Darboux transformations and solutions of nonlocal Hirota and Maxwell-Bloch equations
DDarboux transformations and solutions of nonlocal Hirotaand Maxwell-Bloch equations
Ling An † , Chuanzhong Li †‡ ∗ , Lixiang Zhang † † School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China ‡ College of Mathematics and Systems Science, Shandong University of Science and Technology,Qingdao, 266590, China
Abstract
In this paper, based on the Hirota and Maxwell-Bloch (H-MB) system and its applica-tion in the theory of the femtosecond pulse propagation through an erbium doped fiber,we define two kinds of nonlocal Hirota and Maxwell-Bloch (NH-MB) systems, namely,
P T -symmetric NH-MB system and reverse space-time NH-MB system. Then we con-struct the Darboux transformations of these NH-MB systems. Meanwhile, we derive theexplicit solutions by the Darboux transformations.
Mathematics Subject Classifications (2010) : 37K05, 37K10, 35Q53.
Keywords: nonlocal Hirota and Maxwell-Bloch system, Darboux transformation, ex-plicit solution.
Contents
P T -symmetric NH-MB system 8 N -fold Darboux transformation . . . . . . . . . . . . . . . . . . . . . . . . 10 P T -symmetric NH-MB system 115 Discussions of the reverse space-time NH-MB system 186 Conclusions 20 ∗ Corresponding author:[email protected] a r X i v : . [ n li n . S I] F e b Introduction
For a long time, it is difficult and tedious to find the interaction solutions of the nonlin-ear systems, but it is also meaningful and important. In order to solve the problems of thenonlinear systems, we introduce the symmetry theory [1], which plays an important rolein both integrable and non-integrable systems. The theory of symmetries involves a widerange of contents. Recently, many scholars pay more attention to the nonlocal symmetry,which is proposed by Vinogradov and Krasilshchik in 1980 [2]. And some studies haveshown that the nonlocal symmetry method [3–5] is one of the best tools for solving thenonlinear systems. The nonlocal symmetry method is to establish a relationship betweenthe local equations and their corresponding nonlocal equations by selecting appropriatesymmetry, so as to study their properties and solutions. Due to the forms of the symme-tries are different, there are great differences in the coupling of time and space betweenthese local and nonlocal equations. Therefore new physical phenomena may appear andnew physical applications may be produced.Here, we use AKNS system as an example to explain the origin of nonlocal equationsin detail. We all know the soliton equations have many peculiar properties [6], the mostfundamental property of them is that they all can be represented by the integrabilityconditions of a pair of linear problems, as shown below ϕ t = M ϕ, (1.1) ϕ x = N ϕ, (1.2)where ϕ = ( ϕ , ϕ , · · · ϕ n ) T is a n -dimensional column vector, M and N are n -ordermatrices, whose elements depend on the spectral parameters λ and the potential u .In order to make the Eqs. (1.1) and (1.2) hold at the same time, ϕ must satisfy thecompatibility condition ϕ xt = ϕ tx , thus M and N must satisfy M x − N t + [ M, N ] = 0 , (1.3)then Eq. (1.3) is called Lie group structural equation or zero curvature equation.If we choose different M and N , we can derive different soliton equations. For AKNSsystem, we usually take˜ M = (cid:32) − iλ q ( x, t ) r ( x, t ) iλ (cid:33) , ˜ N = (cid:32) A ( x, t ) B ( x, t ) C ( x, t ) − A ( x, t ) (cid:33) , (1.4)where A , B and C are functions containing spectral parameters λ , functions q ( x, t ) , r ( x, t )and their derivatives.Recently, a series of new nonlocal equations have been reported and studied. The mostdistinctive kind of nonlocal equations are the equations with P T -symmetry [7, 8]. Anequation with
P T -symmetry means that the equation is invariant under the joint actionof
P T -operator [3]. In 2013, Ablowitz and Musslimani proposed the first
P T -symmetricequation. They chose the relation r ( x, t ) = q ∗ ( − x, t ), then they got the P T -symmetricnonlinear Schr¨ o dinger (NLS) equation [4] iq t ( x, t ) = q xx ( x, t ) − σq ( x, t ) q ∗ ( − x, t ) , σ = ∓ . (1.5)Let’s rewrite the above equation appropriately iq t ( x, t ) = q xx ( x, t ) + V [ q, x, t ] q ( x, t ) , (1.6)where V [ q, x, t ] = − σq ( x, t ) q ∗ ( − x, t ) is called a self-induced potential of this equation,and the self-induced potential satisfies the condition of P T -symmetry V [ q, x, t ] = V ∗ [ q, − x, t ] . o we call Eq. (1.5) the P T -symmetric NLS equation.In 2016 [9], Ablowitz and Musslimani introduced the integrable nonlocal nonlinearequations systematically, and summarized some nonlocal symmetric forms of AKNS scat-tering problem, which can be divided into the types r ( x, t ) = σq ∗ ( ς x, ς t ) , r ( x, t ) = σq ( ς x, ς t ) , σ, ς , ς = ± . (1.7)According to these symmetries, we can derive different nonlocal equations.In this paper, we will focus on the nonlocal form of the Hirota and the Maxwell-Blochsystem. The nonlinear Schr¨ o dinger equation is a very important equation, which hasmany physical applications. In 1985, Kodama found that the higher-order NLS equationcould be reduced to the Hirota equation via an appropriate transformation. The Maxwellequation is the core of the electromagnetic theory. It was proposed in the theory ofelectromagnetic field dynamics, which was published by British physicist James ClerkMaxwell in 1865 [10]. The Bloch equation is one of the important theoretical foundationsof nuclear magnetic resonance, and it is also the theoretical basis for the study of transientphenomena of coherent light. And it was proposed by Bloch, Hansen and Packard in1946 [11]. With the continuous study of the soliton equations, many scholars begin to focuson the coupling of the equations. For the above equations, if the Maxwell equation andthe Bloch equation coupling, then we can obtain the Maxwell-Bloch (MB) system [12]. Bycombining the NLS equation and the MB system together, then we will get the nonlinearSchr¨ o dinger and the Maxwell-Bloch (NLS-MB) system [13, 14]. Accordingly, on the basisof the NLS equation, we can choose appropriate self steepening and self frequency effects,then the NLS-MB system can be reduced to a coupling system of the Hirota and theMaxwell-Bloch system [15–17]. The forms of the H-MB system are given below E x = β ( E ttt + 6 | E | E t ) + i α ( E tt + 2 | E | E ) + 2 p,p t = 2 Eη + 2 iωp,η t = − ( pE ∗ + p ∗ E ) , (1.8)among them, x and t represent space and time variables respectively, E ( x, t ) and p ( x, t ) arecomplex variables, η ( x, t ) is a real variable corresponding to the extent of the populationinversion, α, β and ω are real constants and ω represents the frequency.In addition, the methods of solving the integrable equations have been developed inrecent years, such as Darboux transformation [18–20], Hirota’s direct method [21, 22],inverse scattering method [23, 24] and so on. The Darboux transformation is one of themost effective methods to solve such equations. In [15], the authors found many kinds ofsolutions of H-MB system by using the Darboux transformation method. Therefore, thispaper will also use the Darboux transformation to solve the NH-MB system.The paper is organized as follows. In section 2, we obtain standard H-MB systemand two kinds of NH-MB systems: P T -symmetric NH-MB system and reverse space-timeNH-MB system. In section 3, we discuss the Darboux transformation of
P T -symmetricNH-MB system, give the concrete forms of the new solutions E (cid:48) , p (cid:48) , η (cid:48) under the one-foldDarboux transformation, and give the determinant form of the n -fold Darboux matrix.In section 4, the solutions of P T -symmetric NH-MB system are discussed. Because ofthe particularity of η , i.e. η ( x, t ) = − η ( − x, t ), the seed solutions satisfying the systemare obviously less than those of standard H-MB system, so we only take two typical seedsolutions. In section 5, we also discuss the Darboux transformation of the reverse space-time NH-MB system and analyze the solutions of this system. Our conclusions are statedin section 6. Zero curvature equation
For the H-MB system which belongs to AKNS system discussed in this paper, similarly,we can take M = (cid:32) − iλ q ( x, t ) r ( x, t ) iλ (cid:33) , N = (cid:32) N ( x, t ) N ( x, t ) N ( x, t ) − N ( x, t ) (cid:33) . (2.1)Substituting (2.1) into the compatibility condition (1.3), we can obtain N t ( x, t ) = q ( x, t ) N ( x, t ) − r ( x, t ) N ( x, t ) ,q x ( x, t ) = N t ( x, t ) + 2 iλN ( x, t ) + 2 q ( x, t ) N ( x, t ) ,r x ( x, t ) = N t ( x, t ) − iλN ( x, t ) − r ( x, t ) N ( x, t ) . (2.2)Next, we make N ( x, t ) = 4 iβλ − iαλ + 2 iβqrλ + (cid:104) β ( qr t − rq t ) − i αqr (cid:105) + iη ( λ + ω ) − ,N ( x, t ) = − βqλ + ( − iβq t + αq ) λ + (cid:104) β ( q tt − q r ) + i αq t (cid:105) − iδp ( λ + ω ) − ,N ( x, t ) = − βrλ + (2 iβr t + αr ) λ + (cid:104) β ( r tt − qr ) − i αr t (cid:105) − iδm ( λ + ω ) − , that is to say N = (cid:32) iβ − iβ (cid:33) λ + (cid:32) − iα − βq − βr iα (cid:33) λ + (cid:32) iβqr − iβq t + αq iβr t + αr − iβqr (cid:33) λ + (cid:32) β ( qr t − rq t ) − i αqr β ( q tt − q r ) + i αq t β ( r tt − qr ) − i αr t − β ( qr t − rq t ) + i αqr (cid:33) + iλ + ω (cid:32) η − δp − δm − η (cid:33) := N λ + N λ + N λ + N + iλ + ω N − . (2.3)Plugging the concrete forms of N ( x, t ) , N ( x, t ) and N ( x, t ) into (2.2), five equalitiesabout functions q ( x, t ) , r ( x, t ) , p ( x, t ) , m ( x, t ) and η ( x, t ) can be obtained q x = β ( q ttt − qrq t ) + i α ( q tt − q r ) + 2 δp,r x = β ( r ttt − qrr t ) − i α ( r tt − qr ) − δm,p t = 2 δqη + 2 iωp,m t = − δrη − iωm,η t = δ ( pr − mq ) . (2.4)If q ( x, t ) , r ( x, t ) , p ( x, t ) and m ( x, t ) are selected properly, the above equalities can bereduced to three equalities. Standard symmetry: r ( x, t ) = σq ∗ ( x, t ) = σE ∗ ( x, t ) , m ( x, t ) = δp ∗ ( x, t ) . (2.5) aking the above symmetric forms into Eq. (2.4), we can obtain E x = β ( E ttt − σEE ∗ E t ) + i α ( E tt − σE E ∗ ) + 2 δp,σE ∗ x = σβ ( E ∗ ttt − σEE ∗ E ∗ t ) − i ασ (cid:0) E ∗ tt − σEE ∗ (cid:1) − p ∗ ,p t = 2 δEη + 2 iωp,δp ∗ t = − σδE ∗ η − iδωp ∗ ,η t = σδpE ∗ − p ∗ E, (2.6)when σ = − , δ = 1 and α, β, ω ∈ R , there are E x = β ( E ttt + 6 EE ∗ E t ) + i α ( E tt + 2 E E ∗ ) + 2 p,E ∗ x = β ( E ∗ ttt + 6 EE ∗ E ∗ t ) − i α (cid:0) E ∗ tt + 2 EE ∗ (cid:1) + 2 p ∗ ,p t = 2 Eη + 2 iωp,p ∗ t = 2 E ∗ η − iωp ∗ ,η t = − ( pE ∗ + p ∗ E ) . (2.7a)(2.7b)(2.7c)(2.7d)(2.7e)By taking the conjugate on both sides of equality (2 . b ), we can get (2 . a ). Similarly,the equality (2 . d ) can be reduced to (2 . c ) , then the equalities (2 . a ), (2 . c ) and (2 . e )constitute the H-MB system (1.8).When α = 1 , β = 0, the H-MB system can be converted to the NLS-MB system E x = i (cid:0) E tt + 2 | E | E (cid:1) + 2 p,p t = 2 Eη + 2 iωp,η t = − ( pE ∗ + p ∗ E ) . (2.8)When α = 0 , β = 1, the H-MB system can be converted to the mKdV-MB system E x = E ttt + 6 | E | E t + 2 p,p t = 2 Eη + 2 iωp,η t = − ( pE ∗ + p ∗ E ) . (2.9) P T -symmetric NH-MB system
P T -symmetry: r ( x, t ) = σq ∗ ( − x, t ) = σE ∗ ( − x, t ) , m ( x, t ) = δp ∗ ( − x, t ) . (2.10)Substituting the above symmetric forms into Eq. (2.4), we consider the first and secondequalities of Eq. (2.4), there are E x ( x, t ) = β ( E ttt ( x, t ) − σE ( x, t ) E ∗ ( − x, t ) E t ( x, t ))+ i α ( E tt ( x, t ) − σE ( x, t ) E ∗ ( − x, t )) + 2 δp ( x, t ) , (2.11) E ∗ x ( − x, t ) = σβ ( E ∗ ttt ( − x, t ) − σE ( x, t ) E ∗ ( − x, t ) E ∗ t ( − x, t )) − i ασ (cid:0) E ∗ tt ( − x, t ) − σE ( x, t ) E ∗ ( − x, t ) (cid:1) − p ∗ ( − x, t ) . (2.12)If we do the variable transformations x → − x and t → t for Eq. (2.12), and then take theconjugate on both sides of Eq. (2.12) at the same time, we get E x ( x, t ) = − β ∗ ( E ttt ( x, t ) − σE ( x, t ) E ∗ ( − x, t ) E t ( x, t )) − i α ∗ ( E tt ( x, t ) − σE ( x, t ) E ∗ ( − x, t )) + 2 σp ( x, t ) . (2.13)To make Eq. (2.13) compatible with Eq. (2.11), if and only if σ = δ, β ∗ = − β, α ∗ = − α, that is, α and β need to be pure imaginary numbers. Next we consider the third andfourth equalities of Eq. (2.4), which become p t ( x, t ) = 2 δE ( x, t ) η ( x, t ) + 2 iωp ( x, t ) , (2.14) δp ∗ t ( − x, t ) = − σδE ∗ ( − x, t ) η ( x, t ) − iδωp ∗ ( − x, t ) . (2.15)If we do the same transformation for Eq. (2.15) as for Eq. (2.12), we get p t ( x, t ) = − σE ( x, t ) η ( − x, t ) + 2 iω ∗ p ( x, t ) . (2.16)To make Eq. (2.16) compatible with Eq. (2.14), if and only if ω ∗ = ω, η ( x, t ) = − σδη ( − x, t ) . Combining the above discussion, we can get the constraint conditions σ = δ, ω ∈ R , α, β, are all pure imaginary numbers and η ( x, t ) = − η ( − x, t ) , and under these conditions, ifwe do the same transformation for the fifth equality of Eq. (2.4), we can find that it iscompatible with itself. Then we can get the P T -symmetric NH-MB system E x ( x, t ) = β (cid:0) E ttt ( x, t ) − σE ( x, t ) E ∗ ( − x, t ) E t ( x, t ) (cid:1) + i α (cid:0) E tt ( x, t ) − σE ( x, t ) E ∗ ( − x, t ) (cid:1) + 2 σp ( x, t ) ,p t ( x, t ) =2 σE ( x, t ) η ( x, t ) + 2 iωp ( x, t ) ,η t ( x, t ) = p ( x, t ) E ∗ ( − x, t ) − p ∗ ( − x, t ) E ( x, t ) , (2.17)where α = iε , β = iε , ε , ε , ω ∈ R , and satisfy the requirement of η ( x, t ) = − η ( − x, t ) . Since the
P T -symmetric NH-MB system (2.17) comes out of the new symmetry reduc-tions, it is also an infinite dimensional integrable Hamiltonian system. An infinite numberof conservation laws of Eq. (2.17) can be obtained by expanding the spectral parameter λ at the point of infinity. Here, we gave the general formulas of the conserved quantities I = (cid:90) + ∞−∞ (cid:104) ση ( x, t ) − iσε E ( x, t ) E ∗ ( − x, t ) + ε (cid:0) E ( x, t ) E ∗ t ( − x, t ) − E t ( x, t ) E ∗ ( − x, t ) (cid:1) + iε (cid:0) E ( x, t ) E ∗ tt ( − x, t )+ E tt ( x, t ) E ∗ ( − x, t ) − E t ( x, t ) E ∗ t ( − x, t ) (cid:1)(cid:105) dt,I n +1 = (cid:90) + ∞−∞ (cid:104) ( − n +1 ( iω ) n η ( x, t ) + 2 σp ( x, t ) n − (cid:88) j =0 ( − iω ) j W n − j − ( x, t ) E ( x, t ) + iε W n +2 ( x, t )+ (cid:0) − iε E t ( x, t ) E ( x, t ) + ε (cid:1) W n +1 ( x, t ) + (cid:16) iε (cid:0) E tt ( x, t ) E ( x, t ) − σE ( x, t ) E ∗ ( − x, t ) (cid:1) − ε E t ( x, t )2 E ( x, t ) (cid:17) W n ( x, t ) (cid:105) dt, n ≥ , (2.18) here W ( x, t ) = − σE ( x, t ) E ∗ ( − x, t ) , W ( x, t ) = − σE ( x, t ) E ∗ t ( − x, t ) ,W n ( x, t ) = E ( x, t )( W n − ( x, t ) E ( x, t ) ) t + n − (cid:88) j =0 W j ( x, t ) W n − j − ( x, t ) , n ≥ . Reverse space-time symmetry: r ( x, t ) = σq ( − x, − t ) = σE ( − x, − t ) , m ( x, t ) = δp ( − x, − t ) = σp ( − x, − t ) . (2.19)Taking the above symmetric forms into Eq. (2.4), and using a similar approach to thatof the P T -symmetric NH-MB system, we can get the reverse space-time NH-MB system E x ( x, t ) = β (cid:0) E ttt ( x, t ) − σE ( x, t ) E ( − x, − t ) E t ( x, t ) (cid:1) + i α (cid:0) E tt ( x, t ) − σE ( x, t ) E ( − x, − t ) (cid:1) + 2 σp ( x, t ) ,p t ( x, t ) = 2 σE ( x, t ) η ( x, t ) + 2 iωp ( x, t ) ,η t ( x, t ) = p ( x, t ) E ( − x, − t ) − p ( − x, − t ) E ( x, t ) , (2.20)where α, β ∈ C , ω ∈ R , and satisfying the requirement of η ( x, t ) = η ( − x, − t ). Theconserved quantities associated with Eq. (2.20) are I = (cid:90) + ∞−∞ (cid:104) ση ( x, t ) − σβE ( x, t ) E ( − x, − t ) − iα (cid:0) E ( x, t ) E t ( − x, − t ) − E t ( x, t ) E ( − x, − t ) (cid:1) + β (cid:0) E ( x, t ) E tt ( − x, − t ) + E tt ( x, t ) E ( − x, − t ) − E t ( x, t ) E t ( − x, − t ) (cid:1)(cid:105) dt,I n +1 = (cid:90) + ∞−∞ (cid:104) ( − n +1 ( iω ) n η ( x, t ) + 2 σp ( x, t ) n − (cid:88) j =0 ( − iω ) j W n − j − ( x, t ) E ( x, t )+ βW n +2 ( x, t ) − (cid:0) β E t ( x, t ) E ( x, t ) + iα (cid:1) W n +1 ( x, t )+ (cid:16) β (cid:0) E tt ( x, t ) E ( x, t ) − σE ( x, t ) E ( − x, − t ) (cid:1) + iαE t ( x, t )2 E ( x, t ) (cid:17) W n ( x, t ) (cid:105) dt, n ≥ , (2.21)where W ( x, t ) = − σE ( x, t ) E ( − x, − t ) , W ( x, t ) = − σE ( x, t ) E t ( − x, − t ) ,W n ( x, t ) = E ( x, t )( W n − ( x, t ) E ( x, t ) ) t + n − (cid:88) j =0 W j ( x, t ) W n − j − ( x, t ) , n ≥ . For the other symmetric forms in (1.7), we found that Eq. (2.4) can not be suc-cessfully coupled through calculation and analysis. Here, we take the symmetric form r ( x, t ) = σq ∗ ( − x, − t ) = σE ∗ ( − x, − t ) , m ( x, t ) = δp ∗ ( − x, − t ) , as an example to illus-trate. Substituting the above symmetric forms into Eq. (2.4), there are E x ( x, t ) = β ( E ttt ( x, t ) − σE ( x, t ) E ∗ ( − x, − t ) E t ( x, t ))+ i α ( E tt ( x, t ) − σE ( x, t ) E ∗ ( − x, − t )) + 2 δp ( x, t ) , (2.22) E ∗ x ( − x, − t ) = σβ ( E ∗ ttt ( − x, − t ) − σE ( x, t ) E ∗ ( − x, − t ) E ∗ t ( − x, − t )) − i ασ (cid:0) E ∗ tt ( − x, − t ) − σE ( x, t ) E ∗ ( − x, − t ) (cid:1) − p ∗ ( − x, − t ) . (2.23) p t ( x, t ) = 2 δE ( x, t ) η ( x, t ) + 2 iωp ( x, t ) , (2.24) δp ∗ t ( − x, − t ) = − σδE ∗ ( − x, − t ) η ( x, t ) − iδωp ∗ ( − x, − t ) , (2.25) η t ( x, t ) = σδp ( x, t ) E ∗ ( − x, − t ) − p ∗ ( − x, − t ) E ( x, t ) . (2.26)If we do the variable transformations x → − x and t → − t for Eqs. (2.23) and (2.25), andthen take the conjugate on both sides of them, we can obtain E x ( x, t ) = β ∗ ( E ttt ( x, t ) − σE ( x, t ) E ∗ ( − x, − t ) E t ( x, t )) − i α ∗ ( E tt ( x, t ) − σE ( x, t ) E ∗ ( − x, − t )) + 2 σp ( x, t ) . (2.27) p t ( x, t ) = 2 σE ( x, t ) η ( − x, − t ) − iω ∗ p ( x, t ) . (2.28)In order to make Eq. (2.22) compatible with Eq. (2.27), it can be found that conditions α ∗ = − α, β ∗ = β, σ · δ = 1 , must be satisfied, while conditions ω ∗ = − ω, η ( x, t ) = σ · δη ( − x, − t ) , must be satisfied when Eq. (2.24) compatible with Eq. (2.28). But ω represents thefrequency which must be a real constant. That is, Eq. (2.4) cannot be coupled successfullywhen the symmetric forms r ( x, t ) = σq ∗ ( − x, − t ) = σE ∗ ( − x, − t ) , m ( x, t ) = δp ∗ ( − x, − t )are taken. Therefore, there are only two kinds of the NH-MB system. P T -symmetric NH-MB system
Based on the Darboux transformation of AKNS system, we will study the Darbouxtransformation of
P T -symmetric NH-MB system (2.17) in this section. First we considera similar gauge transformation ϕ (cid:48) = T ϕ = ( λA − S ) ϕ, (3.1)among them, A = (cid:32) a a a a (cid:33) , S = (cid:32) s s s s (cid:33) . (3.2)Assume that the new function ϕ (cid:48) satisfies the equations ϕ (cid:48) t = M (cid:48) ϕ (cid:48) , ϕ (cid:48) x = N (cid:48) ϕ (cid:48) , (3.3)where M (cid:48) , N (cid:48) depends on E (cid:48) , p (cid:48) , η (cid:48) , and the relationships between M (cid:48) , N (cid:48) and E (cid:48) , p (cid:48) , η (cid:48) are the same as those between M, N and
E, p, η .We can also assume that the function ϕ (cid:48) satisfies the equation ϕ (cid:48) tx = ϕ (cid:48) xt . (3.4) et’s substitute (3.1) into (3.3), combined with (1.1) and (1.2), we can get T t = M (cid:48) T − T M, (3.5) T x = N (cid:48) T − T N, (3.6)next we will discuss the Eqs. (3.5) and (3.6) respectively.Firstly, we substitute the forms of A and S into (3.5) to simplify it, combine thesymmetric form (2.10) involved in this system and compare the coefficients of each powerof λ , we can find that when σ = 1, there are s ( x, t ) = ˆ ξs ∗ ( − x, t ) , s ( x, t ) = ˆ ξs ∗ ( − x, t ) , a ( x, t ) = ˆ ξa ∗ ( − x, t ) , ˆ ξ = ± σ = −
1, there are s ( x, t ) = ˇ ξs ∗ ( − x, t ) , s ( x, t ) = − ˇ ξs ∗ ( − x, t ) , a ( x, t ) = ˇ ξa ∗ ( − x, t ) , ˇ ξ = ± . Then we can obtain a E (cid:48) = a E − is , (3.7) a = a = 0 , a t = a t = 0 ,S t = M (cid:48) S − SM = [ M , S ] + i [ S, σ ] S, where M = (cid:32) E ( x, t ) σE ∗ ( − x, t ) 0 (cid:33) , σ = (cid:32) − (cid:33) , M (cid:48) = (cid:32) E (cid:48) ( x, t ) σE ∗ (cid:48) ( − x, t ) 0 (cid:33) . After the same calculations for Eq. (3.6), we find that the conclusions are consistentwith the above conclusions, and we get N (cid:48) − = ( S + ωA ) N − ( S + ωA ) − , (3.8) S x = N (cid:48) S − SN − i ( N (cid:48) − A − AN − ) . In order to facilitate the subsequent calculations and analyses, we take A = I here, so σ = 1 : s ( x, t ) = s ∗ ( − x, t ) , s ( x, t ) = s ∗ ( − x, t ); σ = − s ( x, t ) = s ∗ ( − x, t ) , s ( x, t ) = − s ∗ ( − x, t ) . We assume S = H ∧ H − , (3.9)where ∧ = (cid:32) λ λ (cid:33) ,H = (cid:32) ϕ ( x, t, λ ) ϕ ( x, t, λ ) ϕ ( x, t, λ ) ϕ ( x, t, λ ) (cid:33) := (cid:32) ϕ , ( x, t ) ϕ , ( x, t ) ϕ , ( x, t ) ϕ , ( x, t ) (cid:33) . According to the symmetric form (2.10) involved in this system and combing with therelationships between s , s and s , s , there are λ = λ ∗ , and σ = 1 : ϕ , ( x, t ) = ˆ ζϕ ∗ , ( − x, t ) , ϕ , ( x, t ) = ˆ ζϕ ∗ , ( − x, t ) , ˆ ζ = ± = − ϕ , ( x, t ) = ˇ ζϕ ∗ , ( − x, t ) , ϕ , ( x, t ) = − ˇ ζϕ ∗ , ( − x, t ) , ˇ ζ = ± . Therefore S = 1∆ · (cid:32) λ ϕ , ϕ ∗ , ( − x, t ) − σλ ∗ ϕ , ϕ ∗ , ( − x, t ) − σ ( λ − λ ∗ ) ϕ , ϕ ∗ , ( − x, t )( λ − λ ∗ ) ϕ ∗ , ( − x, t ) ϕ , λ ∗ ϕ , ϕ ∗ , ( − x, t ) − σλ ϕ , ϕ ∗ , ( − x, t ) (cid:33) , (3.10)where ∆ = ϕ , ( x, t ) ϕ ∗ , ( − x, t ) − σϕ , ( x, t ) ϕ ∗ , ( − x, t ).Then we substitute (3.10) into (3.7) and (3.8), the relationships between the newsolutions E (cid:48) , p (cid:48) , η (cid:48) and the old solutions E, p, η of Eq. (2.17) can be obtained E (cid:48) ( x, t ) = E ( x, t ) + 2 σi ( λ − λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) − σϕ , ( x, t ) ϕ ∗ , ( − x, t ) , (3.11) p (cid:48) ( x, t ) = 1∆ (cid:48) (cid:104) − σp ∗ ( − x, t ) · (cid:0) ( λ − λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) (cid:1) + p ( x, t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) − σ ( ω + λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) (cid:1) − η ( x, t ) · ( λ − λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) − σ ( ω + λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) (cid:1)(cid:105) , (3.12) η (cid:48) ( x, t ) = σ ∆ (cid:48) (cid:104) p ∗ ( − x, t ) · ( λ − λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) · (cid:0) ( ω + λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) − σ ( ω + λ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) (cid:1) + p ( x, t ) · ( λ − λ ∗ ) ϕ ∗ , ( − x, t ) ϕ , ( x, t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) − σ ( ω + λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) (cid:1)(cid:105) + η ( x, t ) · (cid:0) − (cid:48) ( λ − λ ∗ ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) ϕ , ( x, t ) ϕ ∗ , ( − x, t ) (cid:1) , (3.13)where ∆ (cid:48) = ( ω + λ )( ω + λ ∗ ) (cid:0) ϕ ( x, t ) ϕ ∗ ( − x, − t ) − σϕ ( x, t ) ϕ ∗ ( − x, − t ) (cid:1) . N -fold Darboux transformation We found that the forms of the solutions of
P T -symmetric NH-MB system (2.17)obtained by using the one-fold Darboux transformation were already very cumbersome,so we introduced the determinant representations of Darboux transformation to representthe solutions of Eq. (2.17).
Theorem 3.1.
For the n -fold Darboux transformation of the P T -symmetric NH-MBsystem (2.17) , its corresponding Darboux matrix form is as follows T n ( λ ; λ , λ , · · · , λ n ) = λ n I − S [ n ] n − λ n − − · · · − S [ n ]1 λ − S [ n ]0 = 1 | ( H [ n ] ) T | | T [ n ]11 | | T [ n ]12 || T [ n ]21 | | T [ n ]22 | := (cid:32) ( T n ) ( T n ) ( T n ) ( T n ) (cid:33) , (3.14) here λ ∗ n − = λ n , T [ n ] ij = (cid:32) ( A n ) ij ( B n ) ij ( C n ) ij ( D n ) ij (cid:33) , i, j = 1 , ,H [ n ] = λ ϕ , λ ϕ , · · · λ n ϕ , n − λ n ϕ , n λ ϕ , λ ϕ , · · · λ n ϕ , n − λ n ϕ , n ... ... . . . ... ... λ n − ϕ , λ n − ϕ , · · · λ n − n ϕ , n − λ n − n ϕ , n λ n − ϕ , λ n − ϕ , · · · λ n − n ϕ , n − λ n − n ϕ , n , ( A n ) ij = (cid:16) λ λ · · · λ n − (cid:17) , j = 1 , (cid:16) λ λ · · · λ n − (cid:17) , j = 2 , ( B n ) ij = (cid:16) λ n (cid:17) , i = j, (cid:16) (cid:17) , i (cid:54) = j, ( C n ) ij = ( H [ n ] ) T , ( D n ) ij = (cid:0) λ n ϕ i, λ n ϕ i, · · · λ n n − ϕ i, n − λ n n ϕ i, n (cid:1) T . When σ = 1 , there are ϕ , n ( x, t ) = ˙ ζϕ ∗ , n − ( − x, t ) , ϕ , n ( x, t ) = ˙ ζϕ ∗ , n − ( − x, t ) , ˙ ζ = ± when σ = − , there are ϕ , n ( x, t ) = − ¨ ζϕ ∗ , n − ( − x, t ) , ϕ , n ( x, t ) = ¨ ζϕ ∗ , n − ( − x, t ) , ¨ ζ = ± . According to the above mentioned Darboux matrix T n , we can get the relationshipsbetween the new solutions E (cid:48) , p (cid:48) , η (cid:48) and the old solutions E, p, η of Eq. (2.17) E [ n ] ( x, t ) = E ( x, t ) + 2 i ( S [ n ] n − ) , p [ n ] = ˜ p [ n ] ∆ [ n ] , η [ n ] = ˜ η [ n ] ∆ [ n ] , (3.15)where ∆ [ n ] = ( T n ) ( T n ) − ( T n ) ( T n ) , ˜ p [ n ] ( x, t ) = p ( x, t ) · (cid:0) ( T n ) (cid:1) − σp ∗ ( − x, t ) · (cid:0) ( T n ) (cid:1) + 2 ση ( x, t ) · ( T n ) ( T n ) , ˜ η [ n ] ( x, t ) = σp ( x, t ) · ( T n ) ( T n ) − p ∗ ( − x, t ) · ( T n ) ( T n ) + η ( x, t ) · (cid:0) ( T n ) ( T n ) + ( T n ) ( T n ) (cid:1) . P T -symmetric NH-MB sys-tem
In this section, we will combine the Darboux transformation to find various types ofsolutions of
P T -symmetric NH-MB system (2.17) by giving different seed solutions.
Case : E = 0 , p = 0 , η = x. Then ϕ = ( ϕ , ϕ ) T satisfies ϕ t = (cid:32) − iλ iλ (cid:33) ϕ, (4.1) x = (cid:32) − ε λ + ε λ + ixλ + ω
00 4 ε λ − ε λ − ixλ + ω (cid:33) ϕ. (4.2)Take a set of special solution of Eqs. (4.1) and (4.2), in the form shown below ϕ = (cid:32) ϕ ϕ (cid:33) = e − iλt +( − ε λ + ε λ ) x + i λ + ω ) x e iλt +(4 ε λ − ε λ ) x − i λ + ω ) x . (4.3)We assume that λ = µ + iµ , there is a relationship between µ , µ and ε , ε ε µ µ = 2 ε (3 µ µ − µ ) , µ µ (cid:54) = 0 . (4.4)By substituting (4.3) and the seed solutions into Eqs. (3.11) − (3.13) and combining (4.4),we can obtain the solutions of Eq. (2.17). (i) defocusing P T -symmetric NH-MB system ( σ = 1) E (cid:48) = − µ · e f · csch ( f ) , (4.5) p (cid:48) = − iµ · e f ( ω + µ ) + µ · (cid:18) ( ω + µ ) · csch ( f ) + iµ csch ( f ) sech ( f ) (cid:19) · x, (4.6) η (cid:48) = (cid:18) µ ( ω + µ ) + µ · csch ( f ) (cid:19) · x, (4.7)where f = − iµ t + i ( ω + µ ) · x ( ω + µ ) + µ − x (cid:0) ε ( µ − µ µ ) − ε ( µ − µ ) (cid:1) ,f = 2 µ t + µ · x ( ω + µ ) + µ . Obviously, by analyzing the forms of the solutions E (cid:48) , p (cid:48) , η (cid:48) , we can find that this is aset of singular solutions when f = 0, and the solutions E (cid:48) , p (cid:48) , η (cid:48) develop singularity at t = x (cid:0) ( ω + µ ) + µ (cid:1) . (ii) focusing P T -symmetric NH-MB system ( σ = − E (cid:48) = 2 µ · e f · sech ( f ) , (4.8) p (cid:48) = − iµ · e f ( ω + µ ) + µ · (cid:18) ( ω + µ ) · sech ( f ) + iµ sech ( f ) csch ( f ) (cid:19) · x, (4.9) η (cid:48) = (cid:18) − µ ( ω + µ ) + µ · sech ( f ) (cid:19) · x, (4.10)where f and f are consistent with (i).We choose ε = − . , µ = − . , µ = 0 . , ω = 1 , then the pictorial representationsof the solutions of the focusing P T -symmetric NH-MB system are shown in Figs. 1 − Solution E of the focusing P T -symmetric NH-MB system
Fig. 2:
Solution p of the focusing P T -symmetric NH-MB system
Fig. 3:
Solution η of the focusing P T -symmetric NH-MB system
By combining the two forms of diagrams, we can see that all of them are roughlyarched. From the density diagram in Fig. 1, solution E is a standard arch, but from thefirst diagram, the right side is obviously higher than the left side; In Fig. 2, we can observe hat the whole thing of solution p is also an arch, but from its density diagram, the leftand right parts will never intersect, which is the difference between solutions E and p ; InFig. 3, the density diagram of solution η along the x = 0 symmetric, and it can be seenfrom the first diagram in Fig. 3 that the wave is bright on the left and dark on the right. Fig. 4: Evolution of solution E Fig. 5: Evolution of solution p By observing Fig. 4 and Fig. 5, we can see that two waves of solutions E and p are approaching each other. The difference is that the two waves of solution E eventuallymerge into one wave, while the two waves of solution p go forward side by side. The heightof their peaks is decreasing over time, and it’s not hard to see that they will eventuallydisappear. Case : E = de ax + ibt , p = ce ax + ibt , η = 0 . Take the above seed solutions into Eq. (2.17), we can derive the relations b = 2 ω, a = 4 ε ω (2 ω + 3 σd ) + ε (2 ω + σd ) + 2 σcd . (4.11)Then the linear Eqs. (1.1) and (1.2) becomes ϕ t = (cid:32) − iλ de ax +2 iωt σde − ax − iωt iλ (cid:33) ϕ, ϕ x = (cid:32) ˆ N ˆ N ˆ N ˆ N (cid:33) ϕ, (4.12)whereˆ N = − ˆ N = − ε λ + ε λ − σε d λ + σ (4 ε ω + 12 ε ) d , ˆ N = (cid:16) − iε λ + (4 iε ω + iε ) λ − iε (2 ω + σd ) − iε ω − iσcd ( λ + ω ) (cid:17) · de ax +2 iωt , ˆ N = σ ˆ N · e − ax +2 iωt ) . ake a special solution of linear system (4.12) with the form of ϕ = (cid:32) ϕ ϕ (cid:33) = e ax +2 iωt · ( e − ( g + γ g ) x − ( iω − R ) t + e − ( g + γ g ) x − ( iω + R ) t ) d · ( γ e − ( g + γ g ) x − ( iω − R ) t + γ e − ( g + γ g ) x − ( iω + R ) t ) , (4.13)where R = 2 (cid:112) σd − ( λ + ω ) ,g = ˆ N , g = ˆ N d · e − ax − iωt ,γ = i ( λ + ω ) − R , γ = i ( λ + ω ) + R . We take λ = µ + iµ , then µ , µ , ε , ε and ω need meeting the constraints ε = ε (4 µ + σd ) µ , µ = ςµ = − ω, ς = ± . In particular, for σ = − µ > d .We do not represent the expressions of p and η here as their expressions are too long. (i) defocusing P T -symmetric NH-MB system ( σ = 1)Let’s take E as an example, which can be represented as E = de ax +2 iωt · (cid:18) − ςµ · γ e Rt + γ e − Rt + γ e g Rx + γ e − g Rx ( d − γ ) e Rt + ( d − γ ) e − Rt + 2 d ( e g Rx + e − g Rx ) (cid:19) . Solutions of the defocusing
P T -symmetric NH-MB system are obtained as Figs. 6 − Fig. 6:
Solution E of the defocusing P T -symmetric NH-MB system
Solution p of the defocusing P T -symmetric NH-MB system
Fig. 8:
Solution η of the defocusing P T -symmetric NH-MB system
The corresponding parameters in above diagrams are d = 2 , c = − , ε = 2 , µ = − , ς = 1 . From the Fig. 6 and Fig. 7, we can observe that the changing forms ofwaves look very similar, in this process from negative infinity to positive infinity alongthe x -axis, the waves of them start from zero, and finally round towards positive infinity.For the wave of solution η in Fig. 8, when x rounds towards infinity, it always tends tozero. Fig. 9 and Fig. 10 are the time evolution diagrams of solutions E and p , respectively. Fig. 9: Evolution of solution E p From Fig. 9 and Fig. 10, we can see that the two waves of solution E are still combinedinto one wave eventually, while the two waves of solution p are close to each other butdo not blend, which is the same as their variation forms in the solutions of the focusing P T -symmetric NH-MB system. The difference is that the heights of their wave peaks nolonger decrease with time, but increase. In addition, the peak of the first wave in Fig. 9is gradually higher than that of the second wave over time. (ii) focusing
P T -symmetric NH-MB system ( σ = − E = de ax +2 iωt · (cid:18) ςµ · γ e Rt + γ e − Rt + γ e g Rx + γ e − g Rx ( d + γ ) e Rt + ( d + γ ) e − Rt + 2 d ( e g Rx + e − g Rx ) (cid:19) . Fig. 11:
Solution E of the focusing P T -symmetric NH-MB system
Solution p of the focusing P T -symmetric NH-MB system
Fig. 13:
Solution η of the focusing P T -symmetric NH-MB system
Solutions are shown in Figs. 11 −
13 with different values of the parameters d, c, ε , µ and ς , respectively. Here we take d = 0 . , c = − . , ε = − , µ = 0 . , ς = 1 . FromFigs. 11-13, we can observe that the changing forms of waves along the x -axis are consistentwith that of (i). It is worth noting that the propagation form of solution η along the t -axisis interesting. From Fig. 13, we can see that the wave on the left is bright and the wave onthe right is dark. But the propagation paths of the two waves are reversed at some point,that is, the wave on the left becomes dark and the wave on the right becomes bright, andthen keep their shapes and continue to propagate. In this section, we also study the reverse space-time NH-MB system (2.20) by using theDarboux transformation, and discuss the solutions of the Eq. (2.20). Similarly, we analyzeEqs. (3.5) and (3.6) respectively. Here, we substitute Eq. (3.10) into Eqs. (3.5) and (3.6),combine the symmetric form (2.19) and Eq. (3.1) for calculation, and then compare thecoefficients of each power of λ , we could find that when σ = 1, there are s ( x, t ) = ˙ ξs ( − x, − t ) , s ( x, t ) = − ˙ ξs ( − x, − t ) , a ( x, t ) = ˙ ξa ( − x, − t ) , ˙ ξ = ± hen σ = −
1, there are s ( x, t ) = ¨ ξs ( − x, − t ) , s ( x, t ) = ¨ ξs ( − x, − t ) , a ( x, t ) = ¨ ξa ( − x, − t ) , ¨ ξ = ± . Obviously, they are different from the relations obtained in the
P T -symmetric NH-MBsystem (2.17), and the direct reason was the different selection of the symmetric form.Then we obtain a E (cid:48) = a E − is , (5.1) a = a = 0 , a t = a t = 0 ,N (cid:48) − = ( S + ωA ) N − ( S + ωA ) − , (5.2)where N − = (cid:32) η ( x, t ) − σp ( x, t ) − p ( − x, − t ) − η ( x, t ) (cid:33) . Similarly, we assume that the matrix S is in the form of formula (3.10). By calculation,we find that if we take A = I , then S = λ σ . At this time, S is meaningless, so we take A = σ here, and then we have σ = 1 : s ( x, t ) = − s ( − x, − t ) , s ( x, t ) = s ( − x, − t ); σ = − s ( x, t ) = − s ( − x, − t ) , s ( x, t ) = − s ( − x, − t ) . Combined with the relationships between s , s and s , s , through a series of calcu-lations, there are λ = − λ ,ϕ ( x, t ) = ζϕ ( − x, − t ) ,ϕ ( x, t ) = − σζϕ ( − x, − t ) , ζ = ± , therefore S = λ ∆ · (cid:32) ϕ , ϕ , ( − x, − t ) − σϕ , ϕ , ( − x, − t ) 2 σϕ , ϕ , ( − x, − t )2 ϕ , ( − x, − t ) ϕ , − ϕ , ϕ , ( − x, − t ) + σϕ , ϕ , ( − x, − t ) (cid:33) , (5.3)where ∆ = ϕ , ( x, t ) ϕ , ( − x, − t ) + σϕ , ( x, t ) ϕ , ( − x, − t ).Then we substitute (5.3) into Eqs. (5.1) and (5.2) to get the relationships between newsolutions E (cid:48) , p (cid:48) , η (cid:48) and old solutions E, p, η of Eq. (2.20), as shown below E (cid:48) ( x, t ) = − E ( x, t ) + 4 iσλ ϕ , ( x, t ) ϕ , ( − x, − t ) ϕ , ( x, t ) ϕ , ( − x, − t ) + σϕ , ( x, t ) ϕ , ( − x, − t ) , (5.4) p (cid:48) ( x, t ) = 1∆ (cid:48) (cid:104) − σp ( − x, − t ) · (cid:0) λ ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:1) + p ( x, t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) + σ ( ω − λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:1) + 4 η ( x, t ) · λ ϕ , ( x, t ) ϕ , ( − x, − t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ , ( − x, − t )+ σ ( ω − λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:1)(cid:105) , (5.5) (cid:48) ( x, t ) = 1∆ (cid:48) (cid:104) σp ( − x, − t ) · λ ϕ , ( x, t ) ϕ , ( − x, − t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) + σ ( ω − λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:1) + 2 σp ( x, t ) · λ ϕ , ( − x, − t ) ϕ , ( x, t ) · (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) + σ ( ω − λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:1)(cid:105) + η ( x, t ) · (cid:18) σ (cid:48) λ ϕ , ( x, t ) ϕ , ( − x, − t ) ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:19) , (5.6)where ∆ (cid:48) = − (cid:0) ( ω + λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) + σ ( ω − λ ) ϕ , ( x, t ) ϕ , ( − x, − t ) (cid:1) − σλ ϕ , ( x, t ) ϕ , ( − x, − t ) ϕ , ( x, t ) ϕ , ( − x, − t ) . Similarly, given the seed solutions, we will get the corresponding new solutions of thereverse space-time NH-MB system (2.20). Here we only discuss two seed solutions ofthe same types as in the section 4. For example, we can take E = 0 , p = 0 , η = 1 or E = ˜ de i (˜ ax +˜ bt ) , p = i ˜ ce i (˜ ax +˜ bt ) , η = 1. However, we find that the ϕ = [ ϕ , ϕ ] T of thesetwo kinds of seed solutions are all in the form of e -index. Combined with Eqs. (5.4) − (5.6),it is obvious that the new solutions forms finally generated are consistent: both E and p are in the form of e -index, while η is a constant variable. In order to achieve the compatibility of the five equalities between Eq. (2.4) arisingfrom the zero curvature condition for the H-MB system, we exploited various possibilityof different combinations between q ( x, t ) and r ( x, t ), p ( x, t ) and m ( x, t ), including parity,time-space inversion, and complex conjugation. In general, each possibility can correspondto a new type of integrable system. However, due to the limitation of some parametersin the H-MB system (1.8), we only get three kinds of integrable systems: the standardH-MB system (1.8) and two kinds of nonlocal H-MB system. The two nonlocal H-MBsystems are P T -symmetric NH-MB system (2.17) and reverse space-time NH-MB system(2.20), respectively.In this paper, the explicit solutions of the focusing and defocusing
P T -symmetric NH-MB systems (2.17) are constructed by using the Darboux transformation. In addition,the solutions of the reverse space-time NH-MB system (2.20) is also discussed briefly.The results show that the defocusing
P T -symmetric NH-MB system shows singularitywhen the seed solution is E = 0 , p = 0 , η = x , and the singularity appears at t = x / (cid:0) ( ω + µ ) + µ (cid:1) , the singularities of nonlocal systems are also the topic worthy to bediscussed [20, 21]. For the reverse space-time NH-MB system (2.20), we take two differentseed solutions respectively, and get the same forms of the new solutions: both E ( x, t ) and p ( x, t ) are in the form of e -index, and η ( x, t ) is always a constant variable.For the two NH-MB systems studied in this paper, there are a variety of interestingissues to be explored. Obviously, more specific scenarios can be explored for the abovecases and further solutions can be built, for instance by taking different seed function inthe Darboux transformation. We also ignore other methods to solve such systems, suchas Hirota’s direct method, inverse scattering transformation and so on. And we can eveninvestigate whether these nonlocal solutions can be realized experimentally. Furthermore,we can construct the Riemann-Hilbert problems related to the resulting nonlocal systems,as Eq. (2.17) and Eq. (2.20). cknowledgements: Chuanzhong Li is supported by the National Natural ScienceFoundation of China under Grant No. 12071237 and K. C. Wong Magna Fund in NingboUniversity.
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