Bifurcation analysis and classification of all single traveling wave solution in fiber Bragg gratings with Radhakrishnan-Kundu-Lakshmanan equation

1.   Introduction

The Radhakrishnan-Kundu-Lakshmanan (RKL) equation in polarization-preserving fibers is given as ^{[1,2,3,4,5,6,7,8,9,10,11]}

where $q = q(t, x)$ stands for the wave profile. The parameters $a$, $b$, $\lambda$ and $\delta$ are real constants.

The RKL equation are usually used to describe the pulse propagation in polarization-preserving fibers. In recent years, many experts have focused on the research of the RKL equation. Moreover, many classical methods are also used to construct the traveling wave solutions ^{[12,13,14,15,16,17,18,19,20]} of the RKL equation (for details, please refer to ^{[1,2,3,4,5,6,7,8,9,10]} and its references). But the coupled systems are usually used to simulate models form the fields of physics, nonlinear optics and engineering technology, see ^[21,22,23]. Therefore, the discussion of the coupled version of RKL equation has very important practical and theoretical significance.

The coupled version of RKL equation in fiber Bragg gratings is described as follows ^[24]

where $u = u(t, x)$ and $v = v(t, x)$ denote the wave profiles, the parameters $a_{j}$, $b_{j}$, $c_{j}$, $\alpha_{j}$, $\beta_{j}$, $\gamma_{j}$, $\sigma_{j}$ and $\lambda_{j}$($j = 1, 2$) are real constants. Equation (1.2) usually describe the dynamics of optical pulses in fiber Bragg gratings, which is also described by a family of nonlinear Schrödinger equations with cubic nonlinear terms. In ^[24], Elsayed et al. obtained the bright, dark and singular solitons solutions of Eq (1.2) by using the extended auxiliary equation method and unified Riccati equation technique. However, as far as I can, the discussion on the work of phase diagrams and single wave solutions has not been reported. This paper will focus on this issue.

The article is organized as follows. In Section 2, we draw the phase portraits and obtain the classification of single traveling wave solution of the coupled version of RKL equation. In Section 3, we present a conclusion.

2.   Bifurcation analysis and single traveling wave solution of (1.2)

2.1.   Mathematical analysis

Substituting the traveling wave transformation

into Eq (1.2), then integrating it again and separating real parts and imaginary parts of Eq (1.2), we have

where $c$ stands for the speed. $k$ represents wave number. $w$ is the phase constant. $\theta(t, x)$ represents phase component of soliton.

Making $V = AU$ ($A\neq1$), and substituting it into Eq (2.2) yields

Integrating both sides of the third and fourth equations of Eq (2.3) at the same time, we can get

From (2.4), we can easily obtain

Then, we can calculate that

Therefore, the first equation of Eq (2.4) can be simplified to

where $l_{1} = \frac{c-\alpha_{1}+3k^{2}\delta_{1}+2a_{1}kA}{\delta_{1}}$, $l_{2} = \frac{\lambda_{1}+\gamma_{1}A^{2}}{\delta_{1}}$, $\delta_{1}\neq0$.

2.2.   Phase portraits

Here, we denote $\frac{dU}{d\xi} = y$, then system (2.7) becomes the following two-dimensional system

with Hamiltonian system

The phase portraits of system (2.8) are shown in Figure 1.

2.3.   Single traveling wave solution

Multiplying $U^{'}$ both sides of Eq (2.7) and integrating once with respect to $\xi$, we can get

where $l_{0}$ is the constant. Then, we take the following transformation

Inserting (2.11) into (2.10), we have

Next, we can get the integral expression of Eq (2.12)

Here, we set $F(\Phi) = \Phi^{2}+p\Phi+q$ and $\Delta = p^{2}-4q$. According to the root of $F(\Phi) = 0$, the solution of Eq (2.13) has the following four cases.

Case 1. $\Delta = 0$ and $\Phi > 0$.

When $p < 0$ and $l_{1} < 0$, the solution of Eq (1.2) is

When $p > 0$ and $l_{1} > 0$, the solution of Eq (1.2) is

When $p = 0$ and $l_{2} > 0$, the solution of Eq (1.2) is

By selecting appropriate parameters, we draw the solution $|u_{1}(t, x)|$ and $|u_{3}(t, x)|$ of two-dimensional and three-dimensional graphics as shown in Figures 2 and 3, respectively.

Case 2. $\Delta > 0$ and $q = 0.$ When $\Phi > -p$ and $p < 0$, the solution of Eq (1.2) is

When $\Phi > -p$ and $p > 0$, the solution of Eq (1.2) is

Case 3.$\Delta > 0$ and $p\neq0$.

Assume that there are constants $\alpha$, $\beta$ and $\gamma$ satisfying $\alpha < \beta < \gamma$, here one of them is zero and two other constants are the roots of $F(\Phi) = 0$. Thus, when $\alpha < \Phi < \beta$, we have

where $m^{2} = \frac{\beta-\alpha}{\gamma-\alpha}$.

Case 4. $\Delta < 0$.

When $\Phi > \beta$, we have

where $m^{2} = \frac{(2l_{0})^{\frac{1}{2}}-l_{1}}{2(2l_{0})^{\frac{1}{2}}}$.

Remark 2.1. From the linear transformation $V = AU$, traveling wave transformation (2.1) and the obtained solution $u(t, x)$, we can easily get the solution $v(t, x)$.

3.   Conclusions

In this paper, the phase diagram is drawn with the help of Maple software and planar dynamic system theory. Moreover, the complete discrimination system of polynomial method has been applied to construct the single traveling wave solutions of the coupled version of RKL equation in fiber Bragg gratings. The solution obtained in the paper is also very effective in physics, which can help physicists understand the propagation of traveling wave in coupled RKL equation. Moreover, we have also depicted two-dimensional and three-dimensional graphs of Eq (1.2).

Conflict of interest

The authors declare no conflict of interest.