Riemann-Hilbert problem associated with the fourth-order dispersive nonlinear Schrödinger equation in optics and magnetic mechanics

Beibei Hu Ling Zhang Qinghong Li Ning Zhang School of Mathematics and Finance, Chuzhou University, Anhui, 239000, China Department of Basical Courses, Shandong University of Science and Technology, Taian 271019, China

Abstract

In this paper, we utilize Fokas method to investigate 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 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 the analysis and symmetry of the scattering matrix, the IBVPs of FODNLS equation is expressed as a matrix Riemann-Hilbert (RH) problem form. Then one can get the potential function solution $u(x,t)$ of the FODNLS equation by solving this matrix RH problem. In addition, we also obtained that some spectral functions admits a key global relationship.

keywords:

Riemann-Hilbert problem, fourth-order dispersive nonlinear Schrödinger equation, initial-boundary value problems, Fokas method.

AMS Subject Classification: 35G31, 35Q15, 35Q55, 37K15

1 Introduction

For a long time, finding a method to solve integrable equations has been a very important research topic in theory and application. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its 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 development equation have emerged. For example, inverse scattering method[1], Hirota method[2], Bäcklund transform[3], Darboux transform(DT)[4] and so on. 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, and it is almost only used to study the pure initial value problem of integrable equations, many real-world phenomena and some studies in the fluctuation process not only need to consider the initial value conditions, 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, which is called the Fokas method. This method can be investigated IBVPs of partial differential equation[5], and in the past 22 years, IBVPs of some classical integrable equations to be discussed via the Fokas method. For example, the modified Korteweg-de Vries(MKdV) equation[6], the nonlinear Schrödinger(NLS) equation[7], the Kaup-Newell equation[8], the stationary axisymmetric Einstein equations[9], the Ablowitz-Ladik system[10], the Kundu-Eckhaus equation[11], the Hirota equation[12, 13]. 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 solving IBVPs of integrable model[14] and using the unified transformation approach to analyzed IBVPs of Degasperis-Procesi equation[15]. After that, more and more individuals began to study the IBVPs of integrable model with higher-order matrix spectral[16, 17, 18, 19, 20, 21, 22, 23, 24]. The authors have also done a slice of works on the application of the Fokas method to an integrable equation with higher-order matrix Lax pairs[25, 26, 27].

In this paper, our work is related to the fourth-order dispersive nonlinear Schrödinger(FODNLS) equation[28, 29] expressed as:

\displaystyle iu_{t}+\alpha_{1}u_{xx}+\alpha_{2}u|u|^{2}+\frac{\varepsilon^{2}}{12}(\alpha_{3}u_{xxxx}+\alpha_{4}|u|^{2}u_{xx}+\alpha_{5}u^{2}\overline{u}_{xx}+\alpha_{6}u_{x}^{2}\overline{u}+\alpha_{7}u|u_{x}|^{2}+\alpha_{8}|u|^{4}u)=0,

(1.1)

where $u$ represents the amplitude of the slowly varying envelope of the wave, $x$ and $t$ are the normalized space and time variables, $\varepsilon^{2}$ is a dimensionless small parameter representing the high-order linear and nonlinear strength, and $\alpha_{j}(j=1,2,\ldots,8)$ is the real 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[30]. 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[31]. In particular, when the parameter value is $\alpha_{1}=\alpha_{3}=1,\alpha_{2}=\alpha_{5}=2,\alpha_{4}=8,\alpha_{6}=\alpha_{8}=6,\alpha_{7}=4$ , and let $\gamma=\frac{\varepsilon^{2}}{12}$ the Eq.(1.1) becomes to

\displaystyle iu_{t}+u_{xx}+2u|u|^{2}+\gamma(u_{xxxx}+8|u|^{2}u_{xx}+2u^{2}\overline{u}_{xx}+6u_{x}^{2}\overline{u}+4u|u_{x}|^{2}+6|u|^{4}u)=0,

(1.2)

which is an integrable model, and many properties have been widely studied, such as, the Lax pair, the infinite conservation laws[32], the breather solution, and the higher-order rogue wave solution based on the DT method[33, 34, 35], the multi-soliton solutions by using Riemann-Hilbert(RH) approach[36], the dark and bright solitary waves and rogue wave solution by using phase plane analysis method[37], the bilinear form and the N-soliton solution via the Hirota approach[38, 39]. However, as far as we know, the FODNLS (1.2) on the hale-line has not been studied, and in the following work, we utilize Fokas method to discuss the IBVPs of the FODNLS equation(1.2) on the half-line domain $\Omega=\{(x,t):0<z<\infty,0<t<T\}$ .

The paper is organized as follows. In section 2, one can introducing eigenfunction to spectral analysis of the Lax pair. In section 3, a slice of key functions $y(\zeta),z(\zeta),Y(\zeta),Z(\zeta)$ are further discussed. In section 4, an important theorem is proposed. And the last section is devoted to conclusions.

2 The spectral analysis

Base on Ablowitz-Kaup-Newell-Segur scheme, the Lax pair of Eq.(1.2) is expressed as[32, 33, 34, 35, 36]


	$\displaystyle\Psi_{x}=(-i\zeta\Lambda+P)\Psi,$		(2.1a)
	$\displaystyle\Psi_{t}=[(8i\gamma\zeta^{4}-2i\zeta^{2})\Lambda-8\gamma\zeta^{3}P-4i\gamma\zeta^{2}A_{1}-2\zeta A_{2}+iA_{3}]\Psi,$		(2.1b)

where $\zeta$ is a complex spectral parameter, $\Psi=(\Psi_{1},\Psi_{2})^{T}$ is the vector eigenfunction, the $2\times 2$ matrices $\Lambda=diag\{1,-1\}$ , and $P,A,B$ and $C$ are defined by

\displaystyle\begin{array}[]{l}P=\left(\begin{array}[]{cc}0&u\\ -\overline{u}&0\end{array}\right),A_{1}=\left(\begin{array}[]{cc}|u|^{2}&u_{x}\\ \overline{u}_{x}&-|u|^{2}\end{array}\right),A_{2}=\left(\begin{array}[]{cc}\gamma(u\overline{u}_{x}-\overline{u}u_{x})&-\gamma u_{xx}-(2\gamma|u|^{2}+1)u\\ -\gamma\overline{u}_{xx}-(2\gamma|u|^{2}+1)\overline{u}&-\gamma(u\overline{u}_{x}-\overline{u}u_{x})\end{array}\right),\\ A_{3}=\left(\begin{array}[]{cc}\gamma(3|u|^{4}-|u_{x}|^{2}+\overline{u}u_{xx}+u\overline{u}_{xx})+|u|^{2}&\gamma u_{xxx}+(6\gamma|u|^{2}+1)u_{x}\\ \gamma\overline{u}_{xxx}+(6\gamma|u|^{2}+1)\overline{u}_{x}&-\gamma(3|u|^{4}-|u_{x}|^{2}+\overline{u}u_{xx}+u\overline{u}_{xx})+|u|^{2}\end{array}\right).\end{array}

(2.12)

2.1 The exact one-form

The Lax pair equations (2.1a)-(2.1b) are rewritten as follows


	$\displaystyle\Psi_{x}+i\zeta\Lambda\Psi=P(x,t,\zeta)\Psi,$		(2.13a)
	$\displaystyle\Psi_{t}-(8i\gamma\zeta^{4}-2i\zeta^{2})\Lambda\Psi=R(x,t,\zeta)\Psi,$		(2.13b)

where

	$\displaystyle R(x,t,\zeta)=-8\gamma\zeta^{3}P-4i\gamma\zeta^{2}A_{1}-2\zeta A_{2}+iA_{3}$
	$\displaystyle\quad=-8\gamma\zeta^{3}P+4i\gamma\zeta^{2}(P^{2}+P_{x})\Lambda+2\gamma\zeta(PP_{x}-P_{x}P-P_{xx}+2P^{3})\Lambda-2\zeta P\Lambda$
	$\displaystyle\qquad+i\gamma(3P^{4}+P_{x}^{2}-P_{xx}P-PP_{xx}-P_{xxx}+6P^{2}P_{x}-P_{x})\Lambda-iP^{2}.$

For the convenience of later calculation, we record $\theta=(8\gamma\zeta^{4}-2\zeta^{2})$ and introduce the following function transformation

\displaystyle\Psi(x,t,\zeta)=G(x,t,\zeta)e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\Lambda},0<x<\infty,0<t<T.

(2.14)

Then, we get


	$\displaystyle G_{x}+i\zeta[\Lambda,G]=PG,$		(2.15a)
	$\displaystyle G_{t}-i(8\gamma\zeta^{4}-2\zeta^{2})[\Lambda,G]=RG,$		(2.15b)

which can be expressed as the following full differential

\displaystyle d(e^{i[\zeta x-(8\gamma\zeta^{4}-2\zeta^{2})t]\hat{\Lambda}}G(x,t,\zeta))=F(x,t,\zeta),

(2.16)

where exact one-form $F(x,t,\zeta)$ is

\displaystyle F(x,t,\zeta)=e^{i[\zeta x-(8\gamma\zeta^{4}-2\zeta^{2})t]\hat{\Lambda}}(P(x,t,\zeta)dx+R(x,t,\zeta)dt)G(x,t,\zeta),

(2.17)

and $\hat{\Lambda}$ represents a matrix operator acting on a second order matrix $\Lambda$ , i.e. $\hat{\Lambda}P=[\Lambda,P]$ and $e^{\hat{\Lambda}}P=e^{\Lambda}Pe^{-\Lambda}$ .

2.2 The analytic and bounded eigenfunctions $G_{j}(x,t,\zeta)^{\prime}s$

We assume that $u(x,t)\in\mathcal{S}$ with $(x,t)\in\Omega=\{(x,t):0<z<\infty,0<t<T\}$ , and use the integral equation containing the exact one-form to define eigenfunctions $\{G_{j}(x,t,\zeta)\}_{1}^{3}$ of Eq.(2.15a)-(2.15b) as follows

\displaystyle G_{j}(x,t,\zeta)=\mathrm{I}+\int_{(x_{j},t_{j})}^{(x,t)}e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\hat{\Lambda}}F(\xi,\tau,\zeta),

(2.18)

where the integration path is $(x_{j},t_{j})\rightarrow(x,t)$ which is a directed smooth curve. 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 are all parallel to the axis shown in Figure 1.

Refer to caption — Figure 1: The smooth curve $\gamma_{1},\gamma_{2},\gamma_{3}$ in the $(x,t)$ -plan

We might as well take $(x_{1},t_{1})=(0,0),(x_{2},t_{2})=(0,T)$ , and $(x_{3},t_{3})=(\infty,t)$ , then we have


	$\displaystyle G_{1}(x,t,\zeta)=\mathrm{I}+\int_{0}^{x}e^{-i\zeta(x-\xi)\hat{\Lambda}}(PG_{1})(\xi,t,\zeta)d\xi$
	$\displaystyle\qquad\qquad\qquad+e^{-i\zeta x\hat{\Lambda}}\int_{0}^{t}e^{i(8\gamma\zeta^{4}-2\zeta^{2})(t-\tau)\hat{\Lambda}}(R_{1}G_{1})(0,\tau,\Lambda)d\tau,$		(2.19a)
	$\displaystyle G_{2}(x,t,\zeta)=\mathrm{I}+\int_{0}^{x}e^{-i\zeta(x-\xi)\hat{\Lambda}}(PG_{2})(\xi,t,\zeta)d\xi$
	$\displaystyle\qquad\qquad\qquad-e^{-i\zeta x\hat{\Lambda}}\int_{t}^{T}e^{i(8\gamma\zeta^{4}-2\zeta^{2})(t-\tau)\hat{\Lambda}}(R_{1}G_{2})(0,\tau,\Lambda)d\tau,$		(2.19b)
	$\displaystyle G_{3}(x,t,\zeta)=\mathrm{I}-\int_{x}^{\infty}e^{-i\zeta(x-\xi)\hat{\Lambda}}(PG_{3})(\xi,t,\zeta)d\xi.$		(2.19c)

On the one hand, any point $(x,t)$ on the integral curve $\{\gamma_{j}\}_{1}^{3}$ satisfies the following inequalities


	$\displaystyle\gamma_{1}:x-\xi\geq 0,t-\tau\geq 0,$		(2.20a)
	$\displaystyle\gamma_{2}:x-\xi\geq 0,t-\tau\leq 0,$		(2.20b)
	$\displaystyle\gamma_{3}:x-\xi\leq 0.$		(2.20c)

On the other hand, it follows from the Eq.(2.18) that the first column of $G_{j}(x,t,\zeta)$ contains $e^{2i\zeta(x-\xi)-2i(8\gamma\zeta^{4}-2\zeta^{2})(t-\tau)}$ . Thus, for $\zeta\in\mathrm{C}$ , we can calculate the bounded analytic region of $[G_{j}(x,t,\zeta)]_{1}$ , that is $\zeta$ must satisfies


	$\displaystyle{[G_{1}]}_{1}(x,t,\zeta):\{\mathrm{Im}\zeta\geq 0\}\cap\{\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\geq 0\},$		(2.21a)
	$\displaystyle{[G_{2}]}_{1}(x,t,\zeta):\{\mathrm{Im}\zeta\geq 0\}\cap\{\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\leq 0\},$		(2.21b)
	$\displaystyle{[G_{3}]}_{1}(x,t,\zeta):\{\mathrm{Im}\zeta\leq 0\}.$		(2.21c)

Similarly, it follows from the Eq.(2.18) that the second column of $G_{j}(x,t,\zeta)$ contains $e^{-2i\zeta(x-\xi)+2i(8\gamma\zeta^{4}-2\zeta^{2})(t-\tau)}$ . Then, for $\zeta\in\mathrm{C}$ , we can also calculate the bounded analytic region of the eigenfunctions $[G_{j}(x,t,\zeta)]_{2}$ , that means $\zeta$ must satisfies


	$\displaystyle{[G_{1}]}_{2}(x,t,\zeta):\{\mathrm{Im}\zeta\leq 0\}\cap\{\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\leq 0\},$		(2.22a)
	$\displaystyle{[G_{2}]}_{2}(x,t,\zeta):\{\mathrm{Im}\zeta\leq 0\}\cap\{\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\geq 0\},$		(2.22b)
	$\displaystyle{[G_{3}]}_{2}(x,t,\zeta):\{\mathrm{Im}\zeta\geq 0\}.$		(2.22c)

where the $[G_{j}]_{k}(x,t,\zeta)$ denotes the $k$ -columns of $G_{j}(x,t,\zeta)$ . After calculation, we get the bounded analytic region of $G_{j}(x,t,\zeta)$ as follows


	$\displaystyle G_{1}(x,t,\zeta)=([G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta),[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)),$		(2.23a)
	$\displaystyle G_{2}(x,t,\zeta)=([G_{2}]_{1}^{L_{2}\cup L_{4}}(x,t,\zeta),[G_{2}]_{2}^{L_{5}\cup L_{7}}(x,t,\zeta)),$		(2.23b)
	$\displaystyle G_{3}(x,t,\zeta)=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta)),$		(2.23c)

where $G_{j}^{L_{i}}(x,t,\zeta)$ represents the bounded analytic region of $\{G_{j}(x,t,\zeta)\}_{1}^{3}$ is $\zeta\in L_{i},i=1,2,\ldots,8$ , and $L_{i},i=1,2,\ldots,8$ are shown in Figure 2.

To establish the RH problem of the FODNLS equation (1.2), we must also define two special functions $\psi(\zeta)$ and $\phi(\zeta)$ with the eigenfunction $\{G_{j}(x,t,\zeta)\}_{1}^{3}$ as follows


	$\displaystyle G_{3}(x,t,\zeta)=G_{1}(x,t,\zeta)e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\hat{\Lambda}}\psi(\zeta),$		(2.24a)
	$\displaystyle G_{2}(x,t,\zeta)=G_{1}(x,t,\zeta)e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\hat{\Lambda}}\phi(\zeta).$		(2.24b)

Set $(x,t)=(0,0)$ in Eq.(2.24a), and let $(x,t)=(0,T)$ in Eq.(2.24b), we obtain the following relationship

\displaystyle\psi(\zeta)=G_{3}(0,0,\zeta),\,\,\phi(\zeta)=G_{2}(0,0,\zeta)=[e^{i(8\gamma\zeta^{4}-2\zeta^{2})T\hat{\Lambda}}G_{1}(0,T,\zeta)]^{-1},

(2.25)

then, we get

\displaystyle G_{3}(x,t,\zeta)=G_{1}(x,t,\zeta)e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\hat{\Lambda}}G_{3}(0,0,\zeta),

(2.26)

and

\displaystyle G_{2}(x,t,\zeta)=G_{1}(x,t,\zeta)e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\hat{\Lambda}}[e^{-i(8\gamma\zeta^{4}-2\zeta^{2})T\hat{\Lambda}}G_{1}(0,T,\zeta)]^{-1},

(2.27)

it follows from the Eqs.(2.26)-(2.27) that

\displaystyle G_{2}(x,t,\zeta)=G_{3}(x,t,\zeta)e^{i[(8\gamma\zeta^{4}-2\zeta^{2})t-\zeta x]\hat{\Lambda}}(\psi(\zeta))^{-1}\phi(\zeta).

(2.28)

Particularly, in the eigenfunction $G_{j}(x,t,\zeta),j=1,2$ , when $x=0$ , we have


	$\displaystyle G_{1}(0,t,\zeta)=([G_{1}]_{1}^{L_{1}\cup L_{3}\cup L_{5}\cup L_{7}}(0,t,\zeta),[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta))$
	$\displaystyle\qquad\qquad\;=\mathrm{I}+\int_{0}^{t}e^{i(8\gamma\zeta^{4}-2\zeta^{2})(t-\tau)\hat{\Lambda}}(RG_{1})(0,\tau,\zeta)d\tau,$		(2.29a)
	$\displaystyle G_{2}(0,t,\zeta)=([G_{2}]_{1}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta),[G_{2}]_{2}^{L_{1}\cup L_{3}\cup L_{5}\cup L_{7}}(0,t,\zeta))$
	$\displaystyle\qquad\qquad\;=\mathrm{I}-\int_{t}^{T}e^{i(8\gamma\zeta^{4}-2\zeta^{2})(t-\tau)\hat{\Lambda}}(RG_{2})(0,\tau,\zeta)d\tau,$		(2.29b)

and in the eigenfunction $G_{1}(x,t,\zeta),G_{3}(x,t,\zeta)$ , when $t=0$ , we have


	$\displaystyle G_{1}(x,0;\zeta)=([G_{1}]_{1}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,0,\zeta),[G_{1}]_{2}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,0,\zeta))$
	$\displaystyle\qquad\qquad\;=\mathrm{I}+\int_{0}^{x}e^{-i\zeta(x-\xi)\hat{\Lambda}}(PG_{1})(\xi,0,\zeta)d\xi,$		(2.30a)
	$\displaystyle G_{3}(x,0;\zeta)=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(z,0,\zeta),[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(z,0,\zeta))$
	$\displaystyle\qquad\qquad\;=\mathrm{I}-\int_{x}^{\infty}e^{-i\zeta(x-\xi)\hat{\Lambda}}(PG_{3})(\xi,0,\zeta)d\xi,$		(2.30b)

Assuming that $u_{0}(x)=u(x,t=0)$ is an initial data of the functions $u(x,t)$ , and $v_{0}(t)=u(x=0,t)$ , $v_{1}(t)=u_{x}(x=0,t)$ , $v_{2}(t)=u_{xx}(x=0,t)$ , $v_{3}(t)=u_{xxx}(x=0,t)$ are boundary datas of the functions $u_{x}(x,t)$ , $u_{xx}(x,t)$ , $u_{xxx}(x,t)$ , at this time, the matrix $P(x,0,\zeta)$ and $R(0,t,\zeta)$ have the following matrix forms, respectively.

\displaystyle P(x,0,\zeta)=\left(\begin{array}[]{cc}0&u_{0}\\ -\overline{u}_{0}&0\end{array}\right),\,\,R(0,t,\zeta)=\left(\begin{array}[]{cc}R_{11}(0,t,\zeta)&R_{12}(0,t,\zeta)\\ R_{21}(0,t,\zeta)&-R_{11}(0,t,\zeta)\end{array}\right),

(2.35)

with

	$\displaystyle R_{11}(0,t,\zeta)=-4i\gamma\zeta^{2}v^{2}_{0}-2\gamma\zeta(v_{0}\bar{v}_{1}-v_{1}\bar{v}_{0})+i\gamma(3\|v_{0}\|^{4}-\|v_{1}\|^{2}+\overline{v_{0}}v_{2}+v_{0}\overline{v}_{2})+i\|v_{0}\|^{2}$
	$\displaystyle R_{12}(0,t,\zeta)=-8\gamma\zeta^{3}v_{0}-4i\gamma\zeta^{2}v_{1}+2\gamma\zeta v_{2}+2\zeta(2\gamma\|v_{0}\|^{2}+1)v_{0}+i\gamma v_{3}+i(6\gamma\|v_{0}\|^{2}+1)v_{1}$
	$\displaystyle R_{21}(0,t,\zeta)=-8\gamma\zeta^{3}\bar{v}_{0}-4i\gamma\zeta^{2}\bar{v}_{1}+2\gamma\zeta\bar{v}_{2}+2\zeta(2\gamma\|v_{0}\|^{2}+1)\bar{v}_{0}+i\gamma\bar{v}_{3}+i(6\gamma\|v_{0}\|^{2}+1)\bar{v}_{1}$

2.3 The other properties of the eigenfunctions

Proposition 2.1

The matrix-value functions $G_{j}(x,t,\zeta)=([G_{j}]_{1}(x,t,\zeta),[G_{j}]_{2}(x,t,\zeta))(j=1,2,3)$ are given in Eq.(2.18) enjoy analytical properties are:

1.

$\mathrm{det}G_{j}(x,t,\zeta)=1$ ;
2.

The $[G_{1}]_{1}(x,t,\zeta)$ is an analytic function for $\zeta\in L_{1}\cup L_{3}$ , and the $[G_{1}]_{2}(x,t,\zeta)$ is also an analytic function for $\zeta\in L_{6}\cup L_{8}$ ;
3.

The $[G_{2}]_{1}(x,t,\zeta)$ is an analytic function for $\zeta\in L_{2}\cup L_{4}$ , and the $[G_{2}]_{2}(x,t,\zeta)$ is also an analytic function for $\zeta\in L_{5}\cup L_{7}$ ;
4.

The $[G_{3}]_{1}(x,t,\zeta)$ is an analytic function for $\zeta\in L_{5}\cup L_{6}\cup L_{7}\cup L_{8}$ , and the $[G_{3}]_{2}(x,t,\zeta)$ is also an analytic function for $\zeta\in L_{1}\cup L_{2}\cup L_{3}\cup L_{4}$ ;
5.

The $[G_{j}]_{1}(x,t,\zeta)\rightarrow(1,0)^{T}$ and $[G_{j}]_{2}(x,t,\zeta)\rightarrow(0,1)^{T}$ , as $\zeta\rightarrow\infty$ .

Proposition 2.2

Indeed, $\psi(\zeta),\phi(\zeta)$ are defined in Eqs.(2.24a)-(2.24b) or Eq.(2.25) is expressed as


	$\displaystyle\psi(\zeta)=\mathrm{I}-\int_{0}^{\infty}e^{i\zeta\xi\hat{\Lambda}}(PG_{3})(\xi,0,\zeta)d\xi,$		(2.36a)
	$\displaystyle\phi^{-1}(\zeta)=\mathrm{I}+\int_{0}^{T}e^{i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}(RG_{1})(0,\tau,\zeta)d\tau.$		(2.36b)

It follows from the symmetry properties of $P(x,t,\zeta)$ and $R(x,t,\zeta)$ that

\displaystyle(G_{j}(x,t,\zeta))_{11}=\overline{(G_{j}(x,t,\overline{\zeta}))_{11}},\,\,(G_{j}(x,t,\zeta))_{21}=-\overline{(G_{j}(x,t,\overline{\zeta}))_{12}},

then

\psi_{11}(\zeta)=\overline{\psi_{11}(\overline{\zeta})},\,\,\psi_{21}(\zeta)=-\overline{\psi_{12}(\overline{\zeta})},

\phi_{11}(\zeta)=\overline{\phi_{11(\overline{\zeta})}},\,\,\phi_{21}(\zeta)=-\overline{\phi_{12}(\overline{\zeta})},

and assume that the $\psi(\zeta)$ and $\phi(\zeta)$ admits the matrix form as follows

\displaystyle\psi(\zeta)=\left(\begin{array}[]{cc}\overline{y(\bar{\zeta})}&z(\zeta)\\ -\overline{z(\bar{\zeta})}&y(\zeta)\end{array}\right),\phi(\zeta)=\left(\begin{array}[]{cc}\overline{Y(\bar{\zeta})}&Z(\zeta)\\ -\overline{Z(\bar{\zeta})}&Y(\zeta)\\ \end{array}\right).

(2.41)

In terms of the Eq.(2.25) and Eqs.(2.36a)-(2.36b), we known that the following properties are ture,

\left(\begin{array}[]{c}z(\zeta)\\ y(\zeta)\\ \end{array}\right)=[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(0,0,\zeta)=\left(\begin{array}[]{c}(G_{3})_{12}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(0,0,\zeta)\\ (G_{3})_{22}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(0,0,\zeta)\\ \end{array}\right),

where the vector function $[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,0,\zeta)$ satisfy the ordinary differential equation as follows

	$\displaystyle\partial_{x}[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,0,\zeta)+2i\zeta\left(\begin{array}[]{cc}1&0\\ 0&0\end{array}\right)[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,0,\zeta)$		(2.44)
	$\displaystyle=P(x,0,\zeta)[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,0,\zeta),\,0<x<\infty,$		(2.45)

where $P(x,0,\zeta)$ is given in Eq(2.35) and

\lim_{x\rightarrow\infty}[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,0,\zeta)=(0,1)^{T}.

\left(\begin{array}[]{c}-e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})T}Z(\zeta)\\ \overline{Y(\bar{\zeta})}\\ \end{array}\right)=[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,T,\zeta)=\left(\begin{array}[]{c}(G_{1})_{12}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta)\\ (G_{1})_{22}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta)\\ \end{array}\right),

where the vector function $[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta)$ satisfy the ordinary differential equation as follows

	$\displaystyle\partial_{t}[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta)-2i(8\gamma\zeta^{4}-2\zeta^{2})\left(\begin{array}[]{cc}1&0\\ 0&0\end{array}\right)[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta)$		(2.48)
	$\displaystyle=R(0,t,\zeta)[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,t,\zeta),\,0<t<T,$		(2.49)

where $R(0,t,\zeta)$ is given in Eq(2.35) and

[G_{1}]_{2}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(0,0,\zeta)=(0,1)^{T}.

2.

$y(-\zeta)=y(\zeta),\,\,z(-\zeta)=-z(\zeta),$

$Y(-\zeta)=Y(\zeta),\,\,Z(-\zeta)=-Z(\zeta).$

For\,\,\zeta\in\mathrm{R},\,\,\mathrm{det}\psi(\zeta)=|y(\zeta)|^{2}+|z(\zeta)|^{2}=1.

For\,\,\zeta\in\mathrm{C},\,\,\mathrm{det}\phi(\zeta)=Y(\zeta)\overline{Y(\bar{\zeta})}+Z(\zeta)\overline{Z(\bar{\zeta})}=1,\,\,(\zeta\in\Gamma_{m},\,if\,T=\infty).

where curve $\Gamma_{m},\,m=1,2,3,4$ are given in Eq.(2.58).

y(\zeta)=1+O(\frac{1}{\zeta}),\,\,z(\zeta)=O(\frac{1}{\zeta}),as\,\zeta\rightarrow\infty,

Y(\zeta)=1+O(\frac{1}{\zeta})+O(\frac{e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})T}}{\zeta}),\,\,Z(\zeta)=O(\frac{1}{\zeta})+O(\frac{e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})T}}{\zeta}),as\,\zeta\rightarrow\infty.

2.4 The basic Riemann-Hilbert problem

In order to facilitate calculation and formula representation, we introduce the symbolic assumptions as follows


	$\displaystyle\omega(x,t,\zeta)=\zeta x-(8\gamma\zeta^{4}-2\zeta^{2})t,$		(2.50a)
	$\displaystyle\rho(\zeta)=y(\zeta)\overline{Y(\bar{\zeta})}+z(\zeta)\overline{Z(\bar{\zeta})},$		(2.50b)
	$\displaystyle\kappa(\zeta)=\overline{y(\bar{\zeta})}\overline{Z(\bar{\zeta})}-\overline{z(\bar{\zeta})}\overline{Y(\bar{\zeta})},$		(2.50c)
	$\displaystyle\delta(\zeta)=\frac{z(\zeta)}{\overline{y(\bar{\zeta})}},\Delta(\zeta)=-\frac{\overline{Z(\bar{\zeta})}}{y(\zeta)\rho(\zeta)},$		(2.50d)

then, we have

	$\displaystyle\overline{Z(\bar{\zeta})}=y(\zeta)\kappa(\zeta)+\overline{z(\bar{\zeta})}\rho(\zeta),$
	$\displaystyle\rho(\zeta)\overline{\rho(\bar{\zeta})}-\kappa(\zeta)\overline{\kappa(\bar{\zeta})}=1,$
	$\displaystyle\rho(\zeta)=1+O(\frac{1}{\zeta}),\kappa(\zeta)=O(\frac{1}{\zeta})\,\,as\,\,\zeta\rightarrow\infty,$
	$\displaystyle\rho(-\zeta)=\rho(\zeta),\kappa(-\zeta)=-\kappa(\zeta),$

and the matrix function $D(x,t,\zeta)$ is defined by


	$\displaystyle D_{+}(x,t,\zeta)=(\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)}{y(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta)),\zeta\in L_{1}\cup L_{3},$		(2.51a)
	$\displaystyle D_{-}(x,t,\zeta)=(\frac{[G_{2}]_{1}^{L_{2}\cup L_{4}}(x,t,\zeta)}{\rho(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta)),\zeta\in L_{2}\cup L_{4},$		(2.51b)
	$\displaystyle D_{+}(x,t,\zeta)=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),\frac{[G_{2}]_{2}^{L_{5}\cup L_{7}}(x,t,\zeta)}{\overline{\rho(\bar{\zeta})}}),\zeta\in L_{5}\cup L_{7},$		(2.51c)
	$\displaystyle D_{-}(x,t,\zeta)=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),\frac{[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)}{\overline{y(\bar{\zeta})}}),\zeta\in L_{6}\cup L_{8}.$		(2.51d)

Obviously, the above definitions indicates that

\mathrm{det}D(x,t,\zeta)=1,\,\,D(x,t,\zeta)\rightarrow\mathrm{I},as\,\,\zeta\rightarrow\infty.

(2.52)

Theorem 2.3

Let that the matrix function $D(x,t,\zeta)$ is defined by Eqs.(2.51a)-(2.51d) and the potential function $u(z,t)\in\mathcal{S}$ , then, the matrix function $D(x,t,\zeta)$ admits the jump relation on the curve $\Gamma_{m},m=1,\ldots,4$ as follows

\displaystyle D_{+}(x,t,\zeta)=D_{-}(x,t,\zeta)Q(x,t,\zeta),\zeta\in\Gamma_{m},m=1,\ldots,4,

(2.53)

where

\displaystyle Q(x,t,\zeta)=\left\{\begin{array}[]{l}Q_{1}(x,t,\zeta),\qquad\qquad\qquad\zeta\in\Gamma_{1}\doteq\{\bar{L}_{1}\cup\bar{L}_{3}\}\cap\{\bar{L}_{2}\cup\bar{L}_{4}\}\\ Q_{2}(x,t,\zeta)=Q_{3}Q_{4}^{-1}Q_{1},\quad\zeta\in\Gamma_{2}\doteq\{\bar{L}_{2}\cup\bar{L}_{4}\}\cap\{\bar{L}_{5}\cup\bar{L}_{7}\}\\ Q_{3}(x,t,\zeta),\qquad\qquad\qquad\zeta\in\Gamma_{3}\doteq\{\bar{L}_{5}\cup\bar{L}_{7}\}\cap\{\bar{L}_{6}\cup\bar{L}_{8}\}\\ Q_{4}(x,t,\zeta),\qquad\qquad\qquad\zeta\in\Gamma_{4}\doteq\{\bar{L}_{6}\cup\bar{L}_{8}\}\cap\{\bar{L}_{1}\cup\bar{L}_{3}\}\end{array}\right.

(2.58)

and

\displaystyle\begin{array}[]{l}Q_{1}(x,t,\zeta)=\left(\begin{array}[]{cc}1&0\\ \Delta(\zeta)e^{2i\omega(\zeta)}&1\end{array}\right),\\ Q_{2}(x,t,\zeta)=\left(\begin{array}[]{cc}1-(\delta(\zeta)+\overline{\Delta(\bar{\zeta})})(\Delta(\zeta)+\overline{\delta(\bar{\zeta})})&(\delta(\zeta)+\overline{\Delta(\bar{\zeta})})e^{-2i\omega(\zeta)}\\ (\Delta(\zeta)+\overline{\delta(\bar{\zeta})})e^{2i\omega(\zeta)}&1\end{array}\right),\\ Q_{3}(x,t,\zeta)=\left(\begin{array}[]{cc}1&\overline{\Delta(\bar{\zeta})}e^{-2i\omega(\zeta)}\\ 0&1\end{array}\right),\\ Q_{4}(x,t,\zeta)=\left(\begin{array}[]{cc}1&-\delta(\zeta)e^{-2i\omega(\zeta)}\\ -\overline{\delta(\bar{\zeta})}e^{2i\omega(\zeta)}&1+|\delta(\zeta)|^{2}\end{array}\right).\end{array}

(2.71)

Proof In terms of the Eqs.(2.24a)-(2.24b) and Eq.(2.41), we have


	$\displaystyle\overline{y(\bar{\zeta})}[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)-\overline{z(\bar{\zeta})}e^{2i\omega(\zeta)}[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)=[G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),$		(2.72a)
	$\displaystyle z(\zeta)e^{-2i\omega(\zeta)}[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)+y(\zeta)[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)=[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta),$		(2.72b)

and


	$\displaystyle\overline{Y(\bar{\zeta})}[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)-\overline{Z(\bar{\zeta})}e^{2i\omega(\zeta)}[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)=[G_{2}]_{1}^{L_{2}\cup L_{4}}(x,t,\zeta),$		(2.73a)
	$\displaystyle Z(\zeta)e^{-2i\omega(\zeta)}[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)+Y(\zeta)[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)=[G_{2}]_{2}^{L_{5}\cup L_{7}}(x,t,\zeta),$		(2.73b)

according to the Eqs.(2.72a)-(2.73b) and Eqs.(2.50a)-(2.50d) yields


	$\displaystyle\rho(\zeta)[G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta)-\kappa(\zeta)e^{2i\omega(\zeta)}[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta)=[G_{2}]_{1}^{L_{2}\cup L_{4}}(x,t,\zeta),$		(2.74a)
	$\displaystyle\overline{\kappa(\bar{\zeta})}e^{-2i\omega(\zeta)}[G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta)+\overline{\rho(\bar{\zeta})}[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta)=[G_{2}]_{2}^{L_{5}\cup L_{7}}(x,t,\zeta).$		(2.74b)

By the Eqs.(2.51a)-(2.51d) and Eq.(2.53), one have


	$\displaystyle(\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)}{y(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta))=(\frac{[G_{2}]_{1}^{L_{2}\cup L_{4}}(x,t,\zeta)}{\rho(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta))Q_{1}(x,t,\zeta),$		(2.75a)
	$\displaystyle([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),\frac{[G_{2}]_{2}^{L_{5}\cup L_{7}}(x,t,\zeta)}{\overline{\rho(\bar{\zeta})}})=(\frac{[G_{2}]_{1}^{L_{2}\cup L_{4}}(x,t,\zeta)}{\rho(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta))Q_{2}(x,t,\zeta),$		(2.75b)
	$\displaystyle([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),\frac{[G_{2}]_{2}^{L_{5}\cup L_{7}}(x,t,\zeta)}{\overline{\rho(\bar{\zeta})}})=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),\frac{[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)}{\overline{y(\bar{\zeta})}})Q_{3}(x,t,\zeta),$		(2.75c)
	$\displaystyle(\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)}{y(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\zeta))=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,t,\zeta),\frac{[G_{1}]_{2}^{L_{6}\cup L_{8}}(x,t,\zeta)}{\overline{y(\bar{\zeta})}})Q_{4}(x,t,\zeta).$		(2.75d)

Therefore, we can derive from the Eqs.(2.75a)-(2.75d) that the jump matrices $\{Q_{i}(x,t,\zeta)\}_{1}^{4}$ meets the Eq.(2.58).

Assumption 2.4

Assuming that the zeros of $\rho(\zeta)$ and $y(\zeta)$ enjoy the assumptions as follows

1.

The spectral function $y(\zeta)$ enjoy $2n$ simple zeros $\{\varsigma_{j}\}_{j=1}^{2n}$ , $2n=2n_{1}+2n_{2}$ , if $\{\varsigma_{j}\}_{1}^{2n_{1}}\in L_{1}\cup L_{3}$ , then $\{\bar{\varsigma}_{j}\}_{1}^{2n_{2}}\in L_{8}\cup L_{6}$ .
2.

The spectral function $\rho(\zeta)$ enjoy $2N$ simple zeros $\{\eta_{j}\}_{j=1}^{2N}$ , $2N=2N_{1}+2N_{2}$ , if $\{\eta_{j}\}_{1}^{2N_{1}}\in L_{5}\cup L_{7}$ , then $\{\bar{\eta}_{j}\}_{1}^{2N_{2}}\in L_{4}\cup L_{2}$ .
3.

The spectral function $y(\zeta)$ and the spectral function $\rho(\zeta)$ do not enjoy the same simple zeros.

Proposition 2.5

The matrix function $D(x,t,\zeta)$ is defined by Eqs.(2.51a)-(2.51d) meets the following residue conditions:


	$\displaystyle\mathrm{Res}\{[D(x,t,\zeta)]_{1},\varsigma_{j}\}=\frac{1}{z(\varsigma_{j})\dot{y}(\varsigma_{j})}e^{2i\omega(\varsigma_{j})}[D(x,t,\varsigma_{j})]_{2},j=1,\cdots,2n_{1}.$		(2.76a)
	$\displaystyle\mathrm{Res}\{[D(x,t,\zeta)]_{2},\bar{\varsigma}_{j}\}=-\frac{1}{\overline{z(\varsigma_{j})}\overline{\dot{y}(\varsigma_{j})}}e^{-2i\omega(\bar{\varsigma_{j}})}[D(x,t,\bar{\varsigma}_{j})]_{1},j=1,\cdots,2n_{2}.$		(2.76b)
	$\displaystyle\mathrm{Res}\{[D(x,t,\zeta)]_{1},\eta_{j}\}=-\frac{\overline{Z(\bar{\eta}_{j})}}{y(\eta_{j})\dot{\rho}(\eta_{j})}e^{2i\omega(\eta_{j})}[D(x,t,\eta_{j})]_{1},j=1,\cdots,2N_{1}.$		(2.76c)
	$\displaystyle\mathrm{Res}\{[D(x,t,\zeta)]_{2},\ \bar{\eta}_{j}\}=\frac{Z(\bar{\eta}_{j})}{\overline{y(\eta_{j})}\overline{\dot{\rho}(\eta_{j})}}e^{-2i\omega(\bar{\eta}_{j})}[D(x,t,\bar{\eta_{j}})]_{2},j=1,\cdots,2N_{2}.$		(2.76d)

where $\dot{\rho}(\zeta)=\frac{d\rho}{d\zeta}$ .

Proof We only manifest that the residue relationship Eq.(2.76a) as follows:

Due to $D(x,t,\zeta)=(\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}}{y(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}})$ , which is means that the zeros $\{\varsigma_{j}\}_{1}^{2n_{1}}$ of $y(\zeta)$ are the poles of $\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}}{y(\zeta)}$ . Then, we have

\mathrm{Res}\{\frac{G_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)}{y(\zeta)},\varsigma_{j}\}=\lim_{\zeta\rightarrow\varsigma_{j}}(\zeta-\varsigma_{j})\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)}{y(\zeta)}=\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\varsigma_{j})}{\dot{y}(\varsigma_{j})}.

(2.77)

Taking $\zeta=\varsigma_{j}$ into the second equation of Eqs.(2.74a)-(2.74b) yields

[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\varsigma_{j})=\frac{1}{y(\varsigma_{j})}e^{2i\omega(\varepsilon_{j})}[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\varsigma_{j}).

(2.78)

According to the Eq.(2.77) and Eq.(2.78), we get

\mathrm{Res}\{\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}}(x,t,\zeta)}{y(\zeta)},\varsigma_{j}\}=\frac{1}{z(\varsigma_{j})\dot{y}(\varsigma_{j})}e^{2i\omega(\varsigma_{j})}[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,t,\varsigma_{j}).

(2.79)

Therefore, the Eq.(2.79) can lead to the Eq.(2.76a), and the remaining three residue relationships Eqs.(2.76b)-(2.76d) can be similarly proved.

2.5 The global relation

In this subsection, we give the spectral functions are not independent but meet a nice global relation. In fact, at the boundary of the region ${(\xi,\tau):0<\xi<\infty,0<\tau<t}$ , the integral of the one-form $F(x,t,\zeta)$ is given by the Eq.(2.17) is vanished. If we assume $G(x,t,\zeta)=G_{3}(x,t,\zeta)$ in the one-form $F(x,t,\zeta)$ is given by the Eq.(2.17), one can get

	$\displaystyle\int_{\infty}^{0}e^{i\zeta\xi\hat{\Lambda}}(PG_{3})(\xi,0,\zeta)d\xi+\int_{0}^{t}e^{-i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}(RG_{3})(0,\tau,\zeta)d\tau+e^{-i(8\gamma\zeta^{4}-2\zeta^{2})t\hat{\Lambda}}\times\int_{0}^{\infty}e^{i\zeta\xi\hat{\Lambda}}(PG_{3})(\xi,t,\zeta)d\xi$
	$\displaystyle=\lim_{x\rightarrow\infty}e^{i\zeta x\hat{\Lambda}}\int_{0}^{t}e^{-i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}(RG_{3})(x,\tau,\zeta)d\tau.$		(2.80)

On the one hand, according to the definition of $\psi(\zeta)$ in Eq.(2.25) and together with the Eq.(2.30b), we known that the first term of the Eq.(2.80) is

\psi(\zeta)-I.

Let $x=0$ in the Eq.(2.26) to get

\displaystyle G_{3}(0,\tau,\zeta)=G_{1}(0,\tau,\zeta)e^{i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}\psi(\zeta),

(2.81)

therefore

\displaystyle e^{-i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}(RG_{3})(0,\tau,\zeta)=[e^{-i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}(RG_{1})(0,\tau,\zeta)]\psi(\zeta).

(2.82)

On the other hand, the Eq.(2.82) and Eq.(2.29a) means that the second term of the Eq.(2.80) is

\displaystyle\int_{0}^{t}e^{-i(8\gamma\zeta^{4}-2\zeta^{2})\tau\hat{\Lambda}}(RG_{3})(0,\tau,\zeta)d\tau=[e^{-i(8\gamma\zeta^{4}-2\zeta^{2})t\hat{\Lambda}}RG_{1}(0,t,\zeta)-I]\psi(\zeta).

For $x\rightarrow\infty$ , setting $u(x,t)\in\mathcal{S}$ , then, the Eq.(2.80) is equivalent to

\displaystyle\phi^{-1}(t,\zeta)\psi(\zeta)+e^{-i(8\gamma\zeta^{4}-2\zeta^{2})t\hat{\Lambda}}\times\int_{0}^{\infty}e^{i\zeta\xi\hat{\Lambda}}(PG_{3})(\xi,t,\zeta)d\xi=I,

(2.83)

where the first column of the Eq.(2.83) is valid for $\zeta\in L_{5}\cup L_{6}\cup L_{7}\cup L_{8}$ and the second column of the Eq.(2.83) is valid for $\zeta\in L_{1}\cup L_{2}\cup L_{3}\cup L_{4}$ , and $\phi(t,\zeta)$ is given by

\displaystyle\phi^{-1}(t,\zeta)=e^{-i(8\gamma\zeta^{4}-2\zeta^{2})t\hat{\Lambda}}G_{1}(0,t,\zeta),

Owing to $\phi(\zeta)=\phi(T,\zeta)$ and letting $t=T$ , then, the Eq.(2.83) is equivalent to

\displaystyle\phi^{-1}(\zeta)\psi(\zeta)+e^{-i(8\gamma\zeta^{4}-2\zeta^{2})T\hat{\Lambda}}\times\int_{0}^{\infty}e^{i\zeta\xi\hat{\Lambda}}(PG_{3})(\xi,T,\zeta)d\xi=I.

(2.84)

Hence, the (12)th-component of the Eq.(2.84) equals

\displaystyle y(\zeta)Z(\zeta)-Y(\zeta)z(\zeta)=e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})T}E(\zeta),

(2.85)

where $E(\zeta)$ expressed as

\displaystyle E(\zeta)=\int_{0}^{\infty}e^{i\zeta\xi}(PG_{3})_{12}(\xi,T,\zeta)d\xi,

(2.86)

which is the so-called global relation.

3 The spectral functions

Definition 3.6

(Related to $y(\zeta)$ , $z(\zeta)$ ) Let $u_{0}(x)=u(x,0)\in\mathcal{S}$ , the map

\mathrm{H_{1}}:\{u_{0}(x)\}\rightarrow\{y(\zeta),z(\zeta)\},

is defined by

\left(\begin{array}[]{c}z(\zeta)\\ y(\zeta)\end{array}\right)=[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(0;\zeta)=\left(\begin{array}[]{c}(G_{3})_{12}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(0;\zeta)\\ (G_{3})_{22}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(0;\zeta)\end{array}\right),\mathrm{Im}\zeta\geq 0,

where $G_{3}(x,\zeta)$ with $P(x,\zeta)$ are given by Eq.(2.30b) and Eq.(2.35), respectively.

Proposition 3.7

The $y(\zeta)$ and $z(\zeta)$ satisfies the properties as follows

(i)

For $\mathrm{Im}\zeta<0$ , $y(\zeta)$ and $z(\zeta)$ are analytic functions,

(ii)

$y(\zeta)=1+O(\frac{1}{\zeta}),z(\zeta)=O(\frac{1}{\zeta}),$ as $\zeta\rightarrow\infty$ ,

(iii)

For $\zeta\in\mathrm{R},\,\,\mathrm{det}\psi(\zeta)=|y(\zeta)|^{2}+|z(\zeta)|^{2}=1$ ,

(iv)

$\mathrm{S_{1}}=\mathrm{H_{1}}^{-1}:\{y(\zeta),z(\zeta)\}\rightarrow\{u_{0}(x)\}$ , it’s defined as follows

u_{0}(x)=2i\lim_{\zeta\rightarrow\infty}(\zeta D^{(x)}(x,\zeta))_{12},

where $D^{(x)}(x,\zeta)$ meet the following RH problem.

Remark 3.8

Assume that


	$\displaystyle D_{+}^{(x)}(x,\zeta)=(\frac{[G_{1}]_{1}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,\zeta)}{y(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{2}\cup L_{3}\cup L_{4}}(x,\zeta)),\,\mathrm{Im}\zeta\geq 0,$		(3.1a)
	$\displaystyle D_{-}^{(x)}(x,\zeta)=([G_{3}]_{1}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,\zeta),\frac{[G_{1}]_{2}^{L_{5}\cup L_{6}\cup L_{7}\cup L_{8}}(x,\zeta)}{\overline{y(\bar{\zeta})}}),\,\mathrm{Im}\zeta\leq 0,$		(3.1b)

hence, $D^{(x)}(x,\zeta)$ admits the RH problem as:

1.

$D^{(x)}(x,\zeta)=\left\{\begin{array}[]{l}D_{+}^{(x)}(x,\zeta),\zeta\in L_{1}\cup L_{2}\cup L_{3}\cup L_{4}\\ D_{-}^{(x)}(x,\zeta),\zeta\in L_{5}\cup L_{6}\cup L_{7}\cup L_{8}\end{array}\right.$ is a slice analytic function.

$D_{+}^{(x)}(x,\zeta)=D_{-}^{(x)}(x,\zeta)(Q^{(x)}(x,\zeta))^{-1}$ , $\zeta\in\mathrm{R}$ , and

\displaystyle Q^{(x)}(x,\zeta)=\left(\begin{array}[]{cc}\frac{1}{y(\zeta)\overline{y(\bar{\zeta})}}&\frac{z(\zeta)}{\overline{y(\bar{\zeta})}}e^{-2i\zeta x}\\ -\frac{\overline{z(\bar{\zeta})}}{y(\zeta)}e^{2i\zeta x}&1\end{array}\right).

(3.4)

3.

$D^{(x)}(x,\zeta)\rightarrow\mathrm{I},\zeta\rightarrow\infty.$
4.

$y(\zeta)$ possess $2n$ simple zeros $\{\varsigma_{j}\}_{1}^{2n}$ , $2n=2n_{1}+2n_{2}$ , let us pretend that $\{\varsigma_{j}\}_{1}^{2n_{1}}$ be part of $L_{1}\cup L_{2}\cup L_{3}\cup L_{4}$ , then, $\{\bar{\varsigma}_{j}\}_{1}^{2n_{2}}$ be part of $L_{5}\cup L_{6}\cup L_{7}\cup L_{8}$ .

$[D_{+}^{(x)}]_{1}(x,\zeta)$ enjoy simple poles for $\zeta=\{\varsigma_{j}\}_{1}^{2n_{1}}$ and the $[D_{-}^{(x)}]_{1}(x,\zeta)$ enjoy simple poles for $\zeta=\{\bar{\varsigma}_{j}\}_{1}^{2n_{2}}$ . In this case, the residue relations define by


	$\displaystyle\mathrm{Res}\{[D^{(x)}(x;\zeta)]_{1},\varsigma_{j}\}=\frac{e^{2i\varsigma_{j}x}}{z(\varsigma_{j})\dot{y}(\varsigma_{j})}[D^{(x)}(x;\varsigma_{j})]_{2},$		(3.5a)
	$\displaystyle\mathrm{Res}\{[D^{(x)}(x;\Lambda)]_{2},\bar{\varsigma}_{j}\}=\frac{e^{-2i\bar{\varsigma}_{j}x}}{\overline{z(\varsigma_{j})}\overline{\dot{y}(\varsigma_{j})}}[D^{(y)}(x;\bar{\varsigma}_{j})]_{1}.$		(3.5b)

Definition 3.9

(Related to $Y(\zeta)$ , $Z(\zeta)$ ). Set $v_{0}(t)$ , $v_{1}(t),v_{2}(t),v_{3}(t)\in\mathcal{S}$ , the map

\mathrm{H_{2}}:\{v_{0}(t),v_{1}(t),v_{2}(t),v_{3}(t)\}\rightarrow\{Y(\zeta),Z(\zeta)\},

is defined by

\left(\begin{array}[]{c}Y(\zeta)\\ Z(\zeta)\end{array}\right)={[G_{2}]_{2}}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(t,\zeta)=\left(\begin{array}[]{c}(G_{2})_{12}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(t,\zeta)\\ (G_{2})_{22}^{L_{2}\cup L_{4}\cup L_{6}\cup L_{8}}(t,\zeta)\end{array}\right),

where $G_{2}(t,\zeta)$ with $R(t,\zeta)$ are given by Eq.(2.29b) and Eq.(2.35), respectively.

Proposition 3.10

The $Y(\zeta)$ and $Z(\zeta)$ satisfies the properties as follows

(i)

For $\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\leq 0$ , $Y(\zeta),Z(\zeta)$ are analytic functions,

(ii)

$Y(\zeta)=1+O(\frac{1}{\zeta})+O(\frac{e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})T}}{\zeta}),Z(\zeta)=O(\frac{1}{\zeta})+O(\frac{e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})T}}{\zeta}),$ as $\zeta\rightarrow\infty$ ,

(iii)

For $\zeta\in\mathrm{C},\,\,\mathrm{det}\phi(\zeta)=Y(\zeta)\overline{Y(\bar{\zeta})}+Z(\zeta)\overline{Z(\bar{\zeta})}=1,\,\,((8\gamma\zeta^{4}-2\zeta^{2})\in\mathrm{R},\,if\,T=\infty)$ ,

(iv)

$\mathrm{S_{2}}=\mathrm{H_{2}}^{-1}:\{Y(\zeta),Z(\zeta)\}\rightarrow\{v_{0}(t),v_{1}(t),v_{2}(t),v_{3}(t)\}$ , is defined by


	$\displaystyle v_{0}(t)=2i(D^{(1)}(t))_{12}=2i\lim_{\zeta\rightarrow\infty}(\zeta D^{(t)}(t,\zeta))_{12},$		(3.6a)
	$\displaystyle v_{1}(t)=4(D^{(2)}(t))_{12}+2iv_{0}(t)(D^{(1)}(t))_{12}=\lim_{\zeta\rightarrow\infty}[4(\zeta^{2}D^{(t)}(t,\zeta))_{12}+2iv_{0}(t)(\zeta D^{(t)}(t,\zeta))_{22}],$		(3.6b)
	$\displaystyle\gamma v_{2}(t)=-8i\gamma(D^{(3)}(t))_{12}+4\gamma v_{0}(t)(D^{(2)}(t))_{22}+2i\gamma(v_{0}^{2}(t)(D^{(1)}(t))_{12}+v_{1}(t)(D^{(1)}(t))_{22})$
	$\displaystyle\qquad\quad+2i(D^{(1)}(t))_{12}-(2\gamma v_{0}^{2}(t)+1)v_{0}(t)$
	$\displaystyle\qquad=\lim_{\zeta\rightarrow\infty}[-8i\gamma(\zeta^{3}D^{(t)}(t,\zeta))_{12}+4\gamma v_{0}(t)(\zeta^{2}D^{(t)}(t,\zeta))_{22}+2i\gamma(v_{0}^{2}(t)(\zeta D^{(t)}(t,\zeta))_{12}+v_{1}(t)(\zeta D^{(t)}(t,\zeta))_{22})$
	$\displaystyle\qquad\quad+2i(\zeta D^{(t)}(t,\zeta))_{12}-(2\gamma v_{0}^{2}(t)+1)v_{0}(t)],$		(3.6c)
	$\displaystyle\gamma v_{3}(t)=-16\gamma(D^{(4)}(t))_{12}+4(D^{(2)}(t))_{12}-8i\gamma v_{0}(t)(D^{(3)}(t))_{22}+4\gamma(v_{0}^{2}(t)(D^{(2)}(t))_{12}+v_{1}(t)(D^{(2)}(t))_{22})$
	$\displaystyle\qquad\quad+i(6\gamma v_{0}^{2}(t)+1)v_{1}(t)-2i[\gamma(v_{0}(t)\bar{v_{1}}(t)-\bar{v_{0}}(t)v_{1}(t))(D^{(1)}(t))_{12}-(\gamma v_{2}(t)+(2\gamma v_{0}^{2}(t)+1)v_{0}(t))(D^{(1)}(t))_{22}]$
	$\displaystyle\qquad=\lim_{\zeta\rightarrow\infty}\{-16\gamma(\zeta^{4}D^{(t)}(t,\zeta))_{12}+4(\zeta^{2}D^{(t)}(t,\zeta))_{12}-8i\gamma v_{0}(t)(\zeta^{3}D^{(t)}(t,\zeta))_{22}$
	$\displaystyle\qquad\quad+4\gamma(v_{0}^{2}(t)(\zeta^{2}D^{(t)}(t,\zeta)_{12}+v_{1}(t)(\zeta^{2}D^{(t)}(t,\zeta))_{22})+i(6\gamma v_{0}^{2}(t)+1)v_{1}(t)$
	$\displaystyle\qquad\quad-2i[\gamma(v_{0}(t)\bar{v_{1}}(t)-\bar{v_{0}}(t)v_{1}(t))(\zeta D^{(t)}(t,\zeta))_{12}-(\gamma v_{2}(t)+(2\gamma v_{0}^{2}(t)+1)v_{0}(t))(\zeta D^{(t)}(t,\zeta))_{22}]\},$		(3.6d)

where $D^{(1)}(t),D^{(2)}(t),D^{(3)}(t),D^{(4)}(t)$ meets the following asymptotic expansion

D^{(t)}(t,\zeta)=\mathrm{I}+\frac{D^{(1)}(t,\zeta)}{\zeta}+\frac{D^{(2)}(t,\zeta)}{\zeta^{2}}+\frac{D^{(3)}(t,\zeta)}{\zeta^{3}}+\frac{D^{(4)}(t,\zeta)}{\zeta^{4}}+O(\frac{1}{\zeta^{5}}),\,as\,\zeta\rightarrow\infty,

where $D^{(t)}(t,\zeta)$ meet the following RH problem:

Remark 3.11

Assume that


	$\displaystyle D_{+}^{(t)}(t,\zeta)=(\frac{[G_{1}]_{1}^{L_{1}\cup L_{3}\cup L_{5}\cup L_{7}}(t,\zeta)}{Y(\zeta)},[G_{3}]_{2}^{L_{1}\cup L_{3}}(t,\zeta)),\,\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\geq 0,$		(3.7a)
	$\displaystyle D_{-}^{(t)}(t,\zeta)=([G_{3}]_{1}^{L_{2}\cup L_{4}}(t,\zeta),\frac{[G_{1}]_{2}^{L_{2}\cup L_{4}}(t,\zeta)}{\overline{Y(\bar{\zeta})}}),\,\mathrm{Im}(8\gamma\zeta^{4}-2\zeta^{2})\leq 0,$		(3.7b)

hence, $D^{(t)}(t,\zeta)$ admits the RH problem as follows.

1.

$D^{(t)}(t,\zeta)=\left\{\begin{array}[]{ll}D_{+}^{(t)}(t,\zeta),&\zeta\in L_{1}\cup L_{4}\cup L_{5}\cup L_{8}\\ D_{-}^{(t)}(t,\zeta),&\zeta\in L_{2}\cup L_{3}\cup L_{6}\cup L_{7}\end{array}\right.$ is a sectionally analytic function.

$D_{+}^{(t)}(t,\zeta)=D_{-}^{(t)}(t,\zeta)Q^{(t)}(t,\zeta)$ , $\zeta\in\Gamma_{n},\,n=1,2,3,4$ , and

\displaystyle Q^{(t)}(t,\zeta)=\left(\begin{array}[]{cc}1&-\frac{Z(\zeta)}{\overline{Y(\bar{\zeta})}}e^{2i(8\gamma\zeta^{4}-2\zeta^{2})t}\\ -\frac{\overline{Z(\bar{\zeta})}}{Y(\zeta)}e^{-2i(8\gamma\zeta^{4}-2\zeta^{2})t}&\frac{1}{Y(\zeta)\overline{Y(\bar{\zeta})}}\end{array}\right).

(3.10)

3.

$D^{(t)}(t,\zeta)\rightarrow\mathrm{I},\zeta\rightarrow\infty.$
4.

$Y(\zeta)$ posses $2k$ simple zeros $\{\nu_{j}\}_{1}^{2k}$ , $2k=2k_{1}+2k_{2}$ , let us pretend that $\{\nu_{j}\}_{1}^{2k_{1}}$ be part of $L_{1}\cup L_{3}$ , then, $\{\bar{\nu}_{j}\}_{1}^{2k_{2}}$ be part of $L_{2}\cup L_{4}$ .

$[D_{+}^{(t)}]_{1}(t,\zeta)$ enjoy simple poles for $\zeta=\{\nu_{j}\}_{1}^{2k_{1}}$ and the $[D_{-}^{(t)}]_{2}(t,\Lambda)$ enjoy simple poles for $\zeta=\{\bar{\nu}_{j}\}_{1}^{2k_{2}}$ . In this case, the residue relations define by


	$\displaystyle\mathrm{Res}\{[D^{(t)}(t,\zeta)]_{1},\nu_{j}\}=\frac{e^{-2(8\gamma\nu_{j}^{4}-2\nu_{j}^{2})t}}{Z(\nu_{j})\dot{Y}(\nu_{j})}[D^{(t)}(t,\nu_{j})]_{2},$		(3.11a)
	$\displaystyle\mathrm{Res}\{[D^{(t)}(t,\zeta)]_{2},\bar{\nu}_{j}\}=\frac{e^{2(8\gamma\bar{\nu}_{j}^{4}-2\bar{\nu}_{j}^{2})t}}{\overline{Z(\bar{\nu}_{j})}\overline{\dot{Y}(\nu_{j})}}[D^{(t)}(t,\nu_{j})]_{1}.$		(3.11b)

4 The Riemann-Hilbert problem

In this part, we give two important results in theorem form.

Theorem 4.12

Set $u_{0}(x)\in\mathcal{S}(\mathrm{R^{+}})$ , and $\psi(\zeta),\phi(\zeta)$ is defined in terms of $y(\zeta)$ , $z(\zeta)$ , $Y(\zeta),Z(\zeta)$ are showed in Eq.(2.41), respectively. And the $y(\zeta)$ , $z(\zeta)$ , $Y(\zeta),Z(\zeta)$ are denotes by functions $u_{0}(z)$ , $v_{j}(t),j=0,\ldots,3$ are showed in Definition 3.1 and Definition 3.4. Assume that the function $y(\zeta)$ possess the possible simple zeros are $\{\varsigma_{j}\}_{j=1}^{2n}$ , and the function $\rho(\zeta)$ possess the possible simple zeros are $\{\eta_{j}\}_{j=1}^{2N}$ . Therefore, the solution of the FODNLS equation (1.2) is

\displaystyle u(x,t)=2i\lim_{\zeta\rightarrow\infty}(\zeta D(x,t,\zeta))_{12},

(4.1)

where $D(x,t,\zeta)$ is the solution of the RH problems as follows:

1.

$D(x,t,\zeta)$ is a piecewise analytic function for $\zeta\in C\backslash\Gamma_{m}\,(m=1,\ldots,4)$ .
2.

$D(x,t,\zeta)$ jump appears on the curves $\Gamma_{m},$ which meets the jump conditions as

$D_{+}(x,t,\zeta)=D_{-}(x,t,\zeta)Q(x,t,\zeta),\,\zeta\in\Gamma_{m},m=1,\ldots,4$ (4.2)
3.

$D(x,t,\zeta)=\mathrm{I}+O(\frac{1}{\zeta}),\,\zeta\rightarrow\infty$ .
4.

$D(x,t,\zeta)$ possess residue relationship are showed in Proposition 2.5.

Thus, the matrix function $D(x,t,\zeta)$ exists and is unique. Furthermore

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).

Proof. In fact, the manifest of this RH problem by following[7].

Theorem 4.13

(The vanishing theorem) If the matrix function $D(x,t,\zeta)\rightarrow 0\,\,(\zeta\rightarrow\infty),$ then, the RH problem in Theorem 4.1 possess only the zero solution.

Proof. Indeed, the derivation of this vanishing theorem is given in[7].

Remark 4.14

So far, we have obtained the RH problem of Eq.(1.2) on the half-line, when $\gamma=0$ , that is the IBVPs of the standard NLS equation case[7]. Different from the standard NLS equation, where the bounded analytical region and the jump curve of the FODNLS equation (1.2) are different. which the jump curve contains not only the coordinate axis, but also the hyperbola on the $x$ -axis, and the analytical region is not symmetrical.

5 Conclusions

In this paper, we utilize Fokas method to investigate integrable FODNLS equation(1.2), which can simulate the nonlinear transmission and interaction of ultrashort pulses in the high-speed optical fiber transmission system, and describe the nonlinear spin excitation phenomenon of one-dimensional Heisenberg ferromagnetic chain with eight poles and dipole interaction. Introduce a slice of important functions to spectral analysis of the Lax pair, established the basic RH problem, and the global relationship between spectral functions is also given. 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.

Acknowledgements

This work has been partially supported by the NSFC (Nos. 11601055, 11805114 and 11975145), the NSF of Anhui Province (No.1408085QA06), the University Excellent Talent Fund of Anhui Province (No. gxyq2019096), the Natural Science Research Projects in Colleges and Universities of Anhui Province (No. KJ2019A0637).

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Riemann-Hilbert problem associated with the fourth-order dispersive nonlinear Schrödinger equation in optics and magnetic mechanics

Abstract

keywords:

1 Introduction

2 The spectral analysis

2.1 The exact one-form

2.2 The analytic and bounded eigenfunctions Gj​(x,t,ζ)′​sG_{j}(x,t,\zeta)^{\prime}s

2.3 The other properties of the eigenfunctions

Proposition 2.1

Proposition 2.2

2.4 The basic Riemann-Hilbert problem

Theorem 2.3

Assumption 2.4

Proposition 2.5

2.5 The global relation

3 The spectral functions

Definition 3.6

Proposition 3.7

Remark 3.8

Definition 3.9

Proposition 3.10

Remark 3.11

4 The Riemann-Hilbert problem

Theorem 4.12

Theorem 4.13

Remark 4.14

5 Conclusions

Acknowledgements

References

References

2.2 The analytic and bounded eigenfunctions $G_{j}(x,t,\zeta)^{\prime}s$