Dynamical behaviors and soliton solutions of a generalized higher-order nonlinear Schrödinger equation in optical fibers

Abstract

Under study in this paper is a generalized higher-order nonlinear Schrödinger (GHNLS) equation with the third-order dispersion (TOD), self-steeping (SS) and stimulated Raman scattering effects , which describes the propagation of ultrashort pulses in optical fibers. Via the phase plane analysis, both the homoclinic and heteroclinic orbits are found in the two-dimensional plane autonomous system reduced from the GHNLS equation, which proves the existence of bright and dark soliton solutions from the viewpoint of nonlinear dynamics. Furthermore, through the method of binary Bell polynomials and auxiliary function, the explicit bright and dark soliton solutions under certain conditions are obtained. Particular analysis is made to study the effects of the higher-order on the double-hump bright and double-hole dark solitons. The results show that the self-phase modulation and SS parameters determine the interval between two humps for the double-hump bright soliton, while the one for the double-hole dark soliton is related with the TOD and SS effects. Moreover, numerical simulations show that the double-hump bright soliton and the double-hole dark soliton are more stable when the amplitude or depth is comparably small.

Appendix

Some necessary notations on the Bell polynomials can be seen as follows:

Let $f=f(x_1,\ldots ,x_n)$ be a $C^{\infty }$ multi-variable function then, the multi-dimensional Bell polynomials can be defined as below [36, 37]:

$$\begin{aligned} Y_{n_1x_1,\ldots ,n_lx_l}(f)&\equiv Y_{n_1,\ldots ,n_l}(f_{r_1x_1,\ldots ,r_lx_l})\nonumber \\&= e^{-f}\partial ^{n_1}_{x_1}\cdots \partial ^{n_l}_{x_l}e^{f}\,, \end{aligned}$$

(37)

where $f_{r_1x_1,\ldots ,r_lx_l}=\partial ^{r_1}_{x_1}\cdots \partial ^{r_l}_{x_l}\,f\, (r_k=0,\ldots ,n_k,k=1,\ldots ,l)$ and $Y_{n_1x_1,\ldots ,n_lx_l}(f)$ denotes the multi-variable polynomial with respect to $f_{r_1x_1,\ldots ,r_lx_l}$. In particular, for $f=f(x,t)$, the associated two-dimensional Bell polynomials are

$$\begin{aligned}&Y_{2x}(f)=f_{2x}+f^{2}_x\,,\quad Y_{x,t}(f)=f_{x,t}+f_xf_t\,,\end{aligned}$$

(38)

$$\begin{aligned}&Y_{3x}(f)=f_{3x}+3f_{2x}f_{x}+f^{3}_x\,,\end{aligned}$$

(39)

$$\begin{aligned}&Y_{2x,t}(f)=f_{2x,t}+f_{2x}f_t+2f_{x,t}f_x+f^{2}_xf_t\,,\quad \ldots \,.\nonumber \\ \end{aligned}$$

(40)

In order to differ the odd- and even-order derivatives of $f_{r_1x_1,\ldots ,r_lx_l}$, the multi-dimensional binary Bell polynomial (${\fancyscript{Y}}$-polynomial) is introduced as follows [37]:

$$\begin{aligned}&{\fancyscript{Y}}_{n_1x_1,\ldots ,n_lx_l}(\nu ,\omega )\nonumber \\&\quad =Y_{n_1x_1,\ldots ,n_lx_l}(f)\Bigg |_{f_{r_1x_1,\ldots ,r_lx_l} \left\{ \begin{array}{l} \nu _{r_1x_1,\ldots ,r_lx_l},\\ (r_1+\cdots +r_l \,\,\mathrm{is}\,\,\mathrm{odd}).\\ \omega _{r_1x_1,\ldots ,r_lx_l},\\ (r_1+\cdots +r_l \,\,\mathrm{is}\,\, \mathrm{even}). \end{array}\right. } \end{aligned}$$

Then, Eqs. (38), (39) and (40) can be rewritten in the form of binary Bell polynomials as

$$\begin{aligned}&{\fancyscript{Y}}_{2x}(\nu ,\omega )=\omega _{2x}+\nu ^{2}_x\,,\quad {\fancyscript{Y}}_{x,t} (\nu ,\omega )=\omega _{x,t}+\nu _x\nu _t\,,\nonumber \\&{\fancyscript{Y}}_{3x}(\nu ,\omega )=\nu _{3x}+3\omega _{2x}\nu _{x}+\nu ^{3}_x\,,\nonumber \\&{\fancyscript{Y}}_{2x,t}(\nu ,\omega )=\nu _{2x,t}+\omega _{2x}\nu _t +2\omega _{x,t}\nu _x+\nu ^{2}_x\nu _t\,,\quad \ldots \,.\nonumber \end{aligned}$$

Note that the ${\fancyscript{Y}}$-polynomial and the Hirota expression $D^{n_1}_{x_1}\cdots D^{n_l}_{x_l}F\cdot G$ can be linked by the identity [37]

$$\begin{aligned}&(FG)^{-1}D^{n_1}_{x_1}\cdots D^{n_l}_{x_l}F\cdot G\nonumber \\&\quad ={\fancyscript{Y}}_{n_1x_1,\ldots ,n_lx_l}(\nu =\ln {F/G},\omega =\ln {FG})\,, \end{aligned}$$

(41)

where the Hirota bilinear operators are defined by [35]

$$\begin{aligned}&D^{n_1}_{x_1}\cdots D^{n_l}_{x_l}(F\cdot G)=\left( \frac{\partial }{\partial x_1}-\frac{\partial }{\partial x^{'}_1}\right) ^{n_1}{\cdots }\nonumber \\&\quad \times \left( \frac{\partial }{\partial x_l} {-}\frac{\partial }{\partial x^{'}_{l}}\right) ^{n_l}F(x_1,\ldots ,x_l)\nonumber \\&\quad \times \, G(x^{'}_1,\ldots ,x^{'}_l)\bigg |_{x^{'}_1{=}x_1,\ldots ,x^{'}_l=x_l}\,.\nonumber \end{aligned}$$

In particular, when $F=G$, Eq. (41) becomes

$$\begin{aligned}&(G)^{-2}D^{n_1}_{x_1}\cdots D^{n_l}_{x_l}G\cdot G\nonumber \\&\quad ={\fancyscript{Y}}_{n_1x_1,\ldots ,n_lx_l}(\nu =0,\omega =2\ln {G})\nonumber \\&\quad =\left\{ \begin{array}{ll} 0, &{}\quad n_1+\cdots +n_l \,\,\mathrm{is}\,\,\mathrm{odd}\,,\\ P_{n_1x_1,\ldots ,n_lx_l}(\omega ),&{}\quad n_1+\cdots +n_l \,\,\mathrm{is}\,\, \mathrm{even}, \end{array}\right. \end{aligned}$$

(42)

where the $P$-polynomials take the even part partitional structure

$$\begin{aligned} P_{2x}(\omega )&= \omega _{2x}\,,\quad P_{x,t}(\omega )=\omega _{xt}\,,\end{aligned}$$

(43)

$$\begin{aligned} P_{4x}(\omega )&= \omega _{4x}+3\,\omega ^2_{2x}\ldots \,. \end{aligned}$$

(44)

Therefore, a nonlinear equation can be transformed into the corresponding bilinear equation via Expressions (41), (43) and (44), once this nonlinear equation is expressible as a linear combination of ${\fancyscript{Y}}$-polynomials.