4ε-8ih2ε+2f2ih+6ε+g2,(21c)Ωδ=∑j=1naj+ibjδ2jaj,bj∈R,and δ is a small parameter and Ω(δ) is the separating function, which contains 2n free parameters aj,bj(j=1,…,n) and aj, bj are real constants.Leth=1+δ2 and expanding the vector function Φ1(δ) at δ=0, we have(22)Φ1δ=Φ10+Φ11δ2+Φ12δ4+⋯+Φ1mδ2m+⋯,where(23)Φ10=φ10ϕ10,Φ11=φ11ϕ11,Φ12=φ12ϕ12,⋮,and φ1i-1,ϕ1i-1T(i=1,2,3) are given in Appendix A.It is clearly observed thatΦ10 is a solution of the Lax pair (4a) and (4b) at q0=eit and λ=i. Hence, substituting Φ10, q0=eit, and λ=i into (12), the first-order rogue wave solution of equation (2) is obtained(24)q1=1+4G1+i4tH1eit,where(25a)H1=2+80xtε+200t2ε2+8t2+8x2,(25b)G1=1-40xtε-100t2ε2-4t2-4x2.There is only a free parameterε 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 ε=0. In Figures 1(c) and 1(d), there is ε=0.2. Fixing x=0, the changes of q10,t in the direction of t axes with the parameters ε=0 are displayed in Figure 1(e). It is observed from Figure 1(e) that the maximum amplitude of q10,x is 3, which occurs at t=0.Figure 1
(a) The first-order rogue wave withε=0; (b) the density plot with ε=0; (c) the first-order rogue wave with ε=0.2; (d) the density plot with ε=0.2; (e) the section of q10,t at x=0 with ε=0.
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(e)Then, considering the following limit(26a)limδ→0⁡T11λ=λ1+δΨδ=limδ→0⁡δ+T11λ=λ1Ψδ=Φ10+T11λ1Φ11=Φ11,(26b)Φ11=φ11ϕ11,where T11 and Φ11 can be calculated by Maple.We neglectΦ11 since its expression is rather cumbersome to write it down. Then, we obtain the second-order rogue wave solution of (2)(27)q2=q1+4φ11·ϕ11∗φ112+ϕ112,where(28a)Δ=1+40xtε+100t2ε2+4t2+4x2,(28b)φ11=13Δeit/2A1+iB1,ϕ11=13Δe-it/2A2+iB2,and A1, B1, A2, and B2 are given in Appendix B.In (27), there are three free parameters ε, a1, and b1. Different values will be taken to demonstrate different second-order nonlinear rogue waves. The corresponding graph of the rogue wave is displayed in Figure 2. In Figures 2(a) and 2(b), we have ε=0 and a1=b1=0. It is a basic form which reaches a single maximum at the center in Figure 2(a). Figure 2(b) looks like “Z” type. By slightly adjusting the parameter ε=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 q10,x is 5, which occurs at t=0.Figure 2
(a) The second-order rogue wave withε=0 and a1=b1=0; (b) the density plot with ε=0 and a1=b1=0; (c) the second-order rogue wave with ε=0.2 and a1=b1=0; (d) the density plot with ε=0.2 and a1=b1=0.
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(d)Then, changing the parametersa1 and b1 in Ω(δ) and letting ε=0, the picture of the rogue waves is exhibited in Figure 3. In Figures 3(a) and 3(b), we have a1=10 and b1=0. In Figures 3(c) and 3(d), there are a1=0 and b1=20. In Figures 3(a)–3(d), the solution q2 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 q2 splits into three first-order rogue wave separated completely as the increase of a1 and b1. When ε=0.1 and a1=b1=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 q2 varies with the change of the parameters ε, a1, and b1.Figure 3
(a) The second-order rogue wave withε=0, a1=10, and b1=0; (b) the density plot with ε=0, a1=10, and b1=0; (c) the second-order rogue wave with ε=0, a1=0, and b1=20; (d) the density plot with ε=0, a1=0, and b1=20; (e) the second-order rogue wave with ε=0.1 and a1=b1=40; (f) the density plot with ε=0.1 and a1=b1=40.
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(f)Likewise, we study the third-order solutionq3 of (2). Considering the limitation below(29a)limδ→0⁡δ+T12λ=λ1δ+T11λ=λ1Ψδ2=Φ10+T11λ1+T12λ1Φ11+T12λ1T11λ1Φ12=Φ12,(29b)Φ12=φ12ϕ12,where T12 and Φ12 are calculated by the Maple.We neglectT12 and Φ12 since the expressions are too boring to write them down in the paper. Then, the third-order rogue wave solution of (2) is obtained(30)q3=q2+4φ12·ϕ12∗φ122+ϕ122.There are five parametersε, a1, b1, a2, and b2 in the solution q3. The corresponding pictures of q3 are described in Figure 4. In Figures 4(a) and 4(b), we have ε=0, a1=b1=0, and a2=b2=0. It is a fundamental pattern and there is a single maximum at the center where the maximum amplitude of q3 is 7. Increasing the parameter ε=0.5 while the other parameters are unchanged, the solution q3 is shown in Figures 4(c) and 4(d).Figure 4
(a) The third-order rogue wave withε=0, a1=b1=0, and a2=b2=0; (b) the density plot with ε=0, a1=b1=0, and a2=b2=0; (c) the third-order rogue wave with ε=0.5, a1=b1=0, and a2=b2=0; (d) the density plot with ε=0.5, a1=b1=0, and a2=b2=0.
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(d)When we select the parameters asε=0.02 and b1=500, the pictures of the solution q3 are illustrated in Figures 5(a) and 5(b). Changing the parameter ε as ε=0.25, the corresponding plots of the solution q3 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).Figure 5
(a) The third-order rogue wave withε=0.02 and b1=500; (b) the density plot with ε=0.02 and b1=500.
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(a) The third-order rogue wave withε=0.25 and b1=500; (b) the density plot with ε=0.25 and b1=500.
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(b)When the parametersε=0.02 and b2=1500, the plots of the solution q3 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.Figure 7
(a) The third-order rogue wave withε=0.02 and b2=1500; (b) the density plot with ε=0.02 and b2=1500.
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(b)If the parametersε=0.02, b1=500, and b2=10000, the pictures of the solution q3 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.Figure 8
(a) The third-order rogue wave withε=0.02, b1=500, and b2=10000; (b) the density plot with ε=0.02, b1=500, and b2=10000.
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(b)With the increase of the order for the rogue waves solutionsqN, 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.


## 4. Conclusions

In the paper, theNth-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.



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