Analytical solutions for the coupled Hirota equations in the firebringent fiber

Highlights

• The analytic bright-soliton solutions for the coupled Hirota equations in the birefringent fiber have been obtained firstly.

• The propagation and interactions of solitons have been discussed analytically and graphically. The effects of the physical parameters have been given.

Abstract

Under investigation in this paper are the coupled Hirota (CH) equations, which describe the collision of two waves in the deep ocean and the propagation of the ultrashort optical pulses in a birefringent fiber. Based on the bilinear method, multi-soliton solutions for the CH equations are given. From the perspective of analysis, the interaction dynamics of solitons are obtained. The head-on and overtaking interactions of two/three solitons are analyzed. The elastic and inelastic interactions of two/three solitons are presented. The soliton velocity can be controlled by adjusting the physical parameters ${\varphi }_j,(j=1,2,\dots N)$ and $ϵ$. The energy exchange occurs between the variables $u$ and $v$ before and after the collision. The local interference of two/three solitons is observed. The closer to the center of the interaction, the more significant the interference. The real and imaginary parts of the parameters ${\varphi }_j,(j=1,2,\dots N)$ have the effects on the numbers of peaks and holes in the collisions. The structure of four eyes and one peak is observed during the two-soliton interaction. The three-soliton interactions are given with two peaks and local interference. It is hoped that the results of this study can provide some reference for the wave interaction in deep ocean, pulse propagation in optical fiber, financial option pricing, valuation of intangible assets in sports and so on.

Introduction

In some significant natural science and engineering technology problems, with the deepening of the research, some new phenomena or problems, only using the linear model approximation, obviously appear larger error, so the influence of nonlinear medium must be considered, and the research of nonlinear model is becoming an important problem to be solved eventually [1], [2], [3], [4]. Due to the rapid development of nonlinear science, nonlinear research has been extended to solid state physics, cosmology, fluid mechanics, condensed matter physics, biology, nonlinear optics and other physical fields [5], [6]. The mathematical models are established for the practical problems in those physical fields. Most of those mathematical models are nonlinear ones, because the nonlinear models are closer to the actual problems than the linear models, and more accurate description of specific problems [7]. Compared with linear models, the nonlinear models can describe the essence and internal laws of the real world in a deeper way than the linear model [8], [9]. At present, there is no unified method to deal with the nonlinear model. The exact solutions of nonlinear evolution equations can more reasonably explain the basic laws of related phenomena. Those basic laws have important application prospects for predicting the changes of natural phenomena and solving complex engineering design [10], [11].

The nonlinear Schrödinger(NLS) equation, one of the nonlinear evolution equations, has been used for the broadband optical pulse in nonlinear fibers [12], [13] and hydrodynamic rogue-waves in an open ocean [14], [15]. The NLS equation can describe the transmission law of optical soliton in optical fiber, especially in the optical fiber communication system, it has its own advantages in the long distance, large capacity and high bit rate information transmission [16], [17]. In reality, the higher-order terms that taking into account the self-steepening, third-order dispersion and other nonlinear effects have to be added to [9], [10]. The NLS equation gradually develops from cubic term to quintic term, and then from negative term to positive term [11]. Moreover, the NLS equation in different physical background has the characteristics of high order, high dimension, coupling and variable coefficient [18].

In this paper, we consider the coupled Hirota (CH) equations, which can model the propagation of ultrashort optical pulses in a firebringent fiber [19] and the interaction of two waves in deep ocean [20]. The CH equations have the following form:

$$\begin{array}{ccc}& & i{u}_t+\frac{1}{2}{u}_{xx}+{\left(\right|u|}^2+{\left|v\right|}^2)u+iϵ[u_{xxx}+{\left(6\right|u|}^2+{3\left|v\right|}^2)u_x+3u{v}^{*}{v}_x]=0,\hfill \\ \hfill & & i{v}_t+\frac{1}{2}{v}_{xx}+{\left(\right|u|}^2+{\left|v\right|}^2)v+iϵ[v_{xxx}+{\left(6\right|v|}^2+{3\left|u\right|}^2)v_x+3v{u}^{*}{u}_x]=0,\hfill \end{array}$$

where $u=u(x,t)$ and $v=v(x,t)$ are both the complex smooth envelope functions, $t$ and $x$ represent the temporal and spatial coordinates, and $ϵ$ is a small dimensionless real parameter. $ϵ$ is also the parameter with the high-order effects, such as the self-steepening, third-order dispersion and delayed nonlinear response [21]. For Eq. (1), Ref. [19] has obtained Bäcklund transformation and one-soliton solution. Ref. [22] has presented the explicit soliton solutions with the inverse scattering method. Ref. [23] has derived the Lax pair and dark-soliton solutions. Ref. [21] has given the rogue wave solutions. Ref. [24] has shown the semirational solutions.

To our knowledge, the bright multi-soliton solutions and dynamics of soliton interaction for Eq. (1) have not been investigated. In this paper, we aim at deriving the bright soliton solutions, analyzing the dynamics of bright-soliton interaction for Eq. (1). The plan of the paper will be as follows: The bright one-soliton solution and two-soliton ones for Eq. (1) will be derived in Section 2. In Section 3, the dynamics of bright-soliton interaction and effect of the physical parameters will be studied analytically and graphically. Results will be concluded in Section 4.