Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation
aa r X i v : . [ n li n . S I] M a y Inverse scattering transform for the integrable nonlocalLakshmanan-Porsezian-Daniel equation
Wei-Kang Xun and Shou-Fu Tian ∗ School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology,Xuzhou 221116, People’s Republic of China
Abstract
In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation isintroduced, and its integrability as an infinite dimensional Hamilton dynamic systemis established. Motivated by the ideas of Ablowitz and Musslimani (2016 Nonlinearity J = J = , , δ on these solutions is furtherconsidered via the graphical analysis. Finally, the eigenvalues and conserved quantitiesare investigated under a few special initial conditions. Keywords: integrable nonlocal Lakshmanan-Porsezian-Daniel equation, inversescattering method, Left-Right Riemann-Hilbert problem, soliton solutions. ✩ Project supported by the Fundamental Research Fund for the Central Universities under the grant No.2019ZDPY07. ∗ Corresponding author.
E-mail addresses : [email protected] and [email protected] (S. F. Tian)
Preprint submitted to Journal of L A TEX Templates May 11, 2020 ontents1 Introduction 32 Lax representation and compatibility condition 53 Infinite number of conserved quantities and conservation laws 6 r ( x , t ) = γ q ∗ ( − x , t ) : eigenfunctions and the scatteringdata 10 ffi cients . . . . . . . . . . . . 136.6 Additional symmetry between the eigenfunctions . . . . . . . . . . . 13
10 Conclusions 30 . Introduction Nonlinear integrable evolution equations exist in all aspects of scientific researchand play a essential role in modern physical branches. There are numerous nonlinearintegrable evolution equations which are applied into fluid mechanics, elasticity, lat-tice dynamics, electromagnetics, etc. For example, the Korteweg-de Vries (KdV) andmodified Korteweg-de Vries (mKdV) equations describe the evolution of weakly dis-persive and small amplitude waves in quadratic and cubic nonlinear media, respectively[1]. The KdV equation is more famous for its application in shallow water waves. Be-sides, the integrable cubic nonlinear Schr¨odinger (NLS) equation which is well-knownfor its application to the evolution of weakly nonlinear and quasi-monochromatic wavetrains in media with cubic nonlinearities [2, 3]. Besides, the Kadomtsev-Petviashvili(KP) equation which aries in plasma physics and internal waves [1, 4] is applied todescribe the evolution of weakly dispersive and small amplitude waves with additionalweak transverse variation [5, 6]. Based on the importance of nonlinear integrable evo-lution equations, these are the focus of scholars’ research from the beginning to theend. In order to solve these equations, many novel and e ff ective methods have beenproduced, such as Hirota bilinear method [7], Darboux and B¨acklund transformation[8] and inverse scattering transform(IST) [1, 4, 9, 10].However, there is a special kind of equation called nonlocal equation among manynonlinear integrable equations. As the name suggests, nonlinear integrable nonlocalequation refers to the nonlinear integrable evolution equation with nonlocal nonlin-ear term, for example, q ( x , t ) is replacled by q ∗ ( − x , t ), q ( − x , − t ) or q ∗ ( − x , − t ). InRef. [11], Ablowitz and Musslimani had found a new class of nonlocal integrableNLS hierarchy with the infinite number of conservation laws by introducing a newsymmetry reduction r ( x , t ) = q ∗ ( − x , t ). According to the di ff erent inversion rela-tion, integrable nonlocal nonlinear equations roughly include the following categories:real (complex) reverse time nonlocal equation, real (complex) reverse space nonlo-cal equation ans real(complex) reverse space-time nonlocal equation [12]. Recentlythere are several new nonlocal system have been analyzed, including multidimen-sional versions of the NLS equation [13, 14], nonlocal reverse-time NLS equations[15], nonlocal mKdV equation [16–18], nonlocal sine-Gordon equation[19], nonlo-cal Davey-Stewartson equation [20, 21], nonlocal Alice-Bob systems [22], nonlocal(2 + ff ective to it, and the most classical and e ff ective method is IST. IST asso-ciates a compatible pair of linear equations with the integrable nonlinear equation. Oneof the equations is used to determine suitably analytic eigenfunctions and transform theinitial data to appropriate scattering data. The other linear equation is used to complete3he evolution of the scattering data. Based on the linear equations (Lax pair), one canfind the exact solutions of origin objective equations successfully [1].In this work, we consider IST for the integrable nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation q t ( x , t ) + iq xx ( x , t ) − i γ q ( x , t ) q ∗ ( − x , t ) − δ H [ q ( x , t )] = , (1.1)where H [ q ( x , t )] = − iq xxxx ( x , t ) + i γ q ∗ ( − x , t ) q ( x , t ) + i γ q ( x , t ) q x ( x , t ) q ∗ x ( − x , t ) + i γ q ∗ ( − x , t ) q ( x , t ) q xx ( x , t ) + i γ q ( x , t ) q ∗ xx ( − x , t ) − iq ∗ ( − x , t ) q ( x , t ) , (1.2)which is an NLS type equation with higher order nonlinear terms, such as fourth-orderdispersion, second-order dispersion, cubic and quintic nonlinearities. The LPD equa-tion describes the nonlinear e ff ect more clearly in Refs. [25–27]. The integrable non-local LPD equation whose the potential functions satisfy r ( x , t ) = q ∗ ( − x , t ) is stud-ied via Darboux transformation in [28]. The authors demonstrated the integrabilityof the nonlocal LPD equation, provided its Lax pair, and presented its rational soli-ton solutions and self-potential function. However, in this work, by using a ingeniousmethod, we analyze infinite number of conserved quantities and conservation laws fornonlocal LPD equation whose potential functions satisfy r ( x , t ) = γ q ∗ ( − x , t ), where γ = ±
1. Furthermore, by using IST method [29], we obtain the time-periodic puresoliton solutions of the integrable nonlocal LPD equation whose potential functionssatisfy r ( x , t ) = − q ∗ ( − x , t ). Generally speaking, the significance of this work is to im-prove the previous research on the soliton solutions and some important properties ofintegrable nonlocal LPD equation reported in [28], which is very helpful for us to betterunderstand and master this class equations.This work is organized as follows. In Section 2, the Lax pair and the compatibilitycondition of nonlocal LPD equation are given. Besides, some other properties of non-local LPD equation are listed in the end of this section. In Section 3, by introducing anovel method, we derive the global conservation laws and the local conservation lawswhich establish the integrability of the objective equation. In Section 4, the direct scat-tering problem of the nonlocal LPD equation is constructed and some other importantsymmetries of the eigenfunctions and the scattering data are discussed. Afterwards,by using the Left-Right RH method, the inverse scattering problem is established andthe potential function is recovered successfully. Next, in Section 8, we discuss thetime-periodic pure soliton solutions under the reflectionless case. Moreover, in orderto understand the soliton solutions, we select J = J = , , δ on the soliton solutions is considered. InSection 9, we consider some special cases of initial conditions and derive the eigenval-ues and conserved quantities. Finally, the conclusions and the acknowledgement aregiven in the last two sections.
2. Lax representation and compatibility condition
We begin our discussion by considering the following scattering problem φ x = L φ, L = − ζ J + U ,φ t = M φ, M = ζ J − ζ U + V + δ V p , (2.1)with J = i − i , V = iq ( x , t ) r ( x , t ) − iq x ( x , t ) − ir x ( x , t ) − iq ( x , t ) r ( x , t ) , U = q ( x , t ) r ( x , t ) 0 , V p ( x , t ) = iA p ( x , t ) B p ( x , t ) − C p ( x , t ) − iA p ( x , t ) , (2.2) A p = − ζ − r ( x , t ) q ( x , t ) ζ − ir ( x , t ) q x ( x , t ) ζ − iq ( x , t ) r ( x , t ) ζ − q ( x , t ) r ( x , t ) + q x ( x , t ) r x ( x , t ) + q ( x , t ) r ( x , t ) + r ( x , t ) q xx ( x , t ) , B p = q ( x , t ) ζ + iq x ( x , t ) ζ − q xx ( x , t ) ζ + r ( x , t ) q ( x , t ) − iq xxx ( x , t ) + iq ( x , t ) r ( x , t ) q x ( x , t ) , C p = − r ( x , t ) ζ − ir x ( x , t ) ζ + r xx ( x , t ) ζ − r ( x , t ) q ( x , t ) ζ + ir xxx ( x , t ) − ir ( x , t ) r x ( x , t ) q ( x , t ) , (2.3)where φ ( x , t ) = ( φ ( x , t ) , φ ( x , t )) T is a two-component vector and the potential func-tions q ( x , t ) and r ( x , t ) are complex functions. Under the symmetry reduction r ( x , t ) = γ q ∗ ( − x , t )( γ = ± L t − M x + [ L , M ] = • Time-reverse symmetry: if q ( x , t ) is a solution, then q ∗ ( x , − t ) is also a solution. • Space-reverse symmetry: if q ( x , t ) is a solution, then q ( − x , t ) is also a solution. • Gauge invariance: if q ( x , t ) is a solution, then e i ρ q ( x , t ) is also a solution withreal and constant ρ . • Spatial translation invariance: if q ( x , t ) is a solution, then q ( x + ix , t ) is also asolution for any real and constant x .5 PT-symmetry: if q ( x , t ) is a solution, then q ∗ ( − x , − t ) is also a solution. It is notedthat V ( x , t ) = γ q ( x , t ) q ∗ ( − x , t ) satisfy the special symmetry V ( x , t ) = V ∗ ( − x , t ),which is referred to as a self-induced potential in the classical optics.
3. Infinite number of conserved quantities and conservation laws
As we all know, a finite number of local conservation laws and global conservationlaws of the nonlinear integrable equation is helpful to establish its integrability as aninfinite dimensional Hamilton dynamic system. In this section, we will explain how toobtain the local and global conservation laws of Eq. (1.1).
The infinite number of conserved quantities of Eq. (1.1) is derived as follows.Before our operation, we suppose that the potential function q ( x , t ) decays rapidly atinfinity. At the same time, the solutions of the scattering transform can be obtained bydefining the four functions which satisfy the following boundary conditionlim x →−∞ φ ( x , t ) = e − i ζ x , lim x →−∞ φ ( x , t ) = e i ζ x , lim x → + ∞ ψ ( x , t ) = e i ζ x , lim x → + ∞ ψ ( x , t ) = e − i ζ x , (3.1)where φ ( x , t ) is not the complex conjugate of φ ( x , t ), and φ ∗ ( x , t ) denotes the complexconjugate of φ ( x , t ). If φ ( x , t ) = ( φ ( x , t ) , φ ( x , t )) T is the solution to Eq. (2.1) whichsatisfies the above boundary conditions, we can obtain that φ ( x , t ) e i ζ x is analytic forIm( ζ ) ≥ x → ±∞ . Substituting φ ( x , t ) = exp ( − i ζ x + ϕ ( x , t ))into Eq. (2.1), we can find that the function µ ( x , t ) = ϕ x ( x , t ) satisfies the Riccatiequation q ∂∂ x µ q ! + µ − qr − i ζµ = . (3.2)For Im( ζ ) >
0, we have lim | ζ |→∞ ϕ ( x , ζ ) =
0. Substituting the expansion µ ( x , ζ ) = P ∞ n = µ n ( x , t )(2 i ζ ) n + into Eq. (3.2) and equating the powers of ζ , we find µ ( x , t ) = − q ( x , t ) r ( x , t ) = − γ q ( x , t ) q ∗ ( − x , t ) ,µ ( x , t ) = − q ( x , t ) r x ( x , t ) = γ q ( x , t ) q ∗ x ( − x , t ) , (3.3)and µ n + = q ( x , t ) ∂∂ x µ n q ! + n − X m = µ m µ n − m − , n ≥ . (3.4)6rom the boundary conditions it followslim x →−∞ φ ( x , ζ ) e i ζ x = , lim x →−∞ ϕ ( x , ζ ) = , (3.5)then we can obtainln a ( ζ ) = ln( φ ( x , t ) e i ζ x ) = ∞ X n = C n (2 i ζ ) n + , C n = Z + ∞−∞ µ n ( x , t ) dx . (3.6)Since φ ( x , t ) e i ζ x is time independent for k with Im ζ >
0, then the above C n is alsotime independent for ζ with Im ζ >
0. According to Eqs. (3.3), (3.4) and (3.6), we canobtain all conserved quantities. More explicitly, the first few conserved quantities arelisted as follows: C = − γ Z + ∞−∞ q ( x , t ) q ∗ ( − x , t ) dx , C = γ Z + ∞−∞ q ( x , t ) q ∗ x ( − x , t ) dx , C = − γ Z + ∞−∞ (cid:16) q ( x , t ) q ∗ xx ( − x , t ) − γ q ( x , t ) q ∗ ( − x , t ) (cid:17) dx , C = γ Z + ∞−∞ (cid:16) q ( x , t ) q ∗ ( − x , t ) + γ q ( x , t ) q x ( x , t ) q ∗ ( − x , t ) − q ( x , t ) q ∗ ( − x , t ) q ∗ x ( − x , t ) (cid:17) dx , C = γ Z + ∞−∞ (cid:16) − q ( x , t ) q ∗ xxxx ( − x , t ) + γ q ( x , t ) q ∗ x ( − x , t ) + γ q ( x , t ) q ∗ ( − x , t ) q ∗ xx ( − x , t ) + γ q ( x , t ) q xx ( x , t ) q ∗ ( − x , t ) − γ q ( x , t ) q ∗ ( − x , t ) q x ( x , t ) q ∗ x ( − x , t ) − q ( x , t ) q ∗ ( − x , t ) (cid:17) dx . (3.7) In order to obtain the local conservation laws, we consider the time-dependent prob-lem φ t = A φ + B φ , (3.8)where A , B denote the (1 , − and (1 , − entry of M in Eq. (2.1), respectively. Accord-ing to the expression of µ and φ , we find ∂ t µ ( x , t ) = ∂ x A nonloc + B nonloc µ ( x , t ) q ( x , t ) ! , (3.9)where A nonloc = − i δζ + i ζ (1 − δγ qq ∗ ( − x , t )) + δγζ (cid:0) q x q ∗ ( − x , t ) − qq ∗ x ( − x , t ) (cid:1) + i (cid:18) γ qq ∗ ( − x , t ) − δ q q ∗ ( − x , t ) − δγ q ∗ x ( − x , t ) q x ( x , t ) + δγ qq ∗ xx ( − x , t ) (cid:19) , B nonloc = δ q ζ + i δ q x ζ + (cid:16) δγ q q ∗ ( − x , t ) − q ( x , t ) − q xx (cid:17) ζ + i γ qq x q ∗ ( − x , t ) + q x − q xxx ! . (3.10)7ubstituting Eq. (3.10) and the expansion of µ into Eq. (3.9), we have ∂ t ∞ X n = µ n ( x , t )(2 i ζ ) n + = ∂ x A nonloc + B nonloc q ( x , t ) ∞ X n = µ n ( x , t )(2 i ζ ) n + , (3.11)from which we obtain ∂ t ( µ n ) = i ∂ x µ n S + µ n + S − δ q x q µ n + + δµ n + ! , n = , , , , . . . , (3.12)where S = γ q x q ∗ ( − x , t ) + q x q − q xxx q , S = + q xx q − δγ qq ∗ ( − x , t ) . (3.13)We can write the conservation laws (3.12) as the form ∂ T ∂ t = − i ∂ X ∂ x , (3.14)where T = µ n and X = − µ n S − µ n + S + δ q x q µ n + − δµ n + ( n = , , , , . . . ) are theso-called densities and fluxes, respectively. The first three local conservation laws are T = µ , X = − S µ − S µ + δ q x q µ − δµ , T = µ , X = − S µ − S µ + δ q x q µ − δµ , T = µ , X = − S µ − S µ + δ q x q µ − δµ , (3.15)where µ = − γ q ( x , t ) q ∗ ( − x , t ) , µ = γ q ( x , t ) q ∗ x ( − x , t ) ,µ = − γ q ( x , t ) q ∗ xx ( − x , t ) + q ( x , t ) q ∗ ( − x , t ) ,µ = γ q ( x , t ) q ∗ xxx ( − x , t ) + q ( x , t ) q x ( x , t ) q ∗ ( − x , t ) − q ( x , t ) q ∗ ( − x , t ) q ∗ x ( − x , t ) ,µ = − γ q ( x , t ) q ∗ xxxx ( − x , t ) + q ( x , t ) q ∗ ( − x , t ) q xx ( x , t ) − q ( x , t ) q x ( x , t ) q ∗ ( − x , t ) q ∗ x ( − x , t ) + q ( x , t ) q ∗ x ( − x , t ) + q ( x , t ) q ∗ ( − x , t ) q ∗ xx ( − x , t ) − γ q ( x , t ) q ∗ ( − x , t ) ,µ = γ q q ∗ xxxxx ( − x , t ) + q q xxx q ∗ ( − x , t ) − q q ∗ ( − x , t ) q ∗ xxx ( − x , t ) − γ q q x q ∗ ( − x , t ) − q q ∗ ( − x , t ) q ∗ xxx ( − x , t ) − q q xx q ∗ ( − x , t ) q ∗ x ( − x , t ) + q q x q ∗ ( − x , t ) q ∗ xx ( − x , t ) − q q ∗ x ( − x , t ) q ∗ xx ( − x , t ) − q q ∗ x ( − x , t ) q ∗ xx ( − x , t ) + q q x q ∗ x ( − x , t ) − γ q q x q ∗ ( − x , t ) + γ q q ∗ x ( − x , t ) q ∗ ( − x , t ) + γ q q ∗ ( − x , t ) q ∗ x ( − x , t ) . (3.16)8 . Direct scattering problem In the following sections, we will consider the scattering problem of the system ofEq. (2.1). For the convenience of the discussion, we define the following Jost functions M ( x , ζ ) = e i ζ x φ ( x , ζ ) , M ( x , ζ ) = e − i ζ x φ ( x , ζ ) , N ( x , ζ ) = e − i ζ x ψ ( x , ζ ) , N ( x , ζ ) = e i ζ x ψ ( x , ζ ) , (4.1)which satisfy the constant boundary condition induced from Eq. (3.1). Furthermore,we can see that the above functions satisfy a linear integral equations and show that M ( x , ζ ), N ( x , ζ ) are analytic in the upper half complex ζ plane whereas M ( x , ζ ), N ( x , ζ )are analytic in the lower half complex ζ plane [30]. Moreover, the large ζ behavior ofthe Jost functions are given by [30] M ( x , ζ ) = − i ζ R x −∞ r ( z ) q ( z ) dz − r ( x )2 i ζ + O (cid:16) ζ − (cid:17) , N ( x , ζ ) = q ( x )2 i ζ − i ζ R + ∞ x r ( z ) q ( z ) dz + O (cid:16) ζ − (cid:17) , M ( x , ζ ) = q ( x )2 i ζ − i ζ R x −∞ r ( z ) q ( z ) dz + O (cid:16) ζ − (cid:17) , N ( x , ζ ) = + i ζ R + ∞ x r ( z ) q ( z ) dz − r ( x )2 i ζ + O (cid:16) ζ − (cid:17) . (4.2)From the boundary condition (3.1), it is seen that the solutions φ ( x , ζ ) and φ ( x , ζ )of the scattering problem (2.1) are linearly dependent. Similarly, the solutions ψ ( x , ζ )and ψ ( x , ζ ) of the scattering problem (2.1) are linearly dependent. Since the scatteringproblem (2.1) is a second order linearly ordinary di ff erential equation, { φ, φ } and { ψ, ψ } are linearly dependent. Moreover, we can express the relation of them as follows φ ( x , ζ ) = a ( ζ ) ψ ( x , ζ ) + b ( ζ ) ψ ( x , ζ ) ,φ ( x , ζ ) = a ( ζ ) ψ ( x , ζ ) + b ( ζ ) ψ ( x , ζ ) , (4.3)where a ( ζ ), a ( ζ ), b ( ζ ) and b ( ζ ) are the scattering data, then we can have a ( ζ ) = W ( φ ( x , ζ ) , ψ ( x , ζ )) , a ( ζ ) = W (cid:16) ψ ( x , ζ ) , φ ( x , ζ ) (cid:17) , b ( ζ ) = W (cid:16) ψ ( x , ζ ) , φ ( x , ζ ) (cid:17) , b ( ζ ) = W (cid:16) φ ( x , ζ ) , ψ ( x , ζ ) (cid:17) , (4.4)where W ( u , v ) = u v − u v represents the Wronskian determinant. Furthermore, itcan be seen that a ( ζ ) and a ( ζ ) are analytic in the upper half complex plane and thelower half complex plane, respectively, while b ( ζ ) and b ( ζ ) cannot be extend o ff thereal ζ axis. In addition, the scattering data satisfy the relation a ( ζ ) a ( ζ ) − b ( ζ ) b ( ζ ) = ζ =
0. 9 . Symmetry reduction r ( x , t ) = γ q ∗ ( − x , t ): eigenfunctions and the scatteringdata Next, we make e ff orts to establish the symmetry properties of the eigenfunctionsunder the symmetry reduction r ( x , t ) = γ q ∗ ( − x , t ), γ = ±
1. We suppose that v ( x , ζ ) = ( v ( x , t ) , v ( x , t )) T is the solution of Eq. (2.1), then (cid:16) v ∗ ( − x , − k ∗ ) , − γ v ∗ ( − x , − k ∗ ) (cid:17) T is alsothe solution of Eq. (2.1). Since the solutions of the scattering problem are uniquely de-termined by the boundary condition (3.1), we have the following important symmetry ψ ( x , ζ ) = − γ φ ∗ ( − x , − k ∗ ) ,ψ ( x , ζ ) = − γ φ ∗ ( − x , − k ∗ ) . (5.1)From Eq. (4.1), we obtain the symmetry of the Jost functions N ( x , ζ ) = − γ M ∗ ( − x , − k ∗ ) , N ( x , ζ ) = − γ M ∗ ( − x , − k ∗ ) . (5.2) According to the symmetry of the eigenfunctions (5.1) and the Wronskian repre-sentations of the scattering data (4.4), we get a ( ζ ) = a ∗ ( − ζ ∗ ) , a ( ζ ) = a ∗ ( − ζ ∗ ) , b ( ζ ) = γ b ∗ ( − ζ ∗ ) , (5.3)which means that if ζ k = ξ k + i η k is a zero of a ( ζ ) in the upper half complex plane, then − ζ ∗ k = − ξ k + i η k is also a zero of a ( ζ ) in the upper half complex plane. Similarly, ζ k is a zero of a ( ζ ) in the lower half complex plane, then − ζ ∗ k is also a zero of a ( ζ ) in thelower half complex plane.
6. Inverse scattering problem: Left-Right RH approach
At first, we recall Eq. (4.3) φ ( x , ζ ) = a ( ζ ) ψ ( x , ζ ) + b ( ζ ) ψ ( x , ζ ) ,φ ( x , ζ ) = a ( ζ ) ψ ( x , ζ ) + b ( ζ ) ψ ( x , ζ ) , (6.1)10he above system can be rewritten as the following matrix form Φ ( x , ζ ) = S L Ψ ( x , ζ ) , (6.2)where Φ ( x , ζ ) = (cid:16) φ ( x , ζ ) , φ ( x , ζ ) (cid:17) T , Ψ ( x , ζ ) = (cid:16) ψ ( x , ζ ) , ψ ( x , ζ ) (cid:17) T and S L ( ζ ) is the leftscattering matrix S L ( ζ ) = a ( ζ ) b ( ζ ) b ( ζ ) a ( ζ ) . (6.3)Following the results reported in [29], we can formulate the corresponding RH prob-lem on the left and obtain the following linear integral equations which represent thefunctions N ( x , ζ ) and N ( x , ζ ): N ( x , ζ ) = + J X l = C l N ( x , ζ l ) e − i ζ l x ζ − ζ l − π i Z ∞−∞ ρ ( ξ ) e − i ξ x N ( x , ξ ) ξ − ( ζ + i d ξ, N ( x , ζ ) = + J X l = C l N ( x , ζ l ) e i ζ l x ζ − ζ l − π i Z ∞−∞ ρ ( ξ ) e i ξ x N ( x , ξ ) ξ − ( ζ − i d ξ, (6.4)where ρ ( ζ ) and ρ ( ζ ) are the left reflection coe ffi cients defined by ρ ( ζ ) = b ( ζ ) a ( ζ ) , ρ ( ζ ) = b ( ζ ) a ( ζ ) , (6.5)and C l and C l are the left norming constants defined by C l = b ( ζ l ) a ′ ( ζ l ) , C l = b ( ζ l ) a ′ ( ζ l ) , (6.6)where a ′ ( ζ l ) and a ′ ( ζ l ) denote the derivative at ζ l and ζ l , respectively. According to Eq. (2.1), we derive the time evolution of the scattering data a ( ζ, t ) = a ( ζ, , b ( ζ, t ) = e (16 i δζ − i ζ ) t b ( ζ, , a ( ζ, t ) = a ( ζ, , b ( ζ, t ) = e ( − i δζ + i ζ ) t b ( ζ, . (6.7)In what follows, we obtain the time evolution of the left reflection coe ffi cients ρ ( ζ ) and ρ ( ζ ) and the left norming constants C l and C l according to Eqs. (6.5) and (6.7) C l = C l (0) e (16 i δζ − i ζ ) t , ρ ( ζ, t ) = e (16 i δζ − i ζ ) t b ( ζ, / a ( ζ, , C l = C l (0) e ( − i δζ + i ζ ) t , ρ ( ζ, t ) = e ( − i δζ + i ζ ) t b ( ζ, / a ( ζ, . (6.8)11 .3. Right scattering problem Next, we consider the following system ψ ( x , ζ ) = α ( ζ ) φ ( x , ζ ) + β ( ζ ) φ ( x , ζ ) ,ψ ( x , ζ ) = α ( ζ ) φ ( x , ζ ) + β ( ζ ) φ ( x , ζ ) , (6.9)where α ( ζ ), α ( ζ ), β ( ζ ) and β ( ζ ) are the right scattering data. Similarly, we can rewritethe above system as the matrix form Ψ ( x , ζ ) = S R Φ ( x , ζ ) , (6.10)where Ψ ( x , ζ ) = (cid:16) ψ ( x , ζ ) , ψ ( x , ζ ) (cid:17) T , Φ ( x , ζ ) = (cid:16) φ ( x , ζ ) , φ ( x , ζ ) (cid:17) T and S R is the rightscattering matrix S R = α ( ζ ) β ( ζ ) β ( ζ ) α ( ζ ) . (6.11)We can formulate the corresponding RH problem on the right and obtain the followinglinear integral equations which govern the functions M ( x , ζ ) and M ( x , ζ ): M ( x , ζ ) = + J X l = B l M ( x , ζ l ) e i ζ l x ζ − ζ l − π i Z ∞−∞ R ( ξ ) e i ξ x M ( x , ξ ) ξ − ( ζ + i d ξ, M ( x , ζ ) = + J X l = B l M ( x , ζ l ) e − i ζ l x ζ − ζ l + π i Z ∞−∞ R ( ξ ) e − i ξ x M ( x , ξ ) ξ − ( ζ − i d ξ, (6.12)where R ( ζ ) and R ( ζ ) are the right reflection coe ffi cients given by R ( ζ ) = β ( ζ ) α ( ζ ) , R ( ζ ) = β ( ζ ) α ( ζ ) , (6.13)and B l and B l are the right norming constants defined by B l = β ( ζ l ) α ′ ( ζ l ) , B l = β ( ζ l ) α ′ ( ζ l ) . (6.14) Similar to the left case, we obtain the time evolution of the right scattering data α ( ζ, t ) = α ( ζ, , β ( ζ, t ) = e (16 i δζ − i ζ ) t β ( ζ, ,α ( ζ, t ) = α ( ζ, , β ( ζ, t ) = e ( − i δζ + i ζ ) t β ( ζ, . (6.15)According to Eqs. (6.15), (6.13) and (6.14), the time evolution of the right reflectioncoe ffi cients and norming constants can be obtained by B l = B l (0) e (16 i δζ − i ζ ) t , R ( ζ, t ) = e (16 i δζ − i ζ ) t β ( ζ, /α ( ζ, , B l = B l (0) e ( − i δζ − i ζ ) t , R ( ζ, t ) = e ( − i δζ + i ζ ) t β ( ζ, /α ( ζ, . (6.16)12 .5. Relationship between the reflection coe ffi cients According to the matrix forms of the left and the right scattering problem, we havethe relationship between the left and the right scattering matrix S R = S − L , more explic-itly, a ( ζ ) = α ( ζ ) , a ( ζ ) = α ( ζ ) ,β ( ζ ) = − b ( ζ ) , β ( ζ ) = − b ( ζ ) . (6.17)Furthermore, we have R ( ζ ) = β ( ζ ) α ( ζ ) = − b ( ζ ) a ( ζ ) = − γ b ∗ ( − ζ ∗ ) a ∗ ( − ζ ∗ ) = − γρ ∗ ( − ζ ∗ ) , R ( ζ ) = β ( ζ ) α ( ζ ) = − b ( ζ ) a ( ζ ) = − γ b ∗ ( − ζ ∗ ) a ∗ ( − ζ ∗ ) = − γρ ∗ ( − ζ ∗ ) . (6.18) Suppose that ζ l is the eigenvalue of a ( ζ ) in the upper complex plane, i.e., a ( k l ) = φ ( x , ζ ) and ψ ( x , zeta ) are linear dependent, φ ( x , ζ l ) = b ( ζ l ) ψ ( x , ζ l ).Moreover, M ( x , ζ l ) = b ( ζ l ) N ( x , ζ l ) e ik l x , M ( x , ζ l ) = b ( ζ l ) N ( x , ζ l ) e ik l x , (6.19)then M ( x , ζ l ) N ( x , ζ l ) = M ( x , ζ l ) N ( x , ζ l ) . (6.20)With the aid of Eq. (5.2), we obtain N ∗ ( − x , ζ l ) N ( x , ζ l ) = N ∗ ( − x , ζ l ) N ( x , ζ l ) . (6.21)Similarly, the other important conclusion is given as follows M ∗ ( − x , ζ l ) M ( x , ζ l ) = M ∗ ( − x , ζ l ) M ( x , ζ l ) . (6.22)
7. Recovery of the potentials
Based on the above results, we can recover the potential functions q ( x , t ) and r ( x , t )successfully. At first, recall from Eq. (6.4) that N ( x , ζ ) = + J X l = C l N ( x , ζ l ) e i ζ l x ζ − ζ l − π i Z ∞−∞ ρ ( ξ ) e i ξ x N ( x , ξ ) ξ − ( ζ − i d ξ. (7.1)The large k behavior of N ( x , ζ ) is determined by N ( x , ζ ) ∼ ζ J X l = C l N ( x , ζ l ) e i ζ l x − π i ζ Z ∞−∞ ρ ( ξ ) e i ξ x N ( x , ξ ) d ξ. (7.2)13ccording to Eq. (4.2), we obtain N ( x , ζ ) ∼ − r ( x )2 i ζ , (7.3)thus, we can recover the potential r ( x ) by r ( x ) ∼ − i J X l = C l N ( x , ζ l ) e i ζ l x − π i Z ∞−∞ ρ ( ξ ) e i ξ x N ( x , ξ ) d ξ . (7.4)With the asymptotic relation (4.2) M ( x , ζ ) ∼ q ( x )2 i ζ , (7.5)and the symmetry relation M ( x , ζ ) = − γ N ∗ ( − x , − ζ ∗ ), we obtain the following asymp-totic relation of q ( x ): q ( x ) ∼ − i γζ N ∗ ( − x , − ζ ∗ ) , γ = ± . (7.6)From Eq. (7.2), we obtain q ( x ) = i γ J X l = C ∗ l N ∗ ( − x , ζ l ) e i ζ ∗ l x + γπ Z ∞−∞ ρ ∗ ( ξ ) e i ξ x N ∗ ( − x , ξ ) d ξ. (7.7)According to Eqs. (7.4) and (7.7), it can be seen that the symmetry relation r ( x ) = γ q ∗ ( − x ) still holds.
8. Soliton solutions
In this section, we mainly discuss the pure soliton solutions of nonlocal integrableLPD equation. It is noted that pure soliton solutions arise when the reflection coe ffi -cients ρ ( ζ ) and ρ ( ζ ) vanish. Besides, it can be proved that these types of soliton solu-tions are only be obtained when γ = − q ( x ) = − i J X l = C ∗ l N ∗ ( − x , ζ l ) e i ζ ∗ l x . (8.1)In order to facilitate the discussion of the properties of soliton solutions, it is nec-essary to obtain the explicit expression of some critical parameters C j = b j a ′ j e (16 i δζ − i ζ ) t , C j = b j a ′ j e ( − i δζ + i ζ ) t , (8.2)14here b j = e θ j , a ′ ( ζ ) = Q Nj = ( ζ − ζ j ) Q Nj = ( ζ − ζ j ) N X l = ( ζ l − ζ l )( ζ − ζ l )( ζ − ζ l ) , b j = e θ j , a ′ ( ζ ) = Q Nj = ( ζ − ζ j ) Q Nj = ( ζ − ζ j ) N X l = ( ζ l − ζ l )( ζ − ζ l )( ζ − ζ l ) , (8.3)with θ j and θ j are the amplitude of b j and b j , respectively [29].Next, we will take some special parameters to give the explicit expression of solitonsolutions and present them graphically with the aid of mathematic software, which ishelpful for studying the properties of soliton solutions. In this subsection, we discuss the one-soliton solutions of the nonlocal LPD equa-tions by taking J = J = ζ = ξ + i η , η > , ζ = ξ + i η , η < . (8.4)Taking J = q ( x ) = − iC ∗ N ∗ ( − x , ζ l ) e i ζ ∗ x , (8.5)where C and C are the norming constants (in x ) whose time evolution is determinedby C ( t ) = C (0) e (16 i δζ − i ζ ) t = e θ ( ζ − ζ ) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t = ie θ ( ζ − ζ ) e ( − i δζ + i ζ ) t , (8.6)and the expression of N ∗ ( − x , ζ ) can be obtained by setting J = N ∗ ( − x , ζ ) = | ζ − ζ | | ζ − ζ | − C ∗ C ∗ e i ( ζ + ζ ∗ ) x , (8.7)then we find the one soliton solution q ( x , t ) = − i C ∗ ( t ) e ik ∗ x | ζ − ζ | | ζ − ζ | − C ∗ ( t ) C ∗ ( t ) e i ( ζ + ζ ∗ ) x . (8.8)The localized structure, the density and the wave propagation of one soliton so-lution is shown in Fig. 1. From Fig. 1, we can learn that the single soliton wavepropagate almost along the axis of x =
0. Moreover, in the process of wave propagate,the amplitude and the width of the single soliton are not changed.15 a ) ( b ) ( c ) Figure 1.
One-soliton solution with parameters δ = θ = π , θ = − π , ζ = . i and ζ = − . i . (a) : the structures of the one-soliton solution, (b) : the density plot, (c) : the wave propagation ofthe one-soliton solution. In this subsection, we consider the soliton solutions of the nonlocal LPD equations(1.1) with J = J =
2. Suppose the corresponding eigenvalues as follows ζ = ξ + i η , ζ = ξ + i η , η , η > ,ζ = ξ + i η , ζ = ξ + i η , η , η < . (8.9)Setting J = q ( x ) = − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x , (8.10)where C j , C j , j = , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t . (8.11)To obtain the functions N ∗ ( − x , ζ ) and N ∗ ( − x , ζ ), we need to solve the following sys-tem M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + β N ∗ ( − x , ζ ) , M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + β N ∗ ( − x , ζ ) , N ∗ ( − x , ζ ) = + α M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , N ∗ ( − x , ζ ) = + α M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , (8.12)16here α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ ,α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ ,α = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ ,α = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ . (8.13)Solving the above system, we get N ∗ ( − x , ζ ) = λ − λ λ λ − λ λ , N ∗ ( − x , ζ ) = λ − λ λ λ − λ λ , (8.14)where λ = − α α − α β ,λ = − α β − β β ,λ = − α α − α β ,λ = − β β − α β . (8.15)Substituting the above equations into Eq. (8.10), we can obtain the formula of two-soliton solutions.( a ) ( b ) ( c )( d ) ( e ) ( f )17 g ) ( h ) ( i ) Figure 2.
Two-soliton solutions with parameters θ = π , θ = π , θ = π , θ = π , ζ = . + . i , ζ = − . + . i , ζ = . − . i and ζ = − . − . i . (a)(b)(c) : the structuresand the wave propagation of the two-soliton solutions with δ = (d)(e)(f) : the structures andthe wave propagation of the two-soliton solutions with δ = (g)(h)(i) : the structures and thewave propagation of the two-soliton solutions with δ = The local structure, the density and the wave propagation of two soliton solution isshown in Fig. 2. It is interesting that Fig. 2 shows the whole process of two solitonsmeet, collide elastically and move away. Furthermore, among these two solitons, one isa ordinary soliton and the other is a breather soliton. Besides, we also find a meaningfulphenomenon by select di ff erent parameter δ . Observe the three density plots carefully,we find that the angle between two solitons will increase as the parameter δ increases,which reveals the influence of parameter δ on the soliton solution graphically.( a ) ( b ) ( c ) Figure 3.
Breather-type solution with parameters δ = θ = π , θ = π , θ = π , θ = π , ζ = . i , ζ = . i , ζ = − . i and ζ = − . i . (a) : the structures of the breather-type solution, (b) : the density plot, (c) : the wave propagation of the breather-type solution. By introducing the appropriate parameters, we get the other interesting discoverywhich is presented in Fig. 3. In Fig. 3, two di ff erent breather-type solitons spreadalternately forward. Furthermore, the periodicity of the solution is clearly reflected.18 .3. Three soliton solutions In this section, we consider the three-soliton solutions of the nonlocal LPD equa-tions (1.1). Suppose the corresponding eigenvalues as follows ζ = ξ + i η , ζ = ξ + i η , ζ = ξ + i η , η , η , η > ,ζ = ξ + i η , ζ = ξ + i η , ζ = ξ + i η , η , η , η < . (8.16)Setting J = J = q ( x ) = − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x , (8.17)where C j , C j , j = , , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t . (8.18)To obtain the functions N ∗ ( − x , ζ ), N ∗ ( − x , ζ ) and N ∗ ( − x , ζ ), we need to solve thefollowing system M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , (8.19)where α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ ,α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ ,α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ , α = C ∗ ( t ) e i ζ ∗ x ζ ∗ − ζ ∗ ,β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ ,β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ ,β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ , β = C ∗ ( t ) e − i ζ ∗ x ζ ∗ − ζ ∗ . (8.20)19olving the above system, we get N ∗ ( − x , ζ j ) = det( A j )det( A ) , j = , , , (8.21)where A = λ λ λ λ λ λ λ λ λ , (8.22)and A = λ λ λ λ λ λ , A = λ λ λ λ λ λ , A = λ λ λ λ λ λ , (8.23)with λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β ,λ = − α β − α β − α β . (8.24)Substituting the above equations into Eq. (8.17), we can obtain the formula of three-soliton solutions.( a ) ( b ) ( c )20 d ) ( e ) ( f )( g ) ( h ) ( i ) Figure 4.
Three-soliton solutions with parameters θ = θ = θ = π , θ = θ = θ = π , ζ = . + . i , ζ = − . + . i , ζ = . i , ζ = . − . i , ζ = − . − . i and ζ = − . i . (a)(b)(c) : the structures and the wave propagation of the three-soliton solutions with δ = . (d)(e)(f) : the structures and the wave propagation of the three-soliton solutions with δ = (g)(h)(i) : the structures and the wave propagation of the three-soliton solutions with δ = In Fig. 4, the local structure, the density and the wave propagation of three soli-ton solutions are shown vividly. Di ff erent from the previous three solitons, the threesolitons here are composed of two arc solitons on both sides and one breathe-type soli-ton in the middle. The two arc solitons propagate forward along the left and righthalf of the circumference respectively, while the breathing solitons propagate forwardalong the diameter of the circumference, and three solitons meet, collide elastically,and move away at the central diameter of the circumference periodically. There is an-other obvious point that by change the value of δ . The period of three soliton solutionshave changed significantly. Specifically, the period will be shortened as the parameter δ increases which can be observed clearly form the graphics.Di ff erent form Fig. 4, the following Fig. 5 shows that the local structure and thedynamic behavior of three ordinary solitons. The three solitons propagation along threedi ff erent to the center( x = , t =
0) and they meet at the center point. Then the threesolitons collide elastically and move away along three di ff erent directions. During thewhole process, the amplitude, energy of three solitons are not changed. It is also worthnoting that the change of parameter δ has an influence on the rebound angle of twosolitons on both sides. 21 a ) ( b ) ( c )( d ) ( e ) ( f )( g ) ( h ) ( i ) Figure 5.
Three-soliton solutions with parameters θ = θ = θ = π , θ = θ = θ = π , ζ = . i , ζ = . i , ζ = . i , ζ = − . i , ζ = − . i and ζ = − . i . (a)(b)(c) : the structures andthe wave propagation of the three-soliton solutions with δ = (d)(e)(f) : the structures and thewave propagation of the three-soliton solutions with δ = (g)(h)(i) : the structures and the wavepropagation of the three-soliton solutions with δ = In this section, we consider the four-soliton solutions of the nonlocal LPD equations(1.1). Suppose the corresponding eigenvalues as follows ζ = ξ + i η , ζ = ξ + i η , ζ = ξ + i η , ζ = ξ + i η , η , η , η , η > ,ζ = ξ + i η , ζ = ξ + i η , ζ = ξ + i η , ζ = ξ + i η , η , η , η , η < . (8.25)22etting J = J = q ( x ) = − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x − iC ∗ N ∗ ( − x , ζ ) e i ζ ∗ x , (8.26)where C j , C j , j = , , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t , C ( t ) = C (0) e (16 i δζ − i ζ ) t , C ( t ) = C (0) e ( − i δζ + i ζ ) t . (8.27)To obtain the functions N ∗ ( − x , ζ ), N ∗ ( − x , ζ ) and N ∗ ( − x , ζ ), we need to solve thefollowing system M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , M ( x , − ζ ∗ ) = α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) + α N ∗ ( − x , ζ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , N ∗ ( − x , ζ ) = + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) + β M ( x , − ζ ∗ ) , (8.28)where α i j = C ∗ j ( t ) e i ζ ∗ j x ζ ∗ i − ζ ∗ j , β i j = C ∗ j ( t ) e − i ζ ∗ j x ζ ∗ i − ζ ∗ j , ≤ i , j ≤ . (8.29)Solving the above system, we get N ∗ ( − x , ζ j ) = det( A j )det( A ) , j = , , , , (8.30)where A = λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , (8.31)23nd A = λ λ λ λ λ λ λ λ λ λ λ λ , A = λ λ λ λ λ λ λ λ λ λ λ λ , (8.32) A = λ λ λ λ λ λ λ λ λ λ λ λ , A = λ λ λ λ λ λ λ λ λ λ λ λ , (8.33)with λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β ,λ = − α β − α β − α β − α β , λ = − α β − α β − α β − α β . (8.34)Substituting the above equations into Eq. (8.26), we can obtain the formula of four-soliton solutions.( a ) ( b ) ( c ) Figure 6.
Four-soliton solution with parameters δ = θ = θ = θ = θ = π , θ = θ = θ = θ = π , ζ = . i , ζ = . i , ζ = . i , ζ = . i , ζ = − . i , ζ = − . i , ζ = − . i and ζ = − . i . (a) : the structures of the four-soliton solution, (b) : the density plot, (c) : the wavepropagation of the four-soliton solution. a ) ( b ) ( c ) Figure 7.
Four-soliton solution with parameters δ = θ = θ = θ = θ = π , θ = θ = θ = θ = π , ζ = . + . i , ζ = − . + . i , ζ = . i , ζ = . i , ζ = . − . i , ζ = − . − . i , ζ = − . i and ζ = − . i . (a) : the structures of the four-soliton solution, (b) : the density plot, (c) : the wave propagation of the four-soliton solution. Figs. 6 and 7 present the local structure and the dynamic behavior of four solitonsolutions. The four solitons include two arc-shaped solitons and two ordinary solitons.Four solitons meet, collide and move away at the center point( x = , t = a ) ( b ) ( c ) Figure 8.
Four-soliton solution with parameters δ = θ = θ = θ = θ = π , θ = θ = θ = θ = π , ζ = . + . i , ζ = − . + . i , ζ = . + . i , ζ = − . + . i , ζ = . − . i , ζ = − . − . i , ζ = . − . i and ζ = − . − . i . (a) : the structures of the four-solitonsolution, (b) : the density plot, (c) : the wave propagation of the four-soliton solution. . Eigenvlaues and conserved quantities under some special initial conditions Before this section, we consider pure soliton solutions of the objective equation(1.1) under the condition ρ ( ξ ) = ρ ( ξ ) =
0. However, as for more general initial condi-tion q ( x ,
0) and r ( x , In what follows, we study the following rectangular initial condition q ( x , = , x ∈ ( −∞ , , h , x ∈ (0 , L ) , , x ∈ ( L , ∞ ) , (9.1)where h and L are real and positive constants. Under the symmetry relation r ( x , = − q ∗ ( − x , r ( x , t ) r ( x , = , x ∈ ( −∞ , − L ) , − h , x ∈ ( − L , , , x ∈ (0 , ∞ ) . (9.2)According to the t -independent scattering problem, we have φ , x = − i ζφ + q ( x , t ) φ ,φ , x = i ζφ + r ( x , t ) φ . (9.3)Instituting the above initial condition into Eq. (9.3) and solving the ordinary di ff erentialequations, we have < x < L , φ ( x , ζ ) φ ( x , ζ ) = h i ζ c e i ζ x + c e − i ζ x c e i ζ x , − L < x < , φ ( x , ζ ) φ ( x , ζ ) = ˜ c e − i ζ x ˜ c e i ζ x + h i ζ ˜ c e − i ζ x . (9.4)In order to match the values of eigenfunctions at the critical point x = x = − L , weobtain ˜ c = , c = h i ζ (cid:16) − e i ζ L (cid:17) , ˜ c = − h i ζ e i ζ L , c = + h i ζ ! ( e i ζ L − . (9.5)26t the same time, according to Eqs. (3.1) and (4.3), when x > L , we have φ ( x , t ) = a ( ζ ) e − i ζ x b ( ζ ) e i ζ x . (9.6)In the process of matching the value of eigenfunction at x = L , we find a ( ζ ) = + h i ζ ! ( e i ζ L − − h i ζ ! ( e i ζ L − e i ζ L ) , b ( ζ ) = − he i ζ L sin( ζ L ) ζ , (9.7)then the eigenvalues, i.e. the zeros of a ( ζ ), can be given implicitly by e i ζ L − ± i ζ h = . (9.8)Besides, the asymptotic behavior of a ( ζ ) for large and small ζ can be derived from Eq.(9.7) a ( ζ ) ∼ − h (2 i ζ ) , ζ → ∞ , a ( ζ ) ∼ − h L , ζ → . (9.9)With the aid of the lagre ζ asymptotic behavior of a ( ζ ) and Eq. (3.6), we find that theconserved quantities satisfy C n = , C n + = − h n + n + , n = , , , . . . . (9.10) In the second example, we consider the following arcuated initial condition q ( x , = , x ∈ ( −∞ , , − x + Lx , x ∈ (0 , L ) , , x ∈ ( L , ∞ ) , (9.11)where L is real and positive constant. Under the symmetry relation r ( x , = − q ∗ ( − x , r ( x , t ) r ( x , = , x ∈ ( −∞ , − L ) , x + Lx , x ∈ ( − L , , , x ∈ (0 , ∞ ) . (9.12)27nstituting the above initial condition into the scattering problem (9.3) and solving theordinary di ff erential equations, we have < x < L , φ ( x , ζ ) φ ( x , ζ ) = c e − i ζ x + c e i ζ x h i ζ ( − x + Lx ) + i ζ ) (2 x − L ) − i ζ ) i c e i ζ x , − L < x < , φ ( x , ζ ) φ ( x , ζ ) = ˜ c e − i ζ x ˜ c e − i ζ x h − i ζ ( x + Lx ) − i ζ ) (2 x + L ) − i ζ ) i + ˜ c e i ζ x . (9.13)Matching the value of the eigenfunction at x = − L , we find˜ c = , ˜ c = e i ζ L ( 2(2 i ζ ) − L (2 i ζ ) ) , c = + e i ζ L i ζ ) − L (2 i ζ ) ! − i ζ ) + L (2 i ζ ) ! , c = e i ζ L i ζ ) − L (2 i ζ ) ! − i ζ ) + L (2 i ζ ) ! . (9.14)Simliar to case 1, when x > L , we have φ ( x , t ) = a ( ζ ) e − i ζ x b ( ζ ) e i ζ x . (9.15)In the process of matching the value of eigenfunction at x = L , we find a ( ζ ) = − " e i ζ L i ζ ) − L (2 i ζ ) ! − i ζ ) + L (2 i ζ ) ! , b ( ζ ) = e i ζ L i ζ ) − L (2 i ζ ) ! − e − i ζ L i ζ ) + L (2 i ζ ) ! , (9.16)then the eigenvalues, i.e. the zeros of a ( ζ ), can be given implicitly by e i ζ L i ζ ) − L (2 i ζ ) ! − i ζ ) + L (2 i ζ ) ! ± = . (9.17)Besides, the asymptotic behavior of a ( ζ ) for large and small ζ can be derived from Eq.(9.7) a ( ζ ) ∼ − L (2 i ζ ) , ζ → ∞ , a ( ζ ) ∼ − L , ζ → . (9.18)With the aid of the lagre ζ asymptotic behavior of a ( ζ ) and Eq. (3.6), we find that theconserved quantities satisfy C n = − L m + m + , n = m + , m = , , , . . . , , else . (9.19)28 .3. Trianglular wave In the second example, we consider the following triangular initial condition q ( x , = , x ∈ ( −∞ , , L − | x − L | , x ∈ (0 , L ) , , x ∈ (2 L , ∞ ) , (9.20)where L is real and positive constant. Under the symmetry relation r ( x , = − q ∗ ( − x , r ( x , t ) r ( x , = , x ∈ ( −∞ , − L ) , − L + | x + L | , x ∈ ( − L , , , x ∈ (0 , ∞ ) . (9.21)Instituting the above initial condition into the scattering problem (9.3) and solving theordinary di ff erential equations, we have − L < x < − L , φ ( x , ζ ) φ ( x , ζ ) = c e − i ζ x c e i ζ x + c e − i ζ x (cid:16) x + L i ζ + i ζ ) (cid:17) , − L < x < , φ ( x , ζ ) φ ( x , ζ ) = c e − i ζ x c e − i ζ x (cid:16) − x i ζ − i ζ ) (cid:17) + c e i ζ x , < x < L , φ ( x , ζ ) φ ( x , ζ ) = c e i ζ x (cid:16) x i ζ − i ζ ) (cid:17) + c e − i ζ x c e i ζ x , L < x < L , φ ( x , ζ ) φ ( x , ζ ) = c e − i ζ x + c e i ζ x (cid:16) L − x i ζ + i ζ ) (cid:17) c e i ζ x . (9.22)Matching the value of the eigenfunction at x = − L , − L , , L , we find c = , c = − i ζ ) e i ζ L , c = , c = i ζ ) (2 e i ζ L − e i ζ L ) , c = − i ζ ) ( e i ζ L − , c = − i ζ ) ( e i ζ L − , c = − i ζ ) ( e i ζ L − , c = − i ζ ) ( e i ζ L − + e i ζ L (2 i ζ ) ( e i ζ L − . (9.23)Simliarly, when x > L , we have φ ( x , t ) = a ( ζ ) e − i ζ x b ( ζ ) e i ζ x . (9.24)In the process of matching the value of eigenfunction at x = L , we find a ( ζ ) = − i ζ ) ( e i ζ L − , b ( ζ ) = − e i ζ L (sin( ζ L )) ζ , (9.25)29hen the eigenvalues, i.e. the zeros of a ( ζ ), can be given implicitly by e i ζ L − ± i ζ = , or e i ζ L − ± ζ = . (9.26)However, since a ( ζ ) = a ∗ ( − ζ ∗ ), the eigenvalues are determined uniquely by e i ζ L − ± i ζ = . (9.27)Besides, the asymptotic behavior of a ( ζ ) for large and small ζ can be derived from Eq.(9.7) a ( ζ ) ∼ − i ζ ) , ζ → ∞ , a ( ζ ) ∼ − L , ζ → . (9.28)With the aid of the large ζ asymptotic behavior of a ( ζ ) and Eq. (3.6), we find that theconserved quantities satisfy C n = − m + , n = m + , m = , , , . . . , , else . (9.29)
10. Conclusions
In this work, a detailed study of the inverse scattering transform for a new nonlocalLPD equation is carried out. Firstly, by an ingenious method, the local and global con-servation laws of nonlocal LPD equation is obtained, which establish the integrabilityas an infinite dimensional Hamilton dynamic system. The direct scattering problemis constructed and some critical symmetries are obtained. Afterwards, with the aidof the novel Left-Right RH approach, the inverse scattering problem is established.Furthermore, the potential function is recovered successfully. By introducing the re-flectionless case, the soliton solutions of the nonlocal LPD equation are given. Inorder to understand the dynamic behavior of soliton solutions more intuitively, we take J = J = , , , δ on soliton solu-tions. Besides, under some special cases of initial condition such as rectangular wave,arc wave and triangular wave, we consider the zeros of the scattering data a ( ζ ) and theconserved quantities. Acknowledgements
This work was supported by the Natural Science Foundation of Jiangsu Provinceunder Grant No. BK20181351, the National Natural Science Foundation of China un-der Grant No. 11975306, the Six Talent Peaks Project in Jiangsu Province under Grant30o. JY-059, the Qinglan Project of Jiangsu Province of China, and the FundamentalResearch Fund for the Central Universities under the Grant Nos. 2019ZDPY07 and2019QNA35.
References [1] M. J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, Vol. 4,Siam, 1981.[2] E. M. Dianov, P. Mamyshev, A. M. Prokhorov, Nonlinear fiber optics, Soviet J.Quantum. Elect. 18 (1) (1988) 1.[3] Y. S. Kivshar, G. Agrawal, Optical solitons: from fibers to photonic crystals,Academic press, 2003.[4] M. J. Ablowitz, P. Clarkson, P. A. Clarkson, Solitons, nonlinear evolution equa-tions and inverse scattering, Vol. 149, Cambridge university press, 1991.[5] M. J. Ablowitz, D. E. Baldwin, Nonlinear shallow ocean-wave soliton interactionson flat beaches, Phys. Rev. E. 86 (3) (2012) 036305.[6] B. B. Kadomtsev, V. I. Petviashvili, On the stability of solitary waves in weaklydispersing media, Sov. Phys. Dokl, 15 (1970) 539.[7] R. Hirota, Direct methods in soliton theory, in: Solitons, Springer, Berlin, 2004.[8] V. Matveev, Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters, Lett. Math. Phys.3 (3) (1979) 213–216.[9] A.S. Fokas, A unified transform method for solving linear and certain nonlinearPDEs, Proc. R. Soc. Lond. Ser. A 453 (1997) 1411-1443.[10] S. Novikov, S. Manakov, L. Pitaevskii, V. E. Zakharov, Theory of solitons: theinverse scattering method, Springer Science & Business Media, 1984.[11] M. J. Ablowitz, Z. H. Musslimani, Integrable nonlocal nonlinear Schr¨odingerequation, Phys. Rev. Lett. 110 (6) (2013) 064105.[12] M. J. Ablowitz, Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud.Appl. Math. 139 (1) (2017) 7–59.[13] A. S. Fokas, Integrable multidimensional versions of the nonlocal nonlinearSchr¨odinger equation, Nonlinearity 29 (2016) 319.3114] W.Q. Peng, S.F. Tian, T.T. Zhang, and Y. Fang, Rational and semi-rational so-lutions of a nonlocal (2 + / Sinh-Gordon Equations with Nonzero Boundary Conditions. Stud. Appl.Math. 141(3) (2018) 267-307.[20] J.G. Rao, Y.S. Zhang, A. S. Fokas and J.S. He, Rogue waves of the nonlocalDavey-Stewartson I equation, Nonlinearity 31 (2018) 4090.[21] Z.X. Zhou, Darboux transformations and global explicit solutions for nonlocalDavey-Stewartson I equation, Stud. Appl. Math. 141 (2018) 186.[22] S.Y. Lou, Alice-Bob systems, P s - T d - C principles and multi-soliton solutions,arXiv:nlin / + ffff