PT -symmetric nonlocal Davey–Stewartson I equation: General lump-soliton solutions on a background of periodic line waves

Journal Pre-proof P T -symmetric nonlocal Davey–Stewartson I equation: General lump-soliton solutions on a background of periodic line waves Jiguang Rao, Jingsong He, Dumitru Mihalache, Yi Cheng

PII: DOI: Reference:

S0893-9659(20)30039-2 https://doi.org/10.1016/j.aml.2020.106246 AML 106246

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Applied Mathematics Letters

Received date : 11 December 2019 Revised date : 18 January 2020 Accepted date : 18 January 2020 Please cite this article as: J. Rao, J. He, D. Mihalache et al., P T -symmetric nonlocal Davey–Stewartson I equation: General lump-soliton solutions on a background of periodic line waves, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2020.106246. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.

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PT -symmetric nonlocal Davey-Stewartson I equation: general lump-soliton solutions on a background of periodic line waves Jiguang Rao 1,2 , Jingsong He 1 , Dumitru Mihalache 3 , Yi Cheng 4 2

1 Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, P. R. China; Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, Guangdong 518060, P. R. China; 3 Horia Hulubei National Institute of Physics and Nuclear Engineering, P.O. Box MG–6, Magurele, RO–077125, Romania; 4 School of Mathematical Sciences, USTC, Hefei, Anhui 230026, P. R. China

Abstract

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We study the general lump-type soliton solutions on a background of periodic line waves in the PT -symmetric nonlocal Davey-Stewartson I (DS I) equation. By using the Kadomtsev-Petviashvili hierarchy reduction method, new families of semi-rational solutions termed as lump-soliton solutions to the nonlocal DS I equation are constructed. Under particular parameters restrictions we obtain: (a) different types of lumps sitting on a background of periodic line waves and (b) different types of lumps interacting with line solitons on a background of periodic line waves. The interactions of lumps and line solitons are inelastic, generating three generic dynamical scenarios: (i) lumps fusing into line solitons, (ii) lumps fissioning from line solitons, and (iii) a combined process consisting of lumps fusing into and fissioning from line solitons. Keywords: PT -symmetric nonlocal DS I equation, lump-soliton solutions, background of periodic line waves, Kadomtsev-Petviashvili hierarchy reduction technique.

1. Introduction

A few years ago, a nonlocal reverse-space nonlocal nonlinear Schr¨odinger (NLS) equation iut (x, t) + u xx (x, t) + 2δu(x, t)u∗ (−x, t)u(x, t) = 0, δ = ±1,

(1)

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was proposed by Ablowitz and Musslimani [1] by considering a new reduction of the Ablowitz-Kaup-NewellSegur (AKNS) hierarchy. Since in the nonlocal NLS equation (1) the solution’s evolution at location x depends on both of the local solution at x and the nonlocal solution at distant position −x, equation (1) is distinctive from the local NLS equation, in which the solution’s evolution only depends on its local solution at x. Besides, equation (1) is related to a hot research area of the contemporary physics: the theoretical and experimental study of parity-time (PT )-symmetric systems [2]. Since the work of Bender and Boettcher [3], the PT -symmetry has attracted lots of researches in nonlinear science [2], namely in optics and photonics, electric circuits, magnetism, mechanical systems, and Bose-Einstein condensation, see, for example, a few recent reviews reporting both theoretical and experimental results in PT -symmetric optical and matter-wave media [4–7]. Due to the physical applications and mathematical interests, various types of soliton solutions have been studied in the nonlocal NLS equation [8–17] and other nonlocal equations[18–20]. To provide multidimensional versions of the nonlocal NLS equation (1), Fokas [21] and Ablowitz and Musslimani [22] proposed the PT -symmetric nonlocal Davey-Stewartson I (DS I) equation: iAt = A xx + Ayy + (AA∗ (−x, −y, t) − 2Q)A,

Q xx − Qyy = (AA∗ (−x, −y, t)) xx ,  = ±1.

(2)

The nonlocal DS I equation (2) is PT -symmetric since it is invariant under the joint transformations of x → −x, y → −y, t → −t and complex conjugation. The integrable properties and solution dynamics to the nonlocal DS I equation (2) have been studied from different points of view, and progresses have been made in the following aspects: the inverse scattering transform for solving the initial value problems [21], the Darboux transformation ∗ Corresponding

author: Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, P. R. China; Email address: [email protected], [email protected] (Jingsong He 1 )

Preprint submitted to AML

January 21, 2020

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for deriving global explicit soliton solutions [23] and rational solutions [24], the Hirota bilinear method and KP hierarchy reduction method for constructing general line solitons on a background of constant or periodic line waves, line rogue waves on a background of constant or periodic line waves, and general lump-type soliton solutions on a constant background [25–27]. However, to the best of our knowledge, the general lump-type soliton solutions on a background of periodic line waves have not been investigated for the PT -symmetric nonlocal DS I equation (2). The difficulty of deriving the lump-type soliton solutions on a background of periodic line waves, lies in constructing the semi-rational solutions simultaneously consisting of lump-type solitons and periodic line waves. In Ref. [27], we constructed lump-soliton solutions given by 2M × 2M determinants by employing the KP hierarchy reduction method. To derive general lump-type soliton solutions on a background of periodic line waves by using the KP hierarchy reduction method, we have to construct the semi-rational tau functions consisting of (2M + 1) × (2M + 1) determinants for the nonlocal DS I equation (2). This letter is organized as follows. In Section 2 we first construct new families of semi-rational solutions to the nonlocal DS I equation (2) via the KP hierarchy reduction method. Then in Section 3 we investigate the evolution scenarios of the general lump-type soliton solutions on a background of periodic line waves in the nonlocal DS I equation (2). Finally, our conclusions are given in Section 4. 2. General lump-type soliton solutions on a background of periodic line waves

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The lump-type solitons setting on a background of periodic line waves are semi-rational solutions, hence we first construct new families of semi-rational solutions to the nonlocal DS I equation (2) by using the KP hierarchy reduction method [28–30]. To this aim, we make the variable transformation √ g A = 2 , Q =  − (2log f ) xx , (3) f where f and g are functions subjects to the symmetric and conjugated condition f (x, y, t)g∗ (−x, −y, t) = f ∗ (−x, −y, t)g(x, y, t).

(4)

Then the nonlocal DS I equation (2) is transformed into the bilinear form : (D2x + D2y − iDt )g · f = 0,

(D2x − D2y − 2) f · f = −2gg∗ (−x, −y, t),

(5)

where D is the Hirota’s bilinear differential operator [28]. According to the Sato theory [31, 32], the following bilinear equations in the KP hierarchy (D2x1 − D x2 )τn+1 · τn = 0,

(D2x−1 + D x−2 )τn+1 · τn = 0,

(6)

(D x1 D x−1 − 2)τn · τn = −2τn+1 τn−1 ,

admit the tau function τn = det (m(n) s, j ), and the matrix elements are given by m(n) s, j =(−

n0 n0 X p s n ξs +η j p s n X 1 0 0 )e (− ) [ c s,k (p s ∂ ps + ξ s + n)n0 −k d j,l (q j ∂q j + η j − n)n0 −l ] + δ s, je cs , qj q j k=0 p + qj s l=0

1 1 1 1 x−2 + x−1 + p s x1 + p2s x2 + p se ξ s , η j = − 2 x−2 + x−1 + q j x1 − q2j x2 + q je η j, ps qj p2s qj

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ξs =

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1≤s, j≤N

0

ξs = −

(7)

2 1 2 1 0 x−2 − x−1 + p s x1 + 2p2s x2 + p se ξ s , η j = 2 x−2 − x−1 + q j x1 − 2q2j x2 + q je η j, ps q p2s qj j

where N and n0 are positive integers, respectively, and p s , q j , c s,k , d j,l ,e cs , e ξs , e η j are freely complex constants, and δ s, j is the Kronecker delta. Furthermore, by making the change of independent variables x−2 =

i  1 i t, x−1 = (x − y), x1 = (x + y), x2 = − t, 2 2 2 2

(8)

and adding the first bilinear equation to the second bilinear equation in (6), and multiplying by  both sides of the 2

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third equation in (6), the bilinear equations in (6) reduce to the following bilinear equations (D2x + D2y − iDt )τn+1 · τn = 0,

(D2x − D2y − 2)τn · τn = −2τn+1 τn−1 .

(9)

Further, if we consider the (2M + 1) × (2M + 1) matrix for the tau function τn with the parameter restrictions N = 2M + 1, qι = p∗ι , e ηι = −e ξι∗ , dι,l = c∗ι,l for ι, κ = 1, 2, · · · 2M, l = 1, 2, · · · n0 , and p M+s = −p s ,e c M+s = ∗ ∗ ∗ e e e −e c s , c M+s, j = c s, j , c s,M+ j = c s, j , ξ M+s = ξ s for s, j = 1, 2, · · · M, and q2M+1 = −p2M+1 , e η2M+1 = −ξ2M+1 , d2M+1,l = c∗2M+1,l ,e c2M+1,2M+1 = ic, where p s , c s, j ,e c j, e ξ s , p2M+1 , c2M+1, j , e ξ2M+1 are complex, and c is real, one can easily obtain the following identities: (−n) (n)∗ (−n) (n)∗ (−n) m(n)∗ M+s,M+ j (−x, −y, t) = −m j,s (x, y, t), m M+s, j (−x, −y, t) = −m M+ j,s (x, y, t), m s,M+ j (−x, −y, t) = −m j,M+s (x, y, t),

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(−n) (n)∗ (−n) (n)∗ (−n) m(n)∗ s,2M+1 (−x, −y, t) = −m2M+1,M+s (x, y, t), m2M+1, j (−x, −y, t) = −m M+ j,2M+1 (x, y, t), m2M+1,2M+1 (−x, −y, t) = −m2M+1,2M+1 (x, y, t), (10) which yield τ∗n (−x, −y, t) = (−1)3M τ−n (x, y, t) and τ0 (x, y, t)τ∗−1 (−x, −y, t) = τ∗0 (−x, −y, t)τ1 (x, y, t). The bilinear equation (9) would translate to the bilinear equation (5) of the nonlocal DS I equation for τ0 (x, y, t) = f (x, y, t), τ1 (x, y, t) = g(x, y, t). By summarizing the above results, we can present the semi-rational solutions to the nonlocal DS I equation in the following Theorem.

Theorem 1. The nonlocal DS I equation (2) has the following semi-rational solutions (3) with f and g given by (2M + 1) × (2M + 1) determinants f = τ0 and g = τ1 , where τn = det (m(n) s, j ), the matrix elements are given by 1≤s, j≤2M+1

m(n) s, j

n0 n0 X 1 p s n ξs +η j p s n X 0 0 n0 −k (− ) [ c s,k (p s ∂ ps + ξ s + n) + δ s, je cs , d j,l (q j ∂q j + η j − n)n0 −l ] =(− ) e qj q j k=0 p + qj s l=0

 1  1 1 1  1  1 1 1 ξ s , η j = (q j + )x + (q j − )y + (q2j − 2 )it − q je ξ s = (p s + )x + (p s − )y − (p2s − 2 )it + p se ξ∗s , 2 ps 2 ps 2 2 qj 2 qj 2 ps qj

1 1  1  1  1  1 0 0 ξ s , η j = (q j − )x + (q j + )y + (q2j + 2 )it − q je ξ∗j , ξ s = (p s − )x + (p s + )y − (p2s + 2 )it + p se 2 ps 2 ps 2 qj 2 qj ps qj (11) and the parameter restrictions are as follows: qι = p∗ι , dι,l = c∗ι,l , p M+s = −p s ,e c M+s = −e c∗s , c M+s, j = c s, j , c s,M+ j = c s, j ,

e ξ M+s = e ξ s , q2M+1 = −p∗2M+1 , d2M+1,n0 −l = c∗2M+1,n0 −l ,e c2M+1,2M+1 = ic

(12)

for ι, κ = 1, 2, · · · 2M, l = 1, 2, · · · n0 , and s, j = 1, 2, · · · M, where p s , p2M+1 , e ξs , e ξ2M+1 , c s, j , c2M+1,n0 −l ,e c s are freely complex parameters, and c is a real parameter, and M and n0 are nonnegative integers.

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Remark 1. When c2M+1,n0 −ν , 0 (ν = 1, 2, · · · n0 − 1), these semi-rational solutions always possess singular points. Besides, when c = 0 and e c s = 0, and further using the gauge invariance of τn , the corresponding semi-rational solutions reduce to rational solutions to the nonlocal DS I equation, which are also singular. Hereafter we take c2M+1,n0 −ν = 0, c2M+1,n0 , 0 and c , 0 to avoid the singular solutions. Remark 2. The general line solitons on a background of periodic line waves in the nonlocal DS I equation (2) have been considered in [27]. We point out here that the line solitons on a background of periodic line waves for the nonlocal DS I equation (2), which were investigated in Ref. [27], correspond to special solutions obtained using the above Theorem, when n0 = 0 or c s,0 = c s,1 = · · · = c s,n0 −1 = 0, c s,n0 , 0. Remark 3. For e c s = 0, c , 0, the exact solutions that are obtained using the above Theorem are lumps sitting on a background of periodic line waves Remark 4. For ce c s , 0, the exact solutions that are obtained using the above Theorem describe the interaction beteeen lumps and line solitons on a background of periodic line waves. Three generic types of interaction phenomena are revealed: (a) lumps fusing into line solitons, (b) lumps fissioning from line solitons, and (c) a combination of fusion and fission phenomena involving lumps and line solitons. 3. Evolution scenarios of lump-soliton solutions In this Section, based on the findings contained in Theorem 1, we describe the scenarios of evolution of general lump-soliton solutions on a background of periodic line waves. 3

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3.1. Fundamental (first-order) lump-soliton solutions The fundamental (first-order) lump-soliton solution (3), composed of (a) either two lumps or two line solitons and (b) two lumps and two line solitons, on a background of periodic line waves, can be given explicitly by taking M = 1, n0 = 1 in Theorem 1: (1) (0) m1,1 m(1) m1,1 m(0) 1,2 1,2 (13) f = (0) , , g = (1) m2,1 m(0) m2,1 m(1) 2,2 2,2 where m(n) s, j are given by (11) with the following parametric restrictions:

q j = p∗j , p2 = −p1 , q3 = −p∗3 , d s,µ = c∗s,µ , c2,µ = c1,µ ,e c2 = −e c∗1 ,

(14)

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for j = 1, 2, and s = 1, 2, 3, and µ = 0, 1. With different parametric choices, these lump-soliton solutions possess three different dynamic behaviours: (1) Two line solitons on a periodic line waves background when c1,0 = 0, c1,1 , 0. We note that these solutions have been discussed in Ref. [27]. (2) Two lumps on a background of periodic line waves when c1,0 , 0,e c1 = 0. In this case, the semi-rational solutions only comprise of two lumps on a background of periodic line waves. Similar to the case of two lumps sitting on a constant background (i.e., a continuous wave background), the waveforms of the two lumps are determined by p2 the parameters p1 and e ξ1 . When e ξ1 = 0, the two lumps have similar waveforms: (i) bright lumps (for 0 < p21I ≤ 13 ),

(ii) four-petaled lumps (for

1 3

p21I p21R

p21I p21R

1R

< 3), and (iii) dark lumps (for ≥ 3), where p1R and p1I are the real and e imaginary parts of p1 , respectively. When ξ1 , 0, the waveforms of the two lumps could be different, and there are another three different wave structures: (iv) bright-dark lumps, (v) bright-four-petaled lumps, and (vi) darkfour-petaled lumps. The upper row of Fig. 1 shows four of the above mentioned six different types of lump-soliton solutions: two bright lumps, two four-petaled lumps, two dark lumps, and one bright lump and one dark lump. <

(i) Four types of two-lump solutions on a background of periodic line waves

(ii) Two lumps fusing into two line solitons on a background of periodic line waves

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(iii) Two lumps fissioning from two line solitons on a background of periodic line waves

Figure 1: (Colour online) The upper row shows two line solitons on a background of periodic line waves at time t = 2: (a) Two bright lumps for  = 1, p1 = 1 + 31 i, p3 = 34 i,e c1 = 0, c1,0 = 1, c1,1 = 1,e c3 = 1, e ξ1 = 0, ξ3 = π5 i; (b) Two four-petaled lumps for  = 1, p1 = 1 + i, p3 = i,e c1 = 0, c1,0 = 1, c1,1 = 1,e c3 = 1, e ξ1 = 0, ξ3 = π2 i; (c) Two dark lumps for  = 1, p1 = 31 + i, p3 = i,e c1 = 0, c1,0 = 1, c1,1 = 1,e c3 = 1, e ξ1 = 0, ξ3 = 2π 3 i; (d) One bright lump and one dark lump for  = 1, p1 = 1 + 31 i, p3 = i,e c1 = 0, c1,0 = 1, c1,1 = 1,e c3 = 1, e ξ1 = 51 , ξ3 = π3 i. The middle row shows the time evolution of two lumps fusing into two line solitons on a background of periodic line waves for  = 1, p1 = −1 + 13 i, p3 = 34 i,e c1 = −1, c1,0 = 1, c1,1 = 1,e c3 = 1, e ξ1 = 0, ξ3 = π2 i, at different values of time t = −5, −1, 0, 5. The bottom line shows the time evolution of two lumps fissioning from two line solitons on a background of periodic line waves for  = 1, p1 = 1 + 31 i, p3 = 43 i,e c1 = 1, c1,0 = 1, c1,1 = 1,e c3 = 1, e ξ1 = 0, ξ3 = π2 i, at different values of time t = −5, 0, 1, 5.

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(3) Two lumps and two line solitons on a background of periodic line waves when c1,0 , 0, c3,0 = 0,e c1 = 0. In this case, the corresponding solutions illustrate the interaction of two lumps and two line solitons on a background of periodic line waves. The interaction gives rise to two different types of dynamic phenomena: (i) two lumps fusing into two line solitons (for p1R < 0) and (ii) two lumps fissioning from two line solitons (for p1R > 0). The middle row of Fig. 1 shows the time evolution of the solutions with p1R < 0: at the initial stage of the evolution, the lump-soliton solution is built of two line solitons and two lumps (see the panels at t = −5); as the evolution progresses, the two lumps inevitably interact with the two line solitons (see the panel at t = −1, 0); at the final stage of the evolution, the two lumps completely disappear into the two line solitons (see the panel at t = 5). In this evolution process, the initial two lumps get absorbed by the two line solitons. The time evolution of the solution with p1R > 0 is shown in the bottom row of Fig. 1, which illustrates a process opposite to that when p1R < 0, shown in the middle row of Fig. 1. In this evolution process, two lumps get created from the two line solitons. 3.2. Multi-lump-soliton solutions

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The multi-lump-soliton solutions can be generated from Theorem 1 by taking M > 1, n0 = 1. This subclass of non-fundamental lump-soliton solutions displays the collision of M individual fundamental lump-soliton solutions, and have two different dynamical behaviours: 2M lumps on periodic line waves background for e c s = 0 (s = 1, 2, · · · M), or the collision of 2M lumps and 2M line solitons on a background of periodic line waves for e c s , 0. The later dynamical behaviour would generate three types of different evolution scenarios: (i) 2M lumps fusing into 2M line solitons on a background of periodic line waves for p sR < 0; (ii) 2M lumps fissioning from 2M line solitons on a background of periodic line waves for p sR > 0; and (iii) a mixture of 2s lumps fusing into and 2(M − s) lumps fissioning from the 2M line solitons for p s,R < 0 and p M−s,R > 0 (1 ≤ s < M). 3.3. Higher-order lump-soliton solutions

Other subclass of non-fundamental lump-soliton solutions termed as higher-order lump-soliton solutions is obtained from Theorem 1 for M = 1, n0 ≥ 2. These higher-order lump-soliton solutions also possess two different dynamical behaviours: 2n0 lumps on a background of periodic line waves (for e c1 = 0), and collision of 2n0 lumps and two line solitons on a background of periodic line waves (for e c1 , 0). The collision of 2n0 lumps and two line solitons gives rise to two opposite inelastic interaction phenomena: (a) 2n0 lumps fusing into two line solitons when p1R < 0 and (b) 2n0 lumps fissioning from two line solitons when p1R > 0. Figure 2 shows the time evolution of the higher-order lump-soliton solution for n0 = 2, p1R > 0. It is seen that four lumps are generated from the two line solitons in this evolution process.

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Figure 2: (Colour online) Four lumps fissioning from two line solitons on a background of periodic line waves. Here the values of the parameters are as follows:  = 1, p1 = −1 + 13 i, p3 = 34 i,e c1 = −1, c1,0 = 1, c1,1 = 0, c1,2 = 1,e c3 = 1, e ξ1 = 0, ξ3 = π2 i.

4. Summary and Discussion

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In this letter, for the PT -symmetric nonlocal DS I equation (2), we have considered the general lump-type soliton solutions on a background of periodic line waves. By using the bilinear KP hierarchy reduction method, we have derived new families of semi-rational solutions termed as lump-soliton solutions to the nonlocal DS I equation, which are given by (2M + 1) × (2M + 1) determinants. The fundamental lump-soliton solutions have two different dynamical behaviours by modifying the input parameters: (a) two lumps sitting on a background of periodic line waves and (b) two lumps fusing into or fissioning from the two line solitons, on a background of periodic line waves, see Fig. 1. There are two subclasses of non-fundamental lump-soliton solutions: the multilump-soliton solutions and higher-order lump-soliton solutions. The multi-lump-soliton solutions display (a) 2M lumps sitting on a background of periodic line waves and (b) 2M lumps fusing into or fissioning from 2M line solitons, on a background of periodic line waves. The higher-order lump-soliton solutions show either 2n0 (n0 ≥ 2) lumps or 2n0 lumps fusing into or fissioning from two line solitons, on a background of periodic line waves.

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Acknowledgment This work was supported by the NSF of China under Grant Nos. 11671219 and 11871446. References

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[6]

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Journal Pre-proof Credit Author Statement Jiguang Rao and Jingsong He constructed the solutions and wrote the paper. DumitruMihalache,and Yi Cheng provided the analysis of the solutions and physical concerns of the model. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. All the authors gave their final approval for publication.

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Jiguang Rao, Jingsong He, Dumitru Mihalache, Yi Cheng

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