Multi-solitons in the model of an inhomogeneous optical fiber

1.   Introduction

Multi-solitons are a type of wave packet with soliton properties in nonlinear systems, which have self-similarity and stable transmission properties. Different from simple solitons, multi-solitons exhibit more complex interactions and nonlinear dynamic properties^[1,2]. In 1972, Zakharov and Shabat ^[3] obtained multi-soliton solutions of the nonlinear Schrödinger equation by the inverse scattering method. Subsequently, different types of multi-soliton solutions for nonlinear integrable systems were extensively studied ^[4,5,6,7].

The soliton control system is a technology that effectively manages and controls the state of nonlinear physical systems by adjusting and controlling solitons ^[8]. The optical fiber communication system controls and regulates the transmission of solitons ^[9,10]. Since the first soliton dispersion management experiment ^[11], the various soliton management mechanisms have been theoretically predicted through different methods ^{[12,13,14,15]}.

The realistic optical fiber is inhomogeneous, which can influence various effects such as the amplification or absorption, group velocity dispersion, and self-phase modulation ^[16]. The problem of soliton control in nonlinear systems is described by the following nonlinear Schrödinger equation with variable coefficients:

where $q(t, z)$ is the complex envelope of the electrical field in a co-moving frame, $D(z)$ is the group velocity dispersion, $R(z)$ is the nonlinearity coefficient, $\Gamma(z)$ is the amplification or absorption coefficient, $z$ is the propagation distance, and $t$ is the retarded time. Equation (1.1) describes the amplification or absorption of pulse propagation in a single-mode optical fiber with distributed dispersion and nonlinearity^[17].

Joshi^[18] obtained the integrability constraint of (1.1) by the Painlevé test. Serkin and Hasegawa ^[19] studied the one-soliton solution of (1.1) from the integrable point of view. Then, Hao et al.^[20] studied soliton solutions of (1.1) by Darboux transformation. Tian and Gao ^[21] obtained soliton solutions of (1.1) by symbolic computation. Lü et al.^[22,23] studied soliton solutions of (1.1) by the Hirota method. Sun et al.^[24] obtained rogue-wave solutions of (1.1) by hierarchy reduction.

In this paper, we obtained the new multi-soliton solutions of (1.1) by using the Riemann-Hilbert method ^{[25,26,27,28,29,30]}. These solutions are useful not only in designing transmission lines for soliton management, but also in some femtosecond laser experiments ^[14,15]. Additionally, the inverse scattering transform presented in this paper will pave a way for investigating the long-time asymptotic behavior of the solution to (1.1) by the nonlinear steepest descent method.

This paper is organized as follows: In Section 2, we provide some elementary preliminaries for constructing a Riemann-Hilbert problem. In Section 3, we solve the Riemann-Hilbert problem with simple pole and obtain soliton solutions for (1.1). Moreover, we discuss the properties of optical solitons by numerical simulations. Section 4 gives our conclusions.

2.   Preliminaries

We first construct a Riemann-Hilbert problem by spectral analysis of Lax pair. Generally, Eq (1.1) is not integrable. We consider the following relationship^[12]:

where the subscript denotes taking the derivative of $z$. Then, Eq (1.1) admits the Lax pair

where $\phi = \phi(t, z, k)$ is a $2\times2$ matrix function, $k\in\mathbb{C}$ is a spectral parameter, and

Here and in the following content, the asterisk indicates complex conjugate.

Considering the initial condition $\lim\limits_{z\rightarrow \pm\infty}q(0, z) = 0$, the Lax pair (2.2) becomes

where $X_0 = -ik\sigma_3$ and $T_0 = kD(z)X_0$. We obtain the Jost solutions $\phi_\pm(t, z, k)$ of Eq (2.3)

where $\theta(t, z, k) = -k(t+kzD(z))$. By making transformation

it has

Moreover, $\mu(t, z, k)$ satisfies the following Lax pair:

where $[\sigma_{3}, \mu] = \sigma_{3}\mu-\mu\sigma_{3}$, $\Delta T = T-T_0$. The Jost solution $\mu(t, z, k)$ can be solved by the following Volterral integrable equations:

where $e^{\hat{\sigma}_3}A = e^{\sigma_3} Ae^{-\sigma_3}$. For convenience, we define $D^+$, $D^-$, and $\Sigma$ on the $\mathbb{C}$-plane as

Proposition 2.1. Suppose that $u(t, z)\in L^1(\mathbb{R})$ and $\mu_{\pm, j}(t, z, k)$ represent the $j$-th column of $\mu_{\pm}(t, z, k)$, Then, the Jost solutions $\mu_\pm(t, z, k)$ have the following properties:

$\bullet$ $\mu_{-, 1}$ and $\mu_{+, 2}$ can be analytically extended to $D^+$ and continuously extended to $D^+\cup \Sigma$.

$\bullet$ $\mu_{+, 1}$ and $\mu_{-, 2}$ can be analytically extended to $D^-$ and continuously extended to $D^-\cup \Sigma$.

Proof. We define $\mu_\pm = \left({\begin{array}{*{20}{c}} \mu_{\pm, 11} & \mu_{\pm, 12}\\ \mu_{\pm, 21} & \mu_{\pm, 22} \end{array}} \right)$. Then, taking $\mu_-$ as an example, Eq (2.6) can be rewritten as

Note that $e^{2ik(t-y)} = e^{2i(t-y)\mathrm{Re}(k)}e^{2(y-t)\mathrm{Im}(k)}$. Since $y-t < 0$ and $\mathrm{Im}(k) > 0$, we obtain that $\mu_{-, 1}$ is analytically extended to $D^+$. Moreover, since $\mathrm{Im}(k) = 0$, when $k\in\Sigma$, it is shown that $\mu_{-, 1}$ is continuously extended to $D^+\cup \Sigma$. In the same way, we also obtain the analyticity and continuity of $\mu_{-, 2}$, $\mu_{+, 1}$, and $\mu_{+, 2}$. This completes the proof. □

According to the method in ^[30], the Jost solution $\mu_\pm(t, z, k)$ admits the symmetry

where $\sigma_2 = \left(\begin{array}{*{20}{c}} 0 & -i\\ i & 0 \end{array} \right)$, and its asymptotic behavior is

Since $\phi_\pm(t, z, k)$ are two fundamental matrix solutions of Lax pair (2.2), one can define a constant scattering matrix $S(k) = (s_{ij}(k))_{2\times2}$ such that

where $s_{ij}(k)$ is called scattering coefficients and $\det S(k) = 1$. From Eq (2.7), one has

According to the analyticities of $\mu_{\pm}(t, z, k)$, we obtain that $s_{11}(k)$ analytic in $D^-$ and $s_{22}(k)$ analytic in $D^+$. From Eq (2.8), the scattering matrix $S(k)$ satisfies $S(k)\rightarrow I$, as $k\rightarrow \pm\infty$.

Now, we construct the Riemann-Hilbert problem for Eq (1.1). Define the following sectionally meromorphic matrices:

Then, a multiplicative matrix Riemann-Hilbert problem is proposed:

where

3.   Multi-soliton solutions

In what follows, we will solve the Riemann-Hilbert problem with simple poles and present the multi-soliton solutions for Eq ({1.1}).

We suppose that $s_{22}(k)$ has $N$ simple zeros $k_n \ (n = 1, 2, \cdots, N)$ in $D^+$, which means $s_{22}(k_n) = 0$ and $s^{\prime}_{22}(k_n)\neq0$. Here and in the following $^\prime$ represents taking the derivative of a function variable.

According to Eq (2.10), one has $s_{22}(k_n) = s_{11}(k^*_n) = 0$. Then, the corresponding discrete spectrum can be collected as

Solving the above Riemann-Hilbert problem requires us to regularize it by subtracting out the asymptotic behaviors and the pole contributions. Then, one has

where

$C_n = -\tilde{C}^*_n = {\frac{b_n}{s_{11}^{\prime}(k^*_n)}}$, and $b_n$ is a constant.

With the help of Plemelj's formula, the solution of Eq (2.12) can be written as

Taking $M = M^-$ and comparing the (1, 2) position element of matrices (3.2), we get

where $\tilde{C}_n = {\frac{b_n}{s_{11}^{\prime}(k^*_n)}}$.

Now, we focus on the potentials $q(t, z)$ with the reflection coefficient $\rho(k) = 0$. By some algebraic calculations, we obtain the multi-soliton solutions formula

where

Now, we consider a periodic distributed amplification system with the varying group velocity dispersion parameter

and the nonlinearity parameter

where $r_0$, $r_1$, and $c$ are the parameters described by the Kerr nonlinearity, and $d_0$ is the parameter related to initial peak power in the system.

Now, as an application of formula (3.3), we first present the one-soliton solution. Let $N = 1$, $k_1 = \alpha+i\beta$, and one has

where $C_1 = {\frac{b_1}{s^{\prime}_{11}(k^*_1)}}$, $\theta(k_1) = -k_1(t+k_1zD(z))$.

Figure 1 exhibits the dynamical structures of the one-soliton solution (3.6). Due to the value of parameter $\gamma$, the soliton group velocity is changed in propagating along the fiber, but the shape of the soliton remains unchanged. This is an important property of solitons.

By setting $N = 2$, from (3.3), we obtain the two-soliton solution

where

Figure 2 exhibits the dynamical structures of the two-soliton solution (3.7). From it we observe that two solitons propagate at the same speed in the fiber and exhibit periodic oscillations.

4.   Conclusions

This paper is concerned with the multi-solitons in an inhomogeneous optical fiber model. The formula of multi-soliton solutions and inverse scattering transform are obtained by the Riemann-Hilbert method. Furthermore, we consider a soliton control system and obtain the one-soliton and two-soliton. Comparing our results with the solutions in ^[20,22], we confirm that the obtained soliton solutions are new. Finally, the inverse scattering transformation presented in this paper will pave a way for the study of the long-time asymptotic behavior of the solution to Eq (1.1).

Author contributions

Jinfang Li: Conceptualization, Investigation, Writing-original draft, Writing-review & editing. Chunjiang Wang: Methodology, Supervision. Li Zhang: Visualization, Data creation, Software, Validation. Jian Zhang: Methodology, Supervision.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by the National Natural Science Foundation of China 12271080.

Conflict of interest

All authors declare no conflicts of interest in this paper.