Riemann–Hilbert Problem Associated with the Fourth-Order Dispersive Nonlinear Schrödinger Equation in Optics and Magnetic Mechanics

In this paper, by using Fokas method, we study the initial-boundary value problems (IBVPs) of the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation on the half-line, which can simulate the nonlinear transmission and interaction of ultrashort pulses in the high-speed optical fiber transmission system, and can also describe the nonlinear spin excitation phenomenon of one-dimensional Heisenberg ferromagnetic chain with eight poles and dipole interaction. By discussing the eigenfunctions of Lax pair of FODNLS equation and analyzing symmetry of the scattering matrix, we get a matrix Riemann–Hilbert (RH) problem from for the IBVPs of FODNLS equation. Moreover, we get the potential function solution u(x, t) of the FODNLS equation by solving this matrix RH problem. In addition, we also obtain that some spectral functions satisfy an important global relation.


Introduction
For a long time, finding the solutions of integrable equations has been a very important research topic in theory and application. The expression of exact solutions and some special solutions about integrable equations can provide important guidelines for the analysis of theirs various properties. However, there is no unified method to solve all integrable equations. With the in-depth study of integrable systems by scholars, a series of methods to solve the classic integrable equations have emerged, such as, inverse scattering transform (IST) method [1], Hirota method [2], Bäcklund transform [3], Darboux transform (DT) [4] and so on [5]. Among them, the IST method is the main analytical method for the exact solution of nonlinear integrable systems. However, due to the IST method is suitable for the limitations of the initial value conditions at infinity, it is almost only used to study the pure initial value problem of integrable equations. About many real-world phenomena and some studies in the fluctuation process, not only the initial value conditions need to be considered, but the boundary value conditions also need to be considered. Naturally, people need to replace the initial value problems with the initial-boundary value problems (IBVPs) in the research process.
In 1997, Fokas proposed a unified transformation method from the initial value problem to the IBVPs based on the IST method idea. This method can be used to investigate IBVPs of partial differential equation [6]. In the past 24 years, IBVPs of some classical integrable equations have be discussed via the Fokas method. For example, the modified Korteweg-de Vries (MKdV) equation [7], the nonlinear Schrödinger (NLS) equation [8], the Kaup-Newell equation [9], the stationary axisymmetric Einstein equations [10], the Ablowitz-Ladik system [11], the Kundu-Eckhaus equation [12], the Hirota equation [13,14] and other equations [15][16][17][18]. In 2012, Lenells extended the Fokas method to the integrable equation with higher-order matrix spectrum, he proposed a more general unified transformation approach to solve IBVPs of integrable models [19] and he used the unified transformation approach to analyze IBVPs of Degasperis-Procesi equation [20]. After that, more and more researchers began to study the IBVPs of integrable model with higher-order matrix spectrum [21][22][23][24][25][26][27][28]. In particular, the long-time asymptotic behavior of the solution of the integrable systems were discussed by the nonlinear steepest descent method proposed by Deift and Zhou [29], see for instance the cases of the three-component coupled mKdV system [30] and the three-component coupled NLS system [31].
In this paper, our work is related to the fourth-order dispersive NLS (FODNLS) equation [32,33] (1.1) iu t + 1 u xx + 2 u|u| 2 + 2 12 3 u xxxx + 4 |u| 2 u xx where u represents the amplitude of the slowly varying envelope of the wave, x and t are the normalized space and time variables, ε 2 is an infinitesimal dimensionless parameter representing the high-order linear and nonlinear strength, and j (j = 1, 2, … , 8) is the actual parameter. The Eq. (1.1) is mainly derived from fiber optics and magnetism. On the one hand, in optics, Eq. (1.1) can simulate the nonlinear propagation and interaction of ultrashort pulses in high-speed fiber-optic transmission systems [34]. On the other hand, in magnetic mechanics, Eq. (1.1) can be used to describe the nonlinear spin excitation of a one-dimensional Heisenberg ferromagnetic chain with octuple and dipole interactions [35]. In particular, when the parameter value is 1 = 3 = 1, 2 = 5 = 2, 4 = 8, 6 = 8 = 6, 7 = 4 , and = 2 12 , Eq. (1.1) becomes to which is an integrable model, and many properties have been widely studied, such as, the Lax pair, the infinite conservation laws [36], the breather solution, and the higher-order rogue wave solution based on the DT method [37][38][39], the multi-soliton solutions through Riemann-Hilbert (RH) approach [40], the dark and bright solitary waves and rogue wave solution through phase plane analysis method [41], the bilinear form and the N-soliton solution via the Hirota approach [42,43]. However, as far as we know, the FODNLS equation (1.2) on the half-line has not been studied. In this paper, we utilize the Fokas method to discuss the IBVPs of the FODNLS equa- The paper is organized as follows. In Sect. 2, the eigenfunction for spectral analysis of the Lax pair are introduced. In Sect. 3, the essential functions y( ), z( ), Y( ), Z( ) are further discussed. In Sect. 4, an important theorem is proposed. And the last section is devoted to conclusions.

The exact one-form
The Lax pair Eqs.

The Analytic and Bounded Eigenfunctions
We assume that u(x, t) ∈ S with (x, t) ∈ Ω = {(x, t) ∶ 0 < x < ∞, 0 < t < T} , and define eigenfunctions {G j (x, t, )} 3 1 of Eqs. (2.5a)-(2.5b) as follows where the integration path is (x j , t j ) → (x, t) , which is a directed smooth curves. It follows from the closed of the exact one-form that the integral of Eq. (1.2) is independent of the integration path. Therefore, one can choose three integral curve parallel to the axis shown in Fig. 1. We might as well take (x 1 , t 1 ) = (0, 0), (x 2 , t 2 ) = (0, T) , and (x 3 , t 3 ) = (∞, t) , then we have On the one hand, any point (x, t) on the integral curves { j } 3 1 satisfies the following inequalities x t t t On the other hand, it follows from the Eq. (2.8) that the first column of G j (x, t, ) contains e 2i (x− )−2i(8 4 −2 2 )(t− ) . Thus, for ∈ C , we can calculate the bounded ana- Then, for ∈ C , we can also calculate the bounded analytic region of the eigenfunctions . By calculation, we get the bounded analytic region of G j (x, t, ) as follows then, we get and it follows from the Eqs. (2.16)-(2.17) that ) . Then, the matrices P(x, 0, ) and R(0, t, ) have the following form with

The Other Properties of the Eigenfunctions
given in Eq. (2.8) possess the following analytical properties:

The Basic Riemann-Hilbert Problem
In order to facilitate calculation, we introduce the symbolic assumptions as follows , ∈ L 5 ∪ L 7 ,

Proposition 2.5 The matrix function
Proof We only verify the residue relationship Eq. (2.35a) as follows: ) , which means that the zeros { j } 2n 1 1 of y( ) are the poles of

The Global Relation
In this subsection, we give the spectral functions which are not independent but satisfy a nice global relation. In fact, at the boundary of the region ( , ) ∶ 0 < < ∞, 0 < < t , the integral of the one-form F(x, t, ) given by Eq. (2.7) vanishes. Let G(x, t, ) = G 3 (x, t, ) , we get

36)
Res G On the one hand, according to the definition of ( ) in Eq. where E( ) is expressed as ( ) − I.   which is the so-called global relation.
, (4) (t) meet the following asymptotic expansion and D (t) (t, ) meets the following RH problem.

Remark 3.6 Assume that
hence, D (t) (t, ) admits the RH problem as follows.

The Riemann-Hilbert Problem
In this part, we give two important results in theorem form. Thus, the matrix function D(x, t, ) exists and is unique. Furthermore Proof In fact, if we assume that y( ) and ( ) have no zeros, then the matrix-valued function D(x, t, ) satisfies a non-sigular RH problem, and then we prove that this non-sigular RH problem has a unique global solution using the symmetric property of the jump matrix Q(x, t, ) and the scattering matrix. If y( ) and ( ) have only a finite number of zeros, we can map this situation to a situation without zeros, which can transform into a system of algebraic equations, this algebraic equations can always be solved uniquely. Similar Ref. [12], we give the following vanishing theorem without proof. ◻   u(x, 0) = u 0 (x), u(0, t) = v 0 (t), u x (0, t) = v 1 (t), u xx (0, t) = v 2 (t), u xxx (0, t) = v 3 (t).

Conclusions
In this paper, we use the Fokas method to construct the RH problem of the FODNLS equation on the half-line. When the parameter = 0 , it can be reduced to the RH problem of the classical nonlinear Schrödinger equation on the half-line. We introduced important functions for spectral analysis of the Lax pair, established the basic RH problem, and gave the global relationship between spectral functions. Furthermore, we can analyze the integrable FODNLS equation (1.2) on a finite interval, and also discuss the asymptotic behavior for the solution of the integrable FODNLS equation (1.2). These two questions will be studied in our future investigation.