Dynamic behaviors of soliton solutions for a three-coupled Lakshmanan–Porsezian–Daniel model

In this paper, we use the Riemann–Hilbert (RH) approach to examine the integrable three-coupled Lakshmanan–Porsezian–Daniel (LPD) model, which describes the dynamics of alpha helical protein with the interspine coupling at the fourth-order dispersion term. Through the spectral analysis of Lax pair, we construct the higher-order matrix RH problem for the three-coupled LPD model, when the jump matrix of this particular RH problem is a 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4$$\end{document} unit matrix, the exact N-soliton solutions of the three-coupled LPD model can be exhibited. As special examples, we also investigate the nonlinear dynamical behaviors of the single-soliton, two-soliton, three-soliton and breather soliton solutions. Finally, an integrable generalized N-component LPD model with its linear spectral problem is discussed.


Introduction
Since the nonlinear evolution equations can be widely used to describe some of the physical phenomena, such as nonlinear optical, quantum mechanics, fluid physics, and plasma physics. The research on the methods of solving nonlinear evolution equations becomes a challenging and vital task, and has attracted more and more people's attention. With the development of soliton theory, a series of methods for solving nonlinear development equations are proposed, such as the inverse scattering method [1], the Hirota's bilinear method [2], the Bäcklund transformation method [3], the Darboux transformation (DT) method [4] and others [5][6][7][8]. Based on these available methods, we have obtained a series of solutions of nonlinear evolution equations, including compaton solutions, peakon solutions, periodic sharp wave solutions, lump solutions, breather solutions, bright soliton, dark soliton, rogue waves, etc. These solutions can further help to understand natural phenomena and laws. In recent years, more and more mathematical physicists have begun to pay attention to Riemann-Hilbert (RH) approach [9,10], which is a new powerful method to solve integrable linear and nonlinear evolution equations [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The main idea of this method is to establish a corresponding matrix RH problem on the Lax pair of integrable equations. Furthermore, the RH approach is also an effective way to examine the initial boundary value problems [28][29][30][31][32] and the long-time asymptotic behaviors [33,34] of the integrable nonlinear evolution equations.
The Lakshmanan-Porsezian-Daniel (LPD) model [35] is one of the most paramount integrable systems in mathematics and physics which reads This model not only can simulate the nonlinear transmission and interaction of ultrashort pulses in highspeed optical fiber transmission systems, but also can describe the nonlinear spin excitation phenomenon of a one-dimensional Heisenberg ferromagnetic chain with octopole and dipole interactions [36]. When ε = 0, Eq. (1.1) can be reduced to the famous nonlinear Schrödinger (NLS) equation. However, a slice of phenomena have been observed by experiment which cannot be explained by Eq. (1.1). For example, the dynamics of the alpha helical protein with the intermittent coupling at the fourth-order dispersion term. In order to explain these phenomena, the three-component case has been introduced as follows [37][38][39]: where q α (x, t), (α = 1, 2, 3) represent amplitude of molecular excitation in the α-th spine, the subscripts x and t represent the partial derivatives with respect to the scaled space variable and retarded time variable, respectively, the * denotes the complex conjugate, and ε is a real parameter standing for the strength of higher-order linear and nonlinear effects. When ε = 0, Eq. (1.2) can be reduced to the three-component NLS equation [40,41]. Indeed, the three-coupled LPD model (1.2) is a completely integrable model, and quiet a few properties have been investigated, for instance, the Lax pair [37], the Hirota bilinear forms, the soliton solu-tions by the DT method and the binary Bell polynomial approach [38], and the semirational rogue waves by the generalized DT method [39]. In this paper, we aim to investigate the soliton solutions of three-coupled LPD model (1.2) via the RH approach, and discuss the dynamic behaviors of the soliton solutions. The organization of this paper is as follows. In Sect. 2, we will construct a specific RH problem based on the inverse scattering transformation. In Sect. 3, we will compute N-soliton solutions of the threecoupled LPD model (1.2) from the given RH problem in Sect. 2, which possesses the identity jump matrix on the real axis. In Sect. 4, the spatial structures and collision dynamics behaviors of soliton solutions will be examined. In Sect. 5, as a promotion, we will briefly explain an integrable generalized multicomponent LPD model.
If C + and C − denote the upper half ξ -plane and the lower half ξ -plane, respectively, then (2.9) Next, we introduce a new function A(x, ξ) = e iξσ 4 x ; then, Eq. (2.5a) exists two fundamental matrix solutions + A and − A which satisfy the following relationship: It follows from the analytic property of − that s 11 allows analytic extensions to C − , and the others s i j , i, j = 1 allow analytic extensions to C + . In order to discuss behavior of Jost solutions for large ξ , we assume that substituting the above expansion Eq. (2.13) into Eq. (2.5a) and comparing the coefficients of the same order of ξ yields On the one hand, we define another new Jost solution for Eq. (2.5a) by (2.16) Taking Eq. (2.6a)-(2.6b) into Eq. (2.10) gives rise to then, H + can be rewritten as the following matrix form: which is analytic for ξ ∈ C − and admits asymptotic behavior for very large ξ as On the other hand, in order to obtain the analytic counterpart of H + denoted by H − in C + , we consider the adjoint scattering equation of Eq. (2.5a) as follows: (2.20) Obviously, the inverse matrices −1 ± is defined as: admit analytic extensions to the C + . In addition, it is not difficult to find that the inverse matrices −1 + and −1 − satisfy the following boundary conditions: Therefore, we can define By similar techniques analysis, we can get that H − is then, H + can be rewritten as the following matrix form: (2.27) In fact, these two matrix functions H + (x, ξ) and H − (x, ξ) can be used to construct the following RH problem: • H + (x, ξ) and H − (x, ξ) are analytic for ξ in C − and C + , respectively. • H + (x, ξ) and H − (x, ξ) satisfy the following jumping relationship on the real x-axis and r 11 s 11 + r 12 s 21 + r 13 s 31 + r 14 s 41 = 1.
Furthermore, since − satisfies the temporal part of spectral equation

The soliton solutions for three-coupled LPD model
In this section, based on the RH problem constructed in Sect. 2, we would like to formulate the N-soliton solutions of three-coupled LPD model (1.2). In fact, the solution to this RH problem will not be unique unless the zeros of det H + and det H − in the upper and lower half of the ξ -plane are also specified, and the kernel structures of H ± at these zeros are provided. It follows from the definitions of H + and H − that which means that the zeros of det H + and det H − are the same as s 11 (ξ ) and r 11 (ξ ), respectively. Furthermore, the scattering data s 11 and r 11 are time independent (2.33), and we can get that the roots of s 11 = 0 and r 11 = 0 are also time independent, since where † represents the Hermitian of a matrix. It is easy to see that because H + (ξ ) is the solution of Eq. (2.5a). For large ξ , we suppose that H + (ξ ) possesses the following asymptotic expansion: Then, the potential functions solutions q 1 (x, t), q 2 (x, t) and q 3 (x, t) of the three-coupled LPD model (1.2) can be reconstructed by where H where v m0 is constant, ρ m (y) and ω m (τ ) are two scalar functions; at the same time, we have In order to obtain multi-soliton solutions for the three-coupled LPD model (1.2), we assume that the jump matrix T (x, ξ) = I is a 4 × 4 unit matrix in Eq. (2.28). In other words, the discrete scattering data r 12 = r 13 = r 14 = s 21 = s 31 = s 41 = 0; consequently, the unique solution to this special RH problem can be described as : where θ n = iξ n x − 8iεξ 4 n t + 2iξ 2 n t, P = (P mn ) N ×N is a N × N matrix with elements given by P mn = a * m a n e θ * m +θ n +(b * m b n +c * m c n + d * m d n )e −(θ * (3.18) Furthermore, the N-soliton solutions Eq. (3.17a)-(3.17c) can be rewritten as the following determinant ratio form: (3.19) where the N × N matrix P is defined by Eq. (3.18), and D 1 , D 2 , D 3 are (N + 1) × (N + 1) matrices given by (3.20)

Dynamic behaviors of soliton solutions
In what follows, we can examine the nonlinear dynamical behaviors of the breather and soliton solutions for the three-coupled LPD model (1.  1 , the single-soliton solution Eqs. (4.1a)-(4.1c) becomes By choosing different parameters, we can obtain some plots which are displayed to particularly describe the dynamic behaviors for single-soliton solution Eqs. (4.2a)-(4.2c) in the following section. Figure 1 illustrates that the single-soliton solution q 1 (x, t) is a line soliton, and its widths, amplitudes and velocities remain invariant during the propagation.
Finally, as another special example, choosing N = 3 in Eqs.
Perspective view of modulus of three-soliton q 1 (x, t). b Density plot of modulus of breather soliton q 1 (x, t) 3 , by choosing different parameters, we can obtain the interaction solutions Eqs. (4.5a)-(4.5c) shown in Figs. 4 and 5. In fact, Fig. 4 displays that soliton transmission and the three solitons interact with each other. Figure 5 illustrates that soliton reflection produces the breather and soliton solution.

Discussions and conclusions
Indeed, as a promotion, the integrable three-coupled LPD model (1.2) can be extended to the integrable generalized N-component LPD model as follows: where q α (x, t), (α = 1, 2, . . . , N ) represent amplitude of molecular excitation in the α-th spine, ε is a real parameter which stands for the strength of higher-order linear and nonlinear effects. Let q = (q 1 , q 2 , ..., q N ) T , Eq. (5.1) has the following vector form: Accordingly, we can also examine the N-soliton solutions to the integrable generalized N-component LPD model (5.1) by the same way in Sect. 3. However, we do not report them here since the procedure is mechanical.
In this work, we utilized the RH approach to study the three-coupled LPD model (1.2). By constructing a special matrix RH problem, we have obtained the multisoliton solution of the LPD model (1.2). In addition, some graphical analysis gave the dynamic characteristics of the soliton solution, including the interaction of single soliton, two solitons, three soliton and breather soliton solutions. It is hoped that our results can help enrich and explain other nonlinear integrable models.