General multi-soliton and higher-order soliton solutions for a novel nonlocal Lakshmanan–Porsezian–Daniel equation

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, a novel integrable nonlocal LPD equation corresponding to a 3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document} Lax pair is proposed. Secondly, the inverse scattering process with a novel left-right 3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document} matrix Riemann–Hilbert(RH) problem is constructed. The analytical properties and symmetry relations for the Jost functions and scattering data are considerably different from the local ones. Due to the special symmetry properties for the nonlocal LPD equation, the zeros of the RHP problem are purely imaginary or occur in pairs. With different types and configuration of zeros, the soliton formula is provided and the rich dynamical behaviors for the three kinds of multi-solitons for the novel nonlocal LPD equation are demonstrated. Third, by a technique of adding perturbed parameters and limiting process, the formula of higher-order solitons for the nonlocal LPD equation is exhibited. Lastly, the plots of diverse higher-order solitons and various solutions corresponding to different combinations of the following zeros: purely imaginary higher-order zeros, purely imaginary simple zeros, pairs of non-purely imaginary simple zeros and pairs of non-purely imaginary higher-order zeros are displayed.


Introduction
The nonlinear Schrödinger (NLS) equation is a key physical model in optical and many other fields which describes the wave propagation in Kerr media. Then the higher-order integrable NLS hierarchy has been studied [1][2][3]. As a higher-order extension of the scalar NLS equation, the nonlinear Lakshmanan-Porsezian-Daniel (LPD) equation with the fourth nonlinear and dispersion terms was introduced [4,5] to describe the effect of discreteness of the lattice on the classical continuum limit of the isotropic Heisenberg ferromag-netic spin chain. Due to the important application in nonlinear optics, abundant optical solitons [6][7][8][9][10][11] and some new exact solutions [12] for LPD equation were explored. Besides, the rich dynamic behaviors of solitons, rogue waves and the interactional solutions for a multi-component LPD equation were studied by Darboux transformation [13][14][15][16] and inverse scattering method [17].
Recently, a series of novel nonlocal NLS equations with clear physical motivations were proposed from the nonlocal reduction in Manakov system [43]. Different from the nonlocal NLS equation proposed in [18], the nonlocal nonlinearity-induced potential of the novel nonlocal NLS equation [43] is real and symmetric in x. In addition, they discovered that the structures of the Riemann-Hilbert problem (RHP) and the form of general soliton solutions were more complicated, which led to some interesting dynamical properties of soliton solutions.
Inspired by the idea of Yang [43], the following novel physically important nonlocal LPD equation is proposed: where q is the potential function of (x, t), α, β are arbitrary real numbers, the superscript * represents complex conjugation. The nonlinearity-induced potentials V k (x, t)(k = 1, 2, 3) with forms So V k are metrical in t and V k (x, t) = V * k (x, −t) (k = 1, . . . , 3). The nonlocal Eq. (1) is invariant under the nonlocal reverse-time transformation t → −t and the conjugate transformation.
The nonlocal LPD Eq. (1) corresponding to the following 3 × 3 Lax pair: where Φ is a three-dimensional column vector with the elements are complex functions of (x, t; λ) and λ is the spectral parameter. Λ = diag(1, 1, −1) and The Lax pair is a 3×3 matrix system with two linear matrix differential equations 4a-4b. The nonlocal LPD Eq. (1) can be obtained by the compatible condition U t − V x + [U, V ] = 0. This proves the integrability of the nonlocal LPD Eq. (1). When α = 1, β = 0, Eq. (1) reduced to the nonlocal NLS equation proposed in [43]. The scattering process of the nonlocal LPD Eq. (1) will be presented with the following vanishing initial condition: where S(R) is the Schwartz space. The inverse scattering transformation (IST) of Manakov system with initial value condition (5) had been established in [43,44], where the RH problem corresponding to the spectral problem (4a) and the N -soliton formula were constructed. However, as for the time evolution part and the symmetry relations, which are quite different from the nonlocal LPD Eq. (1). The constraints of the scattering data and eigenvectors were much more complex to derive, which will lead to richer configurations of zeros and dynamical behavior of soliton solutions. Furthermore, this will lead to much richer distributions of the higher-order zeros and the structures of higher-order solitons. We will first study the formula of soliton solution under different distribution of zeros by inverse scattering process via constructing an RHP. Then we will try to get the formula of the solution corresponding to the higher-order zeros. In addition, the rich structures and characteristics of the solution under varied higher-order zeros will be considered.
The organization is as follows. Section 2 provides the inverse scattering process via an RH problem, the time evolution and the recovery of the potential function. Then the multi-soliton solution formula is obtained when the RHP is reflectless and the zeros are all simple. In Section 3, the dynamics for the general soliton solution with simple zeros are investigated in three cases: (1) n 1 = 0, n 2 = 0; (2) n 1 = 0, n 2 = 0 and (3) n 1 = 0, n 2 = 0. Section 4 concludes the higherorder soliton formula by a limit process and exhibits kinds of higher-order soliton solutions corresponding to different configurations of higher-order zeros and simple zeros. The last section is the conclusion.

Inverse scattering process and multi-soliton solutions
In this section, we will consider the scattering problems of the nonlocal Eq. (1) with initial condition (5). By the initial value condition (5), Define the solution of Eqs. (4a and 4b) with the form Φ = J Φ bg with J is a complex function of (x, t; λ). So J → I, x → ±∞, where I is the third-order identity matrix and J satisfies where [Λ, J ] = ΛJ − J Λ is a commutator. The adjoint Lax pair can be obtained as

The Jost solution and the construction of the RHP
To implement the scattering process of potential function at the fixed point t = 0, that is, the potential function q and the other related functions will be treated as functions only related to the variable x. Then, we will construct the corresponding scattering problem and calculate the initial scattering data at t = 0. To avoid confusion, it is no longer explicitly stated in this subsection. After the initial scattering data at the fixed point t = 0 is obtained, the scattering data at arbitrary time will be obtained through time inverse, and then, the solution q(x, t) at arbitrary time will be reconstructed by using the inverse scattering process.
Recall Φ satisfies the following scattering problem (4b) and the corresponding adjoint scattering problem For simplicity, (x, 0; λ) and (x, 0) will be omitted in this section.
By the integral Eq. (10), to guarantee the integral of the right side coverage. Similar to the method of [44], by the structure of the potential matrix Q, the analytical properties of J k can be obtained.
For J 1,2 is the solution of Eq. (6a), by the definition of (4a), so there exists solution function Φ 1,2 of (4a) that is a diagram matrix. Φ 1,2 are two different nonzero solutions of the same linear Eq. (4a), so there exists a matrix S(λ) = [s k j ] 3×3 who is independent of the variable x and Φ 1 S = Φ 2 , i.e., Then by (11), the analytical properties of the elements of S can be obtained. Furthermore, take S −1 = (ŝ k j ) 3×3 , then recall the analytical properties of matrix J 1,2 , we can obtain the following properties: s 11 , s 21 , s 12 , s 22 ,ŝ 33 are analytical when λ ∈ C − andŝ 11 ,ŝ 21 ,ŝ 12 ,ŝ 22 , s 33 are analytical in λ ∈ C + . Sign J k = J [1] k , J [3] k , J [3] k and J −1 k is the j-th column of matrix J k andĴ [ j] k is the jth row of matrix J −1 k , , j = 1, 2, 3. So the analytical properties of S can be obtained. Then, we have the following lemma.
Note that the trace of the matrix U is iλ. It can be obtained by Abel's identity that det(Φ(x; λ)) = e iλx . Then, combine the definition of J ± , and the definition of S, we have det(J 1,2 (x, λ)) = det(S(λ)) = 1.
To construct a well-defined RHP, define the modified Jost solutions P + and P − with the following forms where H 1 = diag(1, 1, 0), H 2 = diag(0, 0, 1). It can be calculated that det(P + ) = s 33 , det(P − ) =ŝ 33 . So P + and P − are analytical in λ ∈ C + and λ ∈ C − , respectively. By the definition of P ± and J 1,2 , it is obvious that P + is a solution of (6a), and P − is the solution of (7a). Recall the Volterra integral Eq. (10), the following λ asymptotic property can be obtained.
In summary, the modified matrix P satisfies the following matrix RH decomposition problem.

Riemann-Hilbert Problem 1
The sectionally meromorphic matrix P satisfies the following conditions: This is a matrix equation of a sectionally analytical matrix P which is constructed from the initial potential at t = 0. So there should be some relationships between the solution of RHP and the potential function.
To establish the relation between the potential function and the solution of the RHP. The relation of the potential matrix and the solution of the RHP is necessary. Thus, if the RHP 1 is solved, we can get the corresponding potential function q.
Expanding the function P + at λ → ∞ as P + (λ) = I 3 +λ −1 P [1] + (x)+O λ −2 , and inserting the expansion of P + into (6a). Then comparing the same order terms of λ, the potential matrix Q can be given by and the coefficient of λ −1 : P [1] +,x = i[λ, P [2] + ] + Q P [1] + , P [1] −,x = i[λ, P [2] − ] − P [1] − Q. Then, together with the equations (14), we can conclude that the asymptotic expansion of P + at λ = ∞ Similarly, the asymptotic expansion of P − can be obtained as Implementing the direct calculation on the equation above, the asymptotic expansion of the determination det(P + ), i.e., the scattering coefficientŝ 33 and s 33 can be obtained Besides, by Eq. (14) and the form of matrix Q, the potential function q can be recovered by the following formula Imposing the similar process to P − , we can get q = 2i[P [1] − ] 13 . If the RH problem (13) can be solved, then the potential function q can be reconstructed from its solution P ± . By the definition of Q, it can be checked that Q and the related functions satisfy the following two kinds of symmetry relations Proposition 1 The modified Jost function P ± , the scattering data satisfy the symmetry properties -The self-conjugate symmetry properties -The reverse-time symmetry properties Proof Begin from the first symmetry relation of Q, the anti-Hermitic symmetry (16). Take Hermitian conjugate on both sides of the scattering Eq. (6a), we can get J † meet with spectral problem (9), and J −1 also satisfies Eq. (9). So there is a constant matrix C 0 J † (λ * ) = J −1 λC 0 , make x → ±∞, in combination with the asymptotic behavior of J at infinite, then C 0 = I, as a result, J † (λ * ) = J −1 (λ). Consequently, the symmetry relations of P ± and scattering matrix are And the symmetry relations of the scattering dataŝ 33 (λ) = s 33 (λ * ), so if s 33 (λ k ) = 0, then we haveŝ 33 (λ * k ) = 0. Secondly, due to the potential matrix Q also satisfies the symmetry relations Q(x, t) = −Q † (x, t), take conjugate of the nonlocal LPD Eq. (1). Then, multiply Σ from both sides and replace t with −t, then we concluded that Similarly, by combining the asymptotic behavior of function J at infinity, Σ J * (x, −t; −λ * )Σ = J (x, t; λ). And furthermore, by the definitions of P ± and the matrix S, the following symmetric properties can be obtained and By the symmetry properties above, suppose det(P + ) =ŝ 33 possess N simple zeros λ k ∈ C + , k = 0, . . . , N = n 1 +2n 2 .
Denote the eigenvectors for P + and P − corresponding to the eigenvalues corresponding to the eigenvalues λ k ∈ C + andλ k ∈ C − are v k andv k , respectively. Since P + satisfies the scattering problem (6a) and P − satisfies the adjoint scattering problem (7a), we have v k = e iλ k Λxṽ k 0 and v † k =ṽ * k 0 e −iλ k Λ * x . Then, we consider the time evolution of the inverse scattering process. To get the time dependence of the related functions, recall the time-dependent part (6b) of the Lax pair for J , then the time evolution of the scattering matrix can be gotten as so the corresponding Eq. (11) becomes by Eq. (20), we can get that Let x → ±∞ in the above equation, due to V 0 → 0, J 1 (x, t; λ) → I, we get Similarly, the time dependence of S −1 (t; λ) can be gotten Consequently, the zeros of the RHP are dependent with t. Successively, by the scattering problem (6b), the time evolution of the eigenvectors v k can be taken as and the corresponding adjoint problem, respectively. Take the derivative with respect to t of both sides of following equation v k = e (iλ k x+2iλ 2 (α−4βλ 2 )t)Λ v k 0 . In addition, transform t → −t and take conjugation on the kernel equation and combine the symmetry relation (17), the corresponding eigenvector for To sum up, for the eigenvalue λ k ∈ C + , the corresponding eigenvectors are and where v [0] k = (a [1] k , (a [1] k ) * , c [1] k ), k = 1, . . . , n 1 (a [2] k , b [2] k , c [2] k ), k = n 1 + 1, . . . , n 1 + n 2 with c [1] k are real and a [1] k , a [2] k , b [2] k , c [2] k are arbitrary complex numbers.
To solve the above problems with simple zeros, we need to transform the non-regular problem into the regular RH problem by removing simple zeros, and then, the solution of the regular RH problem can be expressed by the Sokhotski-Plemelj formula. By virtue of the above eigenvectors v k andv k , for j = 1, , . . . , n 1 , introduce the matrix Θ j For k = n 1 +1, . . . , n 2 , define the matrixΘ k By direct calculation, we obtain .
Then, introduce the following matrix T with and then, we have the following lemma.

Lemma 2 The above product expression (24) of matrix T can be written as the following expression with the sum of partial fractions
where M is a matrix of N × N with the elements and the determination can be gotten as where . The proof of the lemma 2 can refer to the proof of theorem 2.1 in reference [44]. Here, we omit the process of details.
Furthermore, when the system is reflectless, the reflectless potential function q corresponds to the soliton solution. By the solution formula to the nonregular RHP with simple zeros in [44], then the scattering coefficients of the RHP 1 are all equal to zero, i.e., G = I, so P −1 + (λ) = T −1 (λ), λ ∈ C + . i.e., P + (λ) = T (λ), the sum expansion of T (λ) have been obtained by (24), so we have The following theorem can be obtained.

Theorem 1 The N soliton formula for the nonlocal LPD Eq. (1) is
with v k are shown in Eq. (22).
To illustrate the formula (27), the dynamical behaviors for the general soliton solutions will be studied in next section.
The amplitude is determined by the imaginary part of the spectrum parameter η 1 and parameter a [1] 1 . Furthermore, it can be seen that the parameters α and β do not affect the amplitude of this standing wave. The plot with Fig. 2a. (2) If n 1 = 2, take η 1 = 1 4 , the ring breather wave shown in Fig. 2b with η 2 = 1 3 and the line breather wave in Fig. 2c with η 2 = 4 3 . (3) If n 1 = 3, the ring-line breather wave shown in Fig. 2 (e) with η 1 = 1 4 , η 2 = 1 3 , η 3 = 1 2 is gotten. Besides, it is worth mentioned that the parameter a [1] k are all taken as i 2 . But the shape of the waves is quite different with different a [1] k . For example, for n 1 = 2, when a [1] 1 = a [1] 2 , the ring breather wave will no longer be symmetric with respect to x = 0 and with same parameters as (a) except for b [2] 1 = 0; c with μ 1 = 1 2 + i 4 , μ 2 = 1 2 + i 3 and 2 = 1; d with same parameters as (c) except for a [2] 1 = 0; e with same parameters as (c) except for b [2] 1 = 0 Fig. 4 The 2d-plots of Eq. (27) with different parameters. a b with same eigenvalues except for a [2] 1 = 0; c and d with same , the corresponding plot in Fig. 2 (d) and other rich dynamical behaviors can be obtained with different values combination of a [1] k . Case 2 When n 1 = 0, n 2 = 0. The solution is a 2n 2 -soliton solution which is related to a pair of nonpurely imaginary eigenvalues {μ k , −μ k }, so their center positions for the 2n 2 -soliton are symmetric with respect to t = 0. If n 2 = 1, the fundamental twosoliton solution is gotten in Fig. 3a where And with same parameters except for b [2] 1 = 0, the traveling single soliton is obtained in Fig. 3 (b) which is reduced from the twosoliton solution. Similarly, an interesting phenomenon is observed when n 2 = 2. The general four-soliton can be obtained in Fig. 3c with the nonzero parameters but when the parameter a [2] 1 = 0, one of the soliton disappear after the collision in Fig. 3 (d-e). The other parameters are μ 1 = 1 2 + i 4 , μ 2 = 1 2 + i 3 and a [2] 1 = 0, a [2]

Case 3
When n 1 n 2 = 0, the solution is a interaction wave composed by elementary 2n 2 -soliton and periodic breathing wave. But there is one exception, when n 1 = 1, n 2 = 1, a general two-soliton interact with a standing wave parallel to the X-axis is illustrated in Fig. 4a. Similarly, the interactive wave as shown in Fig. 4b can be obtained when a [2] 1 = 0. Besides, we also give two examples of n 1 = 2, n 2 = 1, then the circle breather interacting with a two soliton is obtained in Fig. 4c with a [2] 1 = 1+ i 2 and Fig. 4d a [2] 1 = 0. Fig. 4e is the ring-line breather interacting with a two-soliton wave. In this case, both amplitudes of the breather and soliton change after collision.

Theorem 2
If det (P + ) possess n 1 + 2n 2 higher-order zeros (19), then the N 0 -th higher-order soliton formula can be gotten as where χ [k],χ[ j] be the k-th column and j-th row of the following χ andχ , respectively.
matrixes with the elements are The proof of the theorem can be referred to the details in [45,47], where the nonlocal NLS equation was corresponded to 2 × 2 linear equation. The process for this nonlocal LPD Eq. (1) is similar except for the symmetry properties. Thus, the detailed process of the higher-order soliton formula for such nonlocal LPD Eq. (1) is omitted here.
Remark The biggest difficulty and difference here is that the symmetry properties for the perturbed parameters. Due to the special nonlocal properties for such nonlocal LPD Eq. (1) corresponding to a 3×3 Lax pair, the symmetry properties for the perturbed parameters are quite different and difficult to conclude.
To illustrate the higher-order soliton formula (32), we give some examples as follows.
When N 1 = 1, N 2 = 0, the RHP (1) only have a purely imaginary higher-order zero ζ 1 . In this case, the solution is a higher-order soliton. If r 1 = 2, at this time, implement the Taylor expansions of the following where the elements m [l 1 ,l 2 ] 1,1 are the coefficients of the following expansion +m [1,2] 1,1 + m [2,2] 1,1¯ + o(¯ ). By the higher-order formula (32), then the 2-order soliton solution for the nonlocal LPD Eq.is .
Let ζ 1 = iξ 1 in the equation above and simplify the equation, so we have 1,1 2 ξ 1 2 + 4. Take 1,1 = 0 and the plot of 2-order soliton for the nonlocal LPD Eq. (1) is shown in Fig. 5 (a). It can be seen that the branches of higher order solitons propagate forward in almost parallel directions with time, and then return to the original direction after the collision near the zero point and continue to propagate forward. Besides, the 3-order and 4-order soliton corresponding to a purely imaginary higher-order zero for the nonlocal LPD Eq. (1) are shown in Fig. 5b and c, respectively. In addition, in Fig. 5d, we also give the mixed higherorder soliton solution of a two-order soliton and a standing wave, which is correspond to a purely imaginary higher-order zero and a purely imaginary simple zero. And Fig. 5e is a higher-order soliton solution which is correspond to two purely imaginary 2-order zeros. Fig. 5c and e shows two different kinds of 4-order soliton corresponded to different spectral configurations. Besides, Fig. 5f and g shows the other two kinds of 4order solitons with different higher-order zeros, respectively. The wave shown in Fig. 5f is correspond to a two second-order purely imaginary zeros and a pair of non-purely imaginary simple zeros. And the solution of Fig. 5g is correspond to a pair of second-order nonpurely imaginary zeros. Finally, Fig. 5e gives a standing wave interact with a fourth-order soliton shown in Fig. 5d, which is correspond to a purely imaginary simple zero and a pair of non-purely imaginary secondorder zeros .

Conclusion
In summary, we proposed a novel nonlocal LPD Eq. (1) corresponding to a 3 × 3 Lax pair (4a and 4b), which is reduced from a two-component coupled LPD Eq. (3) by a reverse-time reduction. Then, the inverse scattering transformation for initial value problem of the nonlocal LPD Eq. (1) was studied with the help of an RHP. The RHP was established after constructing the Jost function with specific analytic properties and asymptotic properties. The symmetry relations were then provided for the Jost function and the scattering data.
The zeros of the RHP were purely imaginary or appear in pairs as (μ k , −μ * k ). Then, the corresponding reflectionless potential formulas for the RHP were given as the following three different types of simple zeros: Case 1: all the zero are purely imaginary; Case 2: all the zero are non-purely imaginary or in pairs; Case 3: n 1 zeros are purely imaginary and n 2 pairs of non-purely imaginary zeros . At the same time, the corresponding plots and dynamical behaviors for the above solutions were given. For case 1, when n 1 = 1, n 2 = 0, the solution is a standing wave which did not travel with time. When n 1 ≤ 2, n 2 = 0, the solutions were breathing with time and kinds of rich structures were given with different parameters. For case 2, when n 1 = 0, n 2 = 0, the 2n 2 solitons occurring symmetrically were obtained with suitable parameters. Besides, the 2n 2 − 1 solitons and special soliton structures shown in Fig. 3c and e can be obtained in this case. For Case 3, when n 1 n 2 = 0, the interaction solutions of a two-soliton solutions and a standing wave were obtained. It is worth mentioning that the amplitude of these two waves changed markedly after the collision. The plots of the symmetric two-solitons interacting with a standing wave and breathers are shown in Fig. 4a, c and e. Besides, with special parameters, the special interacting waves as shown in Fig. 4b and d were obtained.
Ultimately, the higher-order soliton formula was obtained by adding perturbations and taking limit of the perturbations. Then by some matrix transformations, the higher-order formula in determinant form (5) was given. It is worth noting that this formula can be reduced to the solution formula (27) corresponding to simple zeros when the order of all higher-order zeros is one. In addition, through this formula (5), we can also get some solutions when the high-order zeros and the simple zeros exist simultaneously. The corresponding solutions corresponding to different combinations of the following zeros: purely imaginary higher-order zeros, purely imaginary simple zeros, pairs of non-purely imaginary simple zeros and pairs of non-purely imaginary higher-order zeros are shown in Fig. 5, respectively.
Furthermore, the inverse scattering transformation and kinds of soliton solutions for the following nonlocal integrable LPD equation will be considered in the future.
Author contributions All authors contributed to the study conception and theory. Calculation and analysis were performed by Wang Minmin. The first draft of the manuscript was written by Wang Minmin. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding The work is supported by the Future Scientist and Outstanding Scholar Cultivation Program of East China Normal University (No.WLKXJ202001), National Natural Science Foundation of China (No.12175069, 12235007) and Science and Technology Commission of Shanghai Municipality (No.21JC1402500 and No.18dz2271000).

Data Availability
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Conflicts of interest
The authors have no relevant financial or nonfinancial interests to disclose. The authors declare that there is no conflict of interests regarding the publication of this paper.