# Novel Particular Solutions, Breathers, and Rogue Waves for an Integrable Nonlocal Derivative Nonlinear Schrödinger Equation

Authors: Yali Shen; Ruoxia Yao
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

A determinant representation of then-fold Darboux transformation for the integrable nonlocal derivative nonlinear Schödinger (DNLS) equation is presented. Using the proposed Darboux transformation, we construct some particular solutions from zero seed, which have not been reported so far for locally integrable systems. We also obtain explicit breathers from a nonzero seed with constant amplitude, deduce the corresponding extended Taylor expansion, and obtain several first-order rogue wave solutions. Our results reveal several interesting phenomena which differ from those emerging from the classical DNLS equation.


---

## Body

## 1. Introduction

It is well known that the derivative nonlinear Schrödinger (DNLS) equation [1]
(1)iqtx,t=qxxx,t+iεqx,t2q∗x,txε=±1,has many physical applications, e.g., in analyzing the propagation of circular polarized nonlinear Alfvén waves and radiofrequency waves in plasmas [1, 2]. Equation (1) is local; that is, the evolution only depends on the value of its local time and space [3]. In the recent years, some new integrable nonlocal equations have been introduced and several interesting results have been obtained [4–12]. These nonlocal equations are significantly different from the local one due to their particular spatiotemporal coupling, which may stimulate new physical applications, as they describe novel physical effects [3].Since the nonlinearity-induced potentials satisfy thePT-symmetric condition, the nonlocal NLS equation is often referred to as PT-symmetric [4]. The dynamics arising from the nonlocal NLS equation leads to several phenomena, including dark solitons [13] and rogue waves [14, 15]. This also fostered interest in other nonlocal integrable equations [15–18], which are generally obtained by using parity (P^,P^=−x), time inversion (T^,T^=−t), and charge conjugation (C^) symmetries. The symmetry P^−T^−C^ has in turn an important role in quantum physics [19] and in many other fields of physics [20–22]. In addition, an important physical link between the nonlocally integrable reduction of the newly discovered AKNS system and physically interesting equations has been established [23]. Overall, there are several reasons motivating further studies of these nonlocal systems.Recently, Zhou presented an integrable nonlocal DNLS equation [24]
(2)iqtx,t=qxxx,t+εqx,t2q∗−x,txε=±1,where the symbol ∗ denotes complex conjugation. Equation (2) has a PT-symmetric conserved density q∗−x,tqx,t, that is,
(3)iq∗−x,tqx,tt=q∗−x,tqxx,t−q∗−x,txqx,tx+3ε2q2x,tq∗2−x,tx.By the transformationsx⟶−ix,t⟶−t, Equation (2) can be obtained from Equation (1) [3], and as a matter of fact, it is more accurate to call Equation (2) P^C^ symmetric [25]. In addition, Ablowitz and Musslimani have presented a real space-time reversal DNLS equation [5].In recent years, there are many new results in obtaining nonlinear waves. Based on the characteristic lines and phase shift analysis, Yin and Tian studied the transitions and mechanisms of nonlinear waves in the2+1-dimensional Sawada-Kotera (2DSK) equation [26]. According to the characteristic lines of breath, various nonlinear waves are obtained, including quasi-anti-dark soliton, M-shaped soliton, W-shaped soliton, multi-peak soliton, and quasi-periodic wave. The dynamic properties of these nonlinear waves are analyzed in detail by means of characteristic lines. Wang et al. have constructed the three-component coupled Hirota hierarchy and have obtained its soliton solutions by using the ∂¯-dressing method [27]. By using the Riemann-Hilbert method, Li et al. systematically investigated the general n-component nonlinear Schödinger equations. The multisoliton solutions of two-component nonlinear Schödinger equations have been detailedly analyzed by means of parameter modulation. Many interesting new phenomena are presented, including elastic collision, parallel propagation, soliton reflection, and time-periodic propagation [28].In mathematical physics,N-soliton solutions are very helpful in exploring nonlinear wave phenomena. Breather, rogue wave, and lump solutions, etc. are all special reductions of N-soliton solutions [29]. Ma et al. studies the soliton solutions of many famous nonlinear equations and presents an algorithm to verify the Hirota N-soliton conditions [29–31]. The existence of N-soliton solutions of two kinds of generalized KdV equations is verified by the common factorization of Hirota functions of N wave vectors and the comparison of the degrees of polynomials containing common factors [30]. And then, a weight number is used in the above algorithm to verify the Hirota N-soliton condition for the B-type Kadomtsev-Petviashvili equation and three integrable equations in (2+1) dimensions [29, 31]. Breathers and rogue waves have been the subject of large interest in the recent years. Breathers have been found in many physically related models such as Bose-Einstein condensates [32], higher-order integrable systems [33], and reaction-diffusion systems [34]. The observation of rogue waves and the study of their formation and their dynamics have been experimentally performed in diverse physical media, such as optical fibers, water wave tanks, and plasmas [35–37].In this paper, we focus on studying breathers and rogue waves of the nonlocal DNLS Equation (2). We first construct an n-fold Darboux transformation for Equation (2) in Section 2, from which we obtain two particular solutions that are obtained from zero seed. In Section 3, breathers by nonzero seeds with constant amplitude are studied, and their typical dynamics are analyzed. By Taylor expansion on the breather, we obtain and analyze the rogue wave of Equation (2). Section 4 closes the paper with some concluding remarks.


## 2. Darboux Transformation and Particular Solutions

In this part, we first construct the Darboux transformation of Equation (2), which is then exploited to obtain two novel solutions that are not soliton solutions, of which we analyze the asymptotic behavior.Our Darboux transformation is mostly determined by the seed solutions and eigenvalues corresponding to the Lax pair and by the relationship between the potential functionq and r. According to the form of the first-order Darboux matrix, we then provide the n-order Darboux matrix and finally obtain the n-order Darboux transformation of Equation (2).Let us sketch the main steps as follows: first, given an eigenvalueλ in the Lax equation of Equation (2) and being q=0 be a zero seed solution, we seek the eigenfunction corresponding to the seed solution of the Lax equation, then substitute it into the first-order Darboux transformation, and get a novel solution. Using the same steps, we select two different eigenvalues, so that we can get four eigenvalues according to the conjugate relation. We still take the seed solution q=0, and solving the Lax equation, we can obtain four eigenfunctions fii=1,⋯,4. According to the relationship among the potential functions, we can get the other four eigenfunctions gii=1,⋯,4 and bring them into the second-order Darboux transformation to obtain the second-order novel solutions.According to the above method, we can get higher-order new solutions and study more interesting phenomena by analyzing their properties.


### 2.1. Darboux Transformation

The integrability of Equation (2) is guaranteed by the Lax pair [1]
(4)ψx=Uψ=λ2λqλr−λ2ψ,(5)ψt=Vψ=−2iλ4+iλ2qr−2iλ3q+−iqx+iq2rλ−2iλ3r+irx+iqr2λ2iλ4−iλ2qrψ,where λ is the spectral parameter, q=qx,t, r=rx,t, and r=−εq¯∗=−εq∗−x,t,ψ=ψ1x,t,ψ2x,tT. Refering to the construction of Darboux transformation of classical integrable equations [38], we could obtain a Darboux transformation of Equation (2).Consider a canonical transformation(6)ψ1=Tψ.Equations (4) and (5) become
(7)ψx1=Tx+TUT−1ψ1≜U1ψ1,ψt1=Tt+TVT−1ψ1≜V1ψ1.Substituting potential functionsq, r with the new ones q1, r1, we can determine T such that U1 and V1 have the same forms of U and V, respectively.Let(8)T=λI−P,where I is unit matrix and P=pij2×2 with pij=pijx,ti,j=1,2. The relation between the old potentials and new ones is given by
(9)q1=q+2p12,r1=r−2p21.According to the relation betweenq and r(10)r=−εq¯∗=−εq∗−x,t,we have the following constraint
(11)p21x,t=εp12∗−x,t.For the sake of convenience, we use the notationf¯x,t=f−x,t.In order to determineT in Equation (8), one may set
(12)P=HΛH−1,with
(13)H=f1g1f2g2,Λ=λ100λ2,where f1,f2T=f1x,t,f2x,tT is a solution of Equations (4) and (5) corresponding to the seed solution and the eigenvalue λ=λ1. According to (10), we know that g1,g2T=f¯2∗,εf¯1∗T is a solution of Equations (4) and (5) when λ=λ1∗≜λ2.Then, we get(14)P=1Δελ1f1f¯1∗−λ1∗f2f¯2∗λ1∗−λ1f1f¯2∗λ1−λ1∗εf2f¯1∗ελ1∗f1f¯1∗−λ1f2f¯2∗,with Δ=εf1f¯1∗−f2f¯2∗. From (14), we can directly verify the constraint (11). Therefore, from (9) and (14), a new solution of Equation (2) is obtained as follows:
(15)q1=q+2Δλ1∗−λ1f1f¯2∗.Then-fold Darboux transformation of Equation (2) can be written in the following determinant representation:
(16)qn=q+2P2nW2n,where
(17)P2n=f1f2λ1f1λ1f2⋯λ1n−1f1λ1nf1g1g2λ2g1λ2g2⋯λ2n−1g1λ2ng1f3f4λ3f3λ3f4⋯λ3n−1f3λ3nf3g3g4λ4g3λ4g4⋯λ4n−1g3λ4ng3⋮⋮⋮⋮⋱⋮⋮g2n−1g2nλ2ng2n−1λ2ng2n⋯λ2nn−1g2n−1λ2nng2n−1,W2n=f1f2λ1f1λ1f2⋯λ1n−1f1λ1n−1f2g1g2λ2g1λ2g2⋯λ2n−1g1λ2n−1g2f3f4λ3f3λ3f4⋯λ3n−1f3λ3n−1f4g3g4λ4g3λ4g4⋯λ4n−1g3λ4n−1g4⋮⋮⋮⋮⋱⋮⋮g2n−1g2nλ2ng2n−1λ2ng2n⋯λ2nn−1g2n−1λ2nn−1g2n.By using (16), we can obtain multiple particular solutions, multibreathers, and higher-order rogue waves of Equation (2). The determinant representation of the DT provides a powerful tool to calculate this tedious expansion.


### 2.2. One-fold Particular Solution

In order to construct one-fold particular solutions of Equation (2) using the above Darboux transformation, one may start from the zero seed solution q=0 in Equations (4) and (5). Then, the eigenfunctions corresponding to the seed solution are given by
(18)f1=eλ12x−2iλ14t,f2=e−λ12x+2iλ14t.Settingλ1=a+ib and substituting (18) into (15), one can easily get the one-fold particular solution
(19)q1=−4ibe2xa2−b2−4ita4−6a2b2+b4εe−4iab4ita2−4itb2−x−e4iab4ita2−4itb2−x.Solution (19) is a complex function whose modulus is given by
(20)q1=4be2xa2−b22cosh32abta2−b2−2εcos8abx,with singularities at x,t∣2cosh32abta2−b2−2εcos8abx=0. From (20), it can be found that when a2>b2, q1⟶+∞ as x⟶+∞, and q1⟶0 as x⟶−∞. We also see that ∣q1∣⟶0 as t⟶±∞. By setting ε=1,a=0.2,b=0.01, Figures 1(a) and 1(b) illustrate the behaviour q1 given in (20). By choosing t=0 in (20) and using the previous parameters, one can get a one-fold particular solution with singularity. Figure 1(c) illustrates this case. For t=1, we can see that ∣q1∣ reaches the maximum at x=0. The maximum amplitude is 15.66400782.Figure 1
The one-fold particular solution evolution graph (a) and contour plot (b) for (20) with ε=1,a=0.2,b=0.01. (c) The waveform at t=0 and (d) the waveform at t=1.
(a)(b)(c)(d)


### 2.3. Two-fold Particular Solution

In order to proceed further, we still choose zero seed solutionq=0 and use the 2-fold Darboux transformation, to obtain a two-fold particular solution of Equation (2) as follows:
(21)q2=q+2P4W4,where
(22)P4=f1f2λ1f1λ12f1g1g2λ2g1λ22g1f3f4λ3f3λ32f3g3g4λ4g3λ42g3,W4=f1f2λ1f1λ1f2g1g2λ2g1λ2g2f3f4λ3f3λ3f4g3g4λ4g3λ4g4.Letλ1=λ2∗=a+ib and λ3=λ4∗=c+id. By solving Equations (4) and (5), we obtain the eigenfunctions
(23)f1=ea+ib2x−2ia+ib4t,f2=e−a+ib2x+2ia+ib4t,f3=ec+id2x−2ic+id4t,f4=e−c+id2x+2ic+id4t.According to (10), we get g1=f¯2∗,g2=εf¯1∗,g3=f¯4∗,g4=εf¯3∗. Hence, a two-fold particular solution can be given explicitly by Equation (21). We take some special parameter values for the solution (21) and show the resulting behavior in Figure 2. Figure 2(a) reveals that there are two crests above the x−t plane. We also found q2⟶+∞ as x⟶−∞, and q2⟶0 as x⟶+∞ as it can be seen from Figure 2(a). Figures 2(b) and 2(c) show beautiful butterfly contour graphs.Figure 2
The two-fold particular solution evolution graph of (21) with parameters: ε=1,a=0.1,b=0.2,c=0.01,d=0.35. (a) ∣q2∣. (c) Req2. (e) Imq2. (b, d, f) Contour plots corresponding to (a, c, e), respectively.
(a)(b)(c)(d)(e)(f)


## 2.1. Darboux Transformation

The integrability of Equation (2) is guaranteed by the Lax pair [1]
(4)ψx=Uψ=λ2λqλr−λ2ψ,(5)ψt=Vψ=−2iλ4+iλ2qr−2iλ3q+−iqx+iq2rλ−2iλ3r+irx+iqr2λ2iλ4−iλ2qrψ,where λ is the spectral parameter, q=qx,t, r=rx,t, and r=−εq¯∗=−εq∗−x,t,ψ=ψ1x,t,ψ2x,tT. Refering to the construction of Darboux transformation of classical integrable equations [38], we could obtain a Darboux transformation of Equation (2).Consider a canonical transformation(6)ψ1=Tψ.Equations (4) and (5) become
(7)ψx1=Tx+TUT−1ψ1≜U1ψ1,ψt1=Tt+TVT−1ψ1≜V1ψ1.Substituting potential functionsq, r with the new ones q1, r1, we can determine T such that U1 and V1 have the same forms of U and V, respectively.Let(8)T=λI−P,where I is unit matrix and P=pij2×2 with pij=pijx,ti,j=1,2. The relation between the old potentials and new ones is given by
(9)q1=q+2p12,r1=r−2p21.According to the relation betweenq and r(10)r=−εq¯∗=−εq∗−x,t,we have the following constraint
(11)p21x,t=εp12∗−x,t.For the sake of convenience, we use the notationf¯x,t=f−x,t.In order to determineT in Equation (8), one may set
(12)P=HΛH−1,with
(13)H=f1g1f2g2,Λ=λ100λ2,where f1,f2T=f1x,t,f2x,tT is a solution of Equations (4) and (5) corresponding to the seed solution and the eigenvalue λ=λ1. According to (10), we know that g1,g2T=f¯2∗,εf¯1∗T is a solution of Equations (4) and (5) when λ=λ1∗≜λ2.Then, we get(14)P=1Δελ1f1f¯1∗−λ1∗f2f¯2∗λ1∗−λ1f1f¯2∗λ1−λ1∗εf2f¯1∗ελ1∗f1f¯1∗−λ1f2f¯2∗,with Δ=εf1f¯1∗−f2f¯2∗. From (14), we can directly verify the constraint (11). Therefore, from (9) and (14), a new solution of Equation (2) is obtained as follows:
(15)q1=q+2Δλ1∗−λ1f1f¯2∗.Then-fold Darboux transformation of Equation (2) can be written in the following determinant representation:
(16)qn=q+2P2nW2n,where
(17)P2n=f1f2λ1f1λ1f2⋯λ1n−1f1λ1nf1g1g2λ2g1λ2g2⋯λ2n−1g1λ2ng1f3f4λ3f3λ3f4⋯λ3n−1f3λ3nf3g3g4λ4g3λ4g4⋯λ4n−1g3λ4ng3⋮⋮⋮⋮⋱⋮⋮g2n−1g2nλ2ng2n−1λ2ng2n⋯λ2nn−1g2n−1λ2nng2n−1,W2n=f1f2λ1f1λ1f2⋯λ1n−1f1λ1n−1f2g1g2λ2g1λ2g2⋯λ2n−1g1λ2n−1g2f3f4λ3f3λ3f4⋯λ3n−1f3λ3n−1f4g3g4λ4g3λ4g4⋯λ4n−1g3λ4n−1g4⋮⋮⋮⋮⋱⋮⋮g2n−1g2nλ2ng2n−1λ2ng2n⋯λ2nn−1g2n−1λ2nn−1g2n.By using (16), we can obtain multiple particular solutions, multibreathers, and higher-order rogue waves of Equation (2). The determinant representation of the DT provides a powerful tool to calculate this tedious expansion.


## 2.2. One-fold Particular Solution

In order to construct one-fold particular solutions of Equation (2) using the above Darboux transformation, one may start from the zero seed solution q=0 in Equations (4) and (5). Then, the eigenfunctions corresponding to the seed solution are given by
(18)f1=eλ12x−2iλ14t,f2=e−λ12x+2iλ14t.Settingλ1=a+ib and substituting (18) into (15), one can easily get the one-fold particular solution
(19)q1=−4ibe2xa2−b2−4ita4−6a2b2+b4εe−4iab4ita2−4itb2−x−e4iab4ita2−4itb2−x.Solution (19) is a complex function whose modulus is given by
(20)q1=4be2xa2−b22cosh32abta2−b2−2εcos8abx,with singularities at x,t∣2cosh32abta2−b2−2εcos8abx=0. From (20), it can be found that when a2>b2, q1⟶+∞ as x⟶+∞, and q1⟶0 as x⟶−∞. We also see that ∣q1∣⟶0 as t⟶±∞. By setting ε=1,a=0.2,b=0.01, Figures 1(a) and 1(b) illustrate the behaviour q1 given in (20). By choosing t=0 in (20) and using the previous parameters, one can get a one-fold particular solution with singularity. Figure 1(c) illustrates this case. For t=1, we can see that ∣q1∣ reaches the maximum at x=0. The maximum amplitude is 15.66400782.Figure 1
The one-fold particular solution evolution graph (a) and contour plot (b) for (20) with ε=1,a=0.2,b=0.01. (c) The waveform at t=0 and (d) the waveform at t=1.
(a)(b)(c)(d)


## 2.3. Two-fold Particular Solution

In order to proceed further, we still choose zero seed solutionq=0 and use the 2-fold Darboux transformation, to obtain a two-fold particular solution of Equation (2) as follows:
(21)q2=q+2P4W4,where
(22)P4=f1f2λ1f1λ12f1g1g2λ2g1λ22g1f3f4λ3f3λ32f3g3g4λ4g3λ42g3,W4=f1f2λ1f1λ1f2g1g2λ2g1λ2g2f3f4λ3f3λ3f4g3g4λ4g3λ4g4.Letλ1=λ2∗=a+ib and λ3=λ4∗=c+id. By solving Equations (4) and (5), we obtain the eigenfunctions
(23)f1=ea+ib2x−2ia+ib4t,f2=e−a+ib2x+2ia+ib4t,f3=ec+id2x−2ic+id4t,f4=e−c+id2x+2ic+id4t.According to (10), we get g1=f¯2∗,g2=εf¯1∗,g3=f¯4∗,g4=εf¯3∗. Hence, a two-fold particular solution can be given explicitly by Equation (21). We take some special parameter values for the solution (21) and show the resulting behavior in Figure 2. Figure 2(a) reveals that there are two crests above the x−t plane. We also found q2⟶+∞ as x⟶−∞, and q2⟶0 as x⟶+∞ as it can be seen from Figure 2(a). Figures 2(b) and 2(c) show beautiful butterfly contour graphs.Figure 2
The two-fold particular solution evolution graph of (21) with parameters: ε=1,a=0.1,b=0.2,c=0.01,d=0.35. (a) ∣q2∣. (c) Req2. (e) Imq2. (b, d, f) Contour plots corresponding to (a, c, e), respectively.
(a)(b)(c)(d)(e)(f)


## 3. Breather and Rogue Wave Solutions

In this section, we construct breathers and rogue wave solutions of Equation (2) using the Darboux transformation illustrated in the previous one. At first, we construct breather solution from a nonzero seed. Then, we use the extended Taylor expansion method to reveal rogue wave solution of Equation (2) from breather ones.


### 3.1. Breather Solution

To obtain nontrivial periodic solutions, we choose a nonzero seedq=ρekx+iωt, where ρ,k,andω are all real constants and ω=−kk+ερ2. Substituting q=ρekx+iωt into Equations (4) and (5) and solving them, we get the eigenfunctions
(24)f1=ε2λ1ρC1k−2λ12+θ1e1/2k−θ1x+1/2iρ2ε+k+2λ12θ1−k2+ρ2εkt+C2k−2λ12−θ1e1/2k+θ1x−1/2iρ2ε+k+2λ12θ1+k2+ρ2εkt,(25)f2=C1e−1/2k+θ1x+1/2ik2+ρ2εk+ρ2ε+k+2λ12θ1t+C2e−1/2k−θ1x+1/2ik2+ρ2εk−ρ2ε+k+2λ12θ1t,where C1,C2 are constants and θ1=k2+4λ14−4ελ12ρ2−4kλ12.According tog1=f¯2∗,g2=εf¯1∗, we get
(26)g1=C1e1/2k+θ2x−1/2ik2+ρ2εk+ρ2ε+k+2λ22θ2t+C2e1/2k−θ2x−1/2ik2+ρ2εk−ρ2ε+k+2λ22θ2t,g2=12λ2ρC1k−2λ22+θ2e−1/2k−θ2x−1/2iρ2ε+k+2λ22θ2−k2+ρ2εkt+C2k−2λ22−θ2e−1/2k+θ2x+1/2iρ2ε+k+2λ22θ2+k2+ρ2εkt,where θ2=k2+4λ24−4ελ22ρ2−4kλ22 and λ1=λ2∗=a+ib.According to the principle of linear superposition, the new eigenfunctions associated toλ1 of Equations (4) and (5) may be expressed by
(27)F1=f1+g1,F2=f2+g2.Next, we use Equations (24)–(27) to produce a new breather solution of Equation (2)
(28)q1=q+2Δλ1∗−λ1F1F¯2∗,where q=ρekx+iωt and Δ=εF1F¯1∗−F2F¯2∗. By tedious calculations, the breather solution is obtained as follows:
(29)qbreather1=ekx−ik2+ρ2εktρ+2λ2λ2−λ12λ1ρ1+εcoshd2A+4λ12ερ2cosh2d2+A2λ1εε−1AB+4λ1λ2ρ2coshd1coshd2.In factε=±1, but we only take ε=−1 in Equation (29) and the breather solution is given by
(30)qbreather1=ekx−ik2−ρ2ktρ+λ2λ2−λ1A2−4λ12ρ2cosh2d2λ1AB+4λ1λ2ρ2coshd1coshd2,where
(31)A=k−2λ12coshd1−θ1sinhd1,B=k−2λ22coshd2+θ2sinhd2,d1=12θ1x−itk−ρ2+2λ12,d2=12θ2x−itk−ρ2+2λ22,θ1=k2+4λ14+4λ12ρ2−4kλ12,θ2=k2+4λ24+4λ22ρ2−4kλ22,λ1=λ2∗=a+ib.Specifically, choosing some specific parameter valuesk=−0.02,ρ=1,a=0,b=0.1, we get
(32)qbreather1=e−0.02x−0.0204it1+0.2icos20.1x+0.104it+sin20.1x+0.104itcos20.1x+0.104it−sin20.1x+0.104it,Figure3 illustrates this solution. From Figure 3(a), we see that the amplitude of (32) is increasing for x⟶−∞. Moreover, we find that the breather solution (32) has periodic singularities at t=0 and Figure 3(c) depicts this phenomenon. Figure 3(d) depicts the wave shape at t=1.Figure 3
Evolution graph of the breather solution (32), (a) qbreather11, (b) contour plot corresponding to (a) and (c, d) the waveform at t=0 and t=1, respectively.
(a)(b)(c)(d)


### 3.2. Rogue Wave Solution

Motivated by recent findings [37, 39], we shall use the extended Taylor expansion method to construct the rogue wave solutions of Equation (2). At first, we need to find the parameters a=2k−ρ2/2, b=ρ/2 in the eigenvalues λ1, λ2 such that θ1⟶0, θ2⟶0 (i.e., d1⟶0, d2⟶0) in the breather solution (29), and, in turn, to put the breather into the indeterminate form 0/0, which is a critical point to transform a first-order breather into a first-order rogue wave. However, due to the complexity of the breather solution, we perform Taylor expansion at d1=0,d2=0 directly. The Taylor expansion of (30) at d1=0,d2=0 gives a first-order rogue wave solution of Equation (2)
(33)qrogue1=ekx−ik2−ρ2ktρ+λ2λ2−λ1k−2λ12k1−θ1k22−4λ12ρ2k32λ1k−2λ12k1−θ1k2k−2λ22k3+θ2k4+4λ12λ2ρ2k1k3,where
(34)k1=1+θ128x−itk−ρ2+2λ122,k2=θ12x−itk−ρ2+2λ12+θ1348x−itk−ρ2+2λ123,k3=1+θ228x−itk−ρ2+2λ222,k4=θ22x−itk−ρ2+2λ22+θ2348x−itk−ρ2+2λ223,and θj,λjj=1,2 satisfy (31). Next, we show the shape of ∣qrogue1∣ in Figure 4(a) by selecting some special parameter values. From Figure 4(a), we can see that qrogue1⟶+∞ as x⟶+∞, and ∣qrogue1∣⟶0 as x⟶−∞. The evolution of this rogue wave at different times is shown in Figure 4(b), where we can see that qrogue1 achieves its maximum at x=1.63698271,t=0. The maximum amplitude is 4.448628515. It is shown that rogue waves exist in integrable systems.Figure 4
The first-order rogue wave for∣qrogue1∣ with parameter selections: k=0.05,ρ=0.001,a=1,b=0.01. (a) ∣qrogue1∣ and (b) its profiles at different times t=−0.5 (red), t=0 (blue), and t=1 (green).
(a)(b)


## 3.1. Breather Solution

To obtain nontrivial periodic solutions, we choose a nonzero seedq=ρekx+iωt, where ρ,k,andω are all real constants and ω=−kk+ερ2. Substituting q=ρekx+iωt into Equations (4) and (5) and solving them, we get the eigenfunctions
(24)f1=ε2λ1ρC1k−2λ12+θ1e1/2k−θ1x+1/2iρ2ε+k+2λ12θ1−k2+ρ2εkt+C2k−2λ12−θ1e1/2k+θ1x−1/2iρ2ε+k+2λ12θ1+k2+ρ2εkt,(25)f2=C1e−1/2k+θ1x+1/2ik2+ρ2εk+ρ2ε+k+2λ12θ1t+C2e−1/2k−θ1x+1/2ik2+ρ2εk−ρ2ε+k+2λ12θ1t,where C1,C2 are constants and θ1=k2+4λ14−4ελ12ρ2−4kλ12.According tog1=f¯2∗,g2=εf¯1∗, we get
(26)g1=C1e1/2k+θ2x−1/2ik2+ρ2εk+ρ2ε+k+2λ22θ2t+C2e1/2k−θ2x−1/2ik2+ρ2εk−ρ2ε+k+2λ22θ2t,g2=12λ2ρC1k−2λ22+θ2e−1/2k−θ2x−1/2iρ2ε+k+2λ22θ2−k2+ρ2εkt+C2k−2λ22−θ2e−1/2k+θ2x+1/2iρ2ε+k+2λ22θ2+k2+ρ2εkt,where θ2=k2+4λ24−4ελ22ρ2−4kλ22 and λ1=λ2∗=a+ib.According to the principle of linear superposition, the new eigenfunctions associated toλ1 of Equations (4) and (5) may be expressed by
(27)F1=f1+g1,F2=f2+g2.Next, we use Equations (24)–(27) to produce a new breather solution of Equation (2)
(28)q1=q+2Δλ1∗−λ1F1F¯2∗,where q=ρekx+iωt and Δ=εF1F¯1∗−F2F¯2∗. By tedious calculations, the breather solution is obtained as follows:
(29)qbreather1=ekx−ik2+ρ2εktρ+2λ2λ2−λ12λ1ρ1+εcoshd2A+4λ12ερ2cosh2d2+A2λ1εε−1AB+4λ1λ2ρ2coshd1coshd2.In factε=±1, but we only take ε=−1 in Equation (29) and the breather solution is given by
(30)qbreather1=ekx−ik2−ρ2ktρ+λ2λ2−λ1A2−4λ12ρ2cosh2d2λ1AB+4λ1λ2ρ2coshd1coshd2,where
(31)A=k−2λ12coshd1−θ1sinhd1,B=k−2λ22coshd2+θ2sinhd2,d1=12θ1x−itk−ρ2+2λ12,d2=12θ2x−itk−ρ2+2λ22,θ1=k2+4λ14+4λ12ρ2−4kλ12,θ2=k2+4λ24+4λ22ρ2−4kλ22,λ1=λ2∗=a+ib.Specifically, choosing some specific parameter valuesk=−0.02,ρ=1,a=0,b=0.1, we get
(32)qbreather1=e−0.02x−0.0204it1+0.2icos20.1x+0.104it+sin20.1x+0.104itcos20.1x+0.104it−sin20.1x+0.104it,Figure3 illustrates this solution. From Figure 3(a), we see that the amplitude of (32) is increasing for x⟶−∞. Moreover, we find that the breather solution (32) has periodic singularities at t=0 and Figure 3(c) depicts this phenomenon. Figure 3(d) depicts the wave shape at t=1.Figure 3
Evolution graph of the breather solution (32), (a) qbreather11, (b) contour plot corresponding to (a) and (c, d) the waveform at t=0 and t=1, respectively.
(a)(b)(c)(d)


## 3.2. Rogue Wave Solution

Motivated by recent findings [37, 39], we shall use the extended Taylor expansion method to construct the rogue wave solutions of Equation (2). At first, we need to find the parameters a=2k−ρ2/2, b=ρ/2 in the eigenvalues λ1, λ2 such that θ1⟶0, θ2⟶0 (i.e., d1⟶0, d2⟶0) in the breather solution (29), and, in turn, to put the breather into the indeterminate form 0/0, which is a critical point to transform a first-order breather into a first-order rogue wave. However, due to the complexity of the breather solution, we perform Taylor expansion at d1=0,d2=0 directly. The Taylor expansion of (30) at d1=0,d2=0 gives a first-order rogue wave solution of Equation (2)
(33)qrogue1=ekx−ik2−ρ2ktρ+λ2λ2−λ1k−2λ12k1−θ1k22−4λ12ρ2k32λ1k−2λ12k1−θ1k2k−2λ22k3+θ2k4+4λ12λ2ρ2k1k3,where
(34)k1=1+θ128x−itk−ρ2+2λ122,k2=θ12x−itk−ρ2+2λ12+θ1348x−itk−ρ2+2λ123,k3=1+θ228x−itk−ρ2+2λ222,k4=θ22x−itk−ρ2+2λ22+θ2348x−itk−ρ2+2λ223,and θj,λjj=1,2 satisfy (31). Next, we show the shape of ∣qrogue1∣ in Figure 4(a) by selecting some special parameter values. From Figure 4(a), we can see that qrogue1⟶+∞ as x⟶+∞, and ∣qrogue1∣⟶0 as x⟶−∞. The evolution of this rogue wave at different times is shown in Figure 4(b), where we can see that qrogue1 achieves its maximum at x=1.63698271,t=0. The maximum amplitude is 4.448628515. It is shown that rogue waves exist in integrable systems.Figure 4
The first-order rogue wave for∣qrogue1∣ with parameter selections: k=0.05,ρ=0.001,a=1,b=0.01. (a) ∣qrogue1∣ and (b) its profiles at different times t=−0.5 (red), t=0 (blue), and t=1 (green).
(a)(b)


## 4. Summary

In summary, a determinant representation ofn-fold Darboux transformation for the integrable nonlocal DNLS equation has been obtained. The determinant representation provides a powerful tool to calculate otherwise tedious expansion and it is very useful to calculate the higher-order Taylor expansion in an indeterminate form 0/0 from the double degeneration of eigenvalues. Armed with the Darboux transformation, we have got two particular solutions from zero seed, which are not soliton solutions, and then, their dynamics patterns are analyzed in some details. We show that the properties of these solutions crucially depend on the selection of the parameter values. By choosing different parameter values, we obtain several interesting and novel solutions, which are different from those of the previously studied locally integrable systems. Breather solution is given explicitly by a nonzero seed with constant amplitude. The variable x in the breather solution (30) may be referred to as the ‘transverse variable,’ and the breather periodic in the ‘transverse variable’ corresponds to the so-called ‘Akhmediev’ breather [40]. Finally, we have got a rogue wave solution of Equation (2) from the breather by an extended Taylor expansion method. We have analyzed the dynamical features of the rogue wave and have shown the evolution at different times. Our results pave the way for further studies on higher-order rogue waves of the nonlocal DNLS Equation (2) from the determinant representation (16) of the Darboux transformation. More generally, we hope that our results can enrich the study of nonlocal integrable systems.



---

*Source: 7670773-2022-01-10.xml*