1. Introduction
Propagations of envelope solitons in ${ \mathcal P }{ \mathcal T }$-symmetric physical systems have been the focus of many studies in recent years [1–5]. This is due to the interesting features associated with ${ \mathcal P }{ \mathcal T }$ symmetry, including nonreciprocity of light propagation [6], absorption-enhanced transmission [7], and power oscillations [6, 8, 9]. Experimentally, ${ \mathcal P }{ \mathcal T }$-symmetry has been demonstrated in different physical media such as optical waveguide structures and lattices [8, 10, 11], optical metamaterials [12], mechanical oscillators [13], plasmonics [14], microwave cavities [15], and coherent atomic medium [16].
The theoretical description of the ${ \mathcal P }{ \mathcal T }$-symmetric lattice solitons is generally based on the complex nonlinear Schrödinger equation (NLSE). This equation includes the ${ \mathcal P }{ \mathcal T }$-symmetric potential besides the effects of the second-order diffraction/dispersion and Kerr nonlinearity [17, 18]. But with consideration of femtosecond (fs) pulse transmission in dispersive nonlinear media, several higher-order effects such as non-Kerr nonlinearity and self-frequency shift (SSF) become important, which may influence the existence and stability properties of propagating solitons. Recently, the existence and dynamics of solitons have been extensively studied in various ${ \mathcal P }{ \mathcal T }$-symmetric systems. For example, the linear phase transition and stability of solitons have been studied in ${ \mathcal P }{ \mathcal T }$-symmetric photonic lattices with Kerr nonlinearity [17]. Moreover, stable multihump solitons supported by two-dimensional ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices with focusing saturable nonlinearity have been reported [19]. Additionally, soliton solutions of the one-dimensional inhomogeneous cubic-quintic-septimal NLSE with ${ \mathcal P }{ \mathcal T }$-symmetric potential were obtained [20]. Bright solitons in quintic-septimal nonlinear media considering two kinds of ${ \mathcal P }{ \mathcal T }$-symmetric potentials have been also presented [21]. Three-dimensional optical solitons in nonlinear media with ${ \mathcal P }{ \mathcal T }$-symmetric potential, different-order nonlinearities (such as cubic, quintic and septimal nonlinearities), as well as second- and fourth-order dispersions/diffractions have been also found [22].
It is of interest to study whether stable soliton modes exist in other inhomogeneous NLSE models with ${ \mathcal P }{ \mathcal T }$-symmetric potentials. Finding such structures is highly desired in practical applications as they can help one to further understand the transmission process and hence help design new optical systems and devices. Up until now, the existence and propagation dynamics of solitons in an inhomogeneous fiber medium exhibiting self-frequency shift, weak nonlocal nonlinearity, cubic-quintic-septic nonlinearities and ${ \mathcal P }{ \mathcal T }$-symmetry potential have not been studied yet. Especially, nonlinearly chirped solitons formed in polynomial law nonlinear media under ${ \mathcal P }{ \mathcal T }$-symmetric potentials have been absent. Therefore, this work will search for stable solitons with nonlinear chirp and analyze their evolutional dynamics within the framework of a generalized NLSE model with higher-order effects and ${ \mathcal P }{ \mathcal T }$-symmetric potential depending on the spatial coordinate.
This paper is structured as follows. The generalized higher-order NLSE with varied coefficients modeling femtosecond light pulse transmission in an inhomogeneous highly nonlinear fiber medium will be cited in section 2 . Moreover, the particular cases of the equation are also discussed. In section 3 , we present a similarity transformation reducing the space-modulated higher-order NLSE to the related constant-coefficients one. We also present here the self-similar variables and constraints satisfied by the distributed coefficients in the model under consideration. In section 4 , we construct the analytical chirped bright and kink self-similar soliton solutions of the model and determine the nonlinear chirp associated with these self-similar structures. As an application of the obtained solutions, we investigate the dynamical behaviors of chirped similaritons in two types of periodic distributed amplification systems in section 5 . The stability of these nonlinearly chirped self-similar solutions is studied by numerical simulation in section 6 . Finally, the results are summarized in section 7 .
2. Theoretical model
The following space-modulated higher-order NLSE characterization of fs light through an inhomogeneous polynomial law fiber with weak nonlocality, SSF and ${ \mathcal P }{ \mathcal T }$-symmetric potential is considered [23, 24]:
$\begin{eqnarray}\begin{array}{c}{\rm{i}}\displaystyle \frac{{\rm{\partial }}u}{{\rm{\partial }}x}-\displaystyle \frac{{\beta }_{2}(x)}{2}\displaystyle \frac{{{\rm{\partial }}}^{2}u}{{\rm{\partial }}{t}^{2}}\\ \,+\,{\gamma }_{1}(x)| u{| }^{2}u+{\gamma }_{2}(x)| u{| }^{4}u+{\gamma }_{3}(x)| u{| }^{6}u\\ \,+\,{\rm{i}}s(x)u\displaystyle \frac{{\rm{\partial }}(| u{| }^{2})}{{\rm{\partial }}t}+\chi (x)u\displaystyle \frac{{{\rm{\partial }}}^{2}(| u{| }^{2})}{{\rm{\partial }}{t}^{2}}\\ \,+\,{V}_{{\rm{PT}}}(x)u=0,\end{array}\end{eqnarray}$
where u(x, t) denotes the complex envelope, x and t represent the propagation distance and retarded time, β2(x) refers to the varying group velocity dispersion (GVD) parameter, γ1(x), γ2(x), and γ3(x) represent the respective cubic, quintic, and septic nonlinear parameters, and s(x) and χ(x) are SSF and weak nonlocal nonlinear coefficients, respectively [25–31]. VPT(x) = V1(x) + iV2(x) represents the complex ${ \mathcal P }{ \mathcal T }$-symmetric potential, in which the real component V1(x) is an even function representing the index guiding, and the imaginary component V2(x) is an odd function denoting the gain (V2(x) < 0) or loss (V2(x) > 0) coefficient. Here, β2(x), γ1(x), γ2(x), γ3(x), s(x), χ(x), V1(x) and V2(x) are all real functions of x.Some special inhomogeneous NLSE models are contained in this model. For example, in the absence of septic nonlinearity, weak nonlocal nonlinearity and index guiding (i.e. γ3(x) = χ(x) = V1(x) = 0), with no request for the oddness of V2(x), and in the limit of vanishing SSF (i.e. s(x) = 0), equation (1 ) reduces to a generalized cubic-quintic NLSE, of which the exact bright, dark and gray analytical nonautonomous soliton solutions are obtained in [32]. With the above conditions and γ2(x) = 0 or γ1(x) = 0, equation (1 ) can reduce to the generalized cubic NLSE [33, 34] or quintic NLSE [35]. When neglecting the cubic nonlinearity, SSF, and weak nonlocal nonlinearity, i.e. γ1(x) = s(x) = χ(x) = 0 and considering the diffraction effect, the model (1 ) becomes the (2+1)-dimensional variable-coefficient NLSE, which paved the way for obtaining Gaussian spatial solitons in quintic-septimal nonlinear materials [36]. Here, we are interested by a special class of solitons that is characterized by a nonlinear chirp, and which can propagate in an inhomogeneous cubic-quintic-septic nonlinear fiber with weak nonlocality under ${ \mathcal P }{ \mathcal T }$-symmetric potential.
3. Similarity transformation
To find the similariton solutions of equation (1 ), one can apply the following transformation [32, 37, 38]:
$\begin{eqnarray}u(x,t)=A(x)E[T(x,t),Z(x)]{{\rm{e}}}^{{\rm{i}}\phi (x,t)},\end{eqnarray}$
where the real functions A(x), φ(x, t) are the pulse amplitude and phase, respectively, while the similarity variable T(x, t) and effective propagation distance Z(x) are two unknown real functions to be determined.Putting equations (2 ) into (1 ), the variable-coefficient NLSE (1 ) converts to the constant-coefficient NLSE as follows:
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial E}{\partial Z}-\displaystyle \frac{d}{2}\displaystyle \frac{{\partial }^{2}E}{\partial {T}^{2}}+{\alpha }_{1}| E{| }^{2}E+{\alpha }_{2}| E{| }^{4}E+{\alpha }_{3}| E{| }^{6}E\\ \quad +\,{\rm{i}}{\alpha }_{4}E\displaystyle \frac{\partial (| E{| }^{2})}{\partial T}+{\alpha }_{5}E\displaystyle \frac{{\partial }^{2}(| E{| }^{2})}{\partial {T}^{2}}=0,\end{array}\end{eqnarray}$
where d, α1, α2, α3, α4, and α5 are real constants.A set of equations can be got from equations (1 )–(3 ), and by solving those equations self-consistently, the phase function φ(x, t) is obtained:
$\begin{eqnarray}\begin{array}{l}\phi (x,t)=\displaystyle \frac{{c}_{0}}{2[1-{c}_{0}D(x)]}{t}^{2}+\displaystyle \frac{{b}_{0}}{1-{c}_{0}D(x)}t\\ \quad +\,\displaystyle \frac{{b}_{0}^{2}D(x)}{2[1-{c}_{0}D(x)]}+{\displaystyle \int }_{0}^{x}{V}_{1}(s){\rm{d}}s+{\phi }_{0},\end{array}\end{eqnarray}$
where φ0, c0 and b0 denote the initial phase, initial chirp and position of wavefront respectively, and $D(x)={\int }_{0}^{x}{\beta }_{2}(s){\rm{d}}s$ denotes the accumulated dispersion.The similarity variable T(x, t) is
$\begin{eqnarray}T(x,t)=\displaystyle \frac{t-{t}_{c}(x)}{W(x)},\end{eqnarray}$
where the width W(x) and center position tc(x) of the soliton are given respectively as $\begin{eqnarray}W(x)={W}_{0}[1-{c}_{0}D(x)],\end{eqnarray}$
$\begin{eqnarray}{t}_{c}(x)={t}_{0}-({b}_{0}+{c}_{0}{t}_{0})D(x).\end{eqnarray}$
Here, W0 and t0 are the initial width and center position of the self-similar pulse, respectively.The effective propagation distance Z(x) reads
$\begin{eqnarray}Z(x)=\displaystyle \frac{D(x)}{{{dW}}_{0}W(x)}.\end{eqnarray}$
The pulse amplitude A(x) takes the form:
$\begin{eqnarray}A(x)={\left[\displaystyle \frac{{\alpha }_{3}}{{\gamma }_{3}(x)}\displaystyle \frac{{\beta }_{2}(x)}{d}\displaystyle \frac{1}{{W}^{2}(x)}\right]}^{1/6}.\end{eqnarray}$
The parameters γ1(x), γ2(x), s(x), χ(x) and V2(x) are given by
$\begin{eqnarray}{\gamma }_{1}(x)={\alpha }_{1}{\left[\displaystyle \frac{{\gamma }_{3}(x)}{{\alpha }_{3}}{\left(\displaystyle \frac{{\beta }_{2}(x)}{d}\right)}^{2}\displaystyle \frac{1}{{W}^{4}(x)}\right]}^{1/3},\end{eqnarray}$
$\begin{eqnarray}{\gamma }_{2}(x)={\alpha }_{2}{\left[{\left(\displaystyle \frac{{\gamma }_{3}(x)}{{\alpha }_{3}}\right)}^{2}\displaystyle \frac{{\beta }_{2}(x)}{d}\displaystyle \frac{1}{{W}^{2}(x)}\right]}^{1/3},\end{eqnarray}$
$\begin{eqnarray}s(x)={\alpha }_{4}{\left[\displaystyle \frac{{\gamma }_{3}(x)}{{\alpha }_{3}}{\left(\displaystyle \frac{{\beta }_{2}(x)}{d}\right)}^{2}\displaystyle \frac{1}{W(x)}\right]}^{1/3},\end{eqnarray}$
$\begin{eqnarray}\chi (x)={\alpha }_{5}{\left[\displaystyle \frac{{\gamma }_{3}(x)}{{\alpha }_{3}}{\left(\displaystyle \frac{{\beta }_{2}(x)}{d}\right)}^{2}{W}^{2}(x)\right]}^{1/3},\end{eqnarray}$
$\begin{eqnarray}{V}_{2}(x)=-\displaystyle \frac{{\gamma }_{3}(x)}{6{\beta }_{2}(x)}\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}\left[\displaystyle \frac{{\beta }_{2}(x)}{{\gamma }_{3}(x)}\right]+\displaystyle \frac{{W}_{0}{c}_{0}{\beta }_{2}(x)}{6W(x)}.\end{eqnarray}$
From the above results, one can obtain the exact similariton solutions to the variable-coefficient NLSE (1 ) by using the exact solutions of equation (3 ) and the transformation (2 ), in which the functions A(x), T(x, t), Z(x), and φ(x, t) are given by equations (9 ), (5 ), (8 ), and (4 ), respectively. For the existence of similariton solutions of equation (1 ), only three of the eight coefficients β2(x), γ1(x), γ2(x), γ3(x), s(x), χ(x), V1(x) and V2(x) in equation (1 ) are free coefficients. For example, if β2(x), γ3(x) and V1(x) are free, the other five parameters will be determined by equations (10 )–(14 ).
4. Nonlinearly chirped self-similar solitons
4.1. Chirped bright solitons
In our recent research [23], exact soliton solutions to the constant-coefficient higher-order NLSE (3 ) including additional terms of the self-steepening effect are presented. Based on this previous work [23], one gets the following chirped bright soliton solution of equation (1 )
$\begin{eqnarray}\begin{array}{l}{u}_{b}(x,t)={\left[\displaystyle \frac{{\alpha }_{3}{\beta }_{2}(x)}{d{\gamma }_{3}(x)}\displaystyle \frac{1}{{W}^{2}(x)}\right]}^{1/6}\\ \quad \times \,\sqrt{\displaystyle \frac{2{f}_{2}}{\sqrt{{f}_{4}^{2}-4{f}_{6}{f}_{2}}\cosh \left(2\sqrt{{f}_{2}}\xi \right)-{f}_{4}}}\exp [{\rm{i}}{{\rm{\Phi }}}_{b}(x,t)],\end{array}\end{eqnarray}$
where ξ = T − qZ is the traveling coordinate with inverse group velocity q = 1/v, $\begin{eqnarray}\begin{array}{l}\xi =\displaystyle \frac{1}{{W}_{0}[1-{c}_{0}{\displaystyle \int }_{0}^{x}{\beta }_{2}(s){\rm{d}}s]}\\ \quad \times \,\left[t-{t}_{0}+({b}_{0}+{c}_{0}{t}_{0}-\displaystyle \frac{q}{{{dW}}_{0}}){\displaystyle \int }_{0}^{x}{\beta }_{2}(s){\rm{d}}s\right],\end{array}\end{eqnarray}$
and f2, f4, and f6 are $\begin{eqnarray}{f}_{2}=\displaystyle \frac{2d\kappa -{q}^{2}}{{d}^{2}},\end{eqnarray}$
$\begin{eqnarray}{f}_{4}=\displaystyle \frac{4{\alpha }_{5}(2d\kappa -{q}^{2})+{d}^{2}{\alpha }_{1}}{{d}^{3}},\end{eqnarray}$
$\begin{eqnarray}{f}_{6}=-\displaystyle \frac{{\alpha }_{3}}{8{\alpha }_{5}},\end{eqnarray}$
where κ is wave number.A parametric condition gives
$\begin{eqnarray}\begin{array}{l}3{\alpha }_{3}{d}^{4}+384{\alpha }_{5}^{3}(2d\kappa -{q}^{2})+96{d}^{2}{\alpha }_{1}{\alpha }_{5}^{2}\\ \quad +\,8({\alpha }_{4}^{2}+2{\alpha }_{2}d){\alpha }_{5}{d}^{2}=0,\end{array}\end{eqnarray}$
and the excitation conditions of bright soliton are given by: $\begin{eqnarray}2d\kappa -{q}^{2}\gt 0,\end{eqnarray}$
$\begin{eqnarray}\left[4{\alpha }_{5}(2d\kappa -{q}^{2})+{d}^{2}{\alpha }_{1}\right]d\lt 0,\end{eqnarray}$
$\begin{eqnarray}0\lt -\displaystyle \frac{{\alpha }_{3}}{2{\alpha }_{5}}\lt \displaystyle \frac{{\left[4{\alpha }_{5}(2d\kappa -{q}^{2})+{d}^{2}{\alpha }_{1}\right]}^{2}}{{d}^{4}(2d\kappa -{q}^{2})}.\end{eqnarray}$
The total phase of field Φb(x, t) is
$\begin{eqnarray}\begin{array}{l}{{\rm{\Phi }}}_{b}(x,t)=\displaystyle \frac{{\alpha }_{4}}{d\sqrt{{f}_{6}}}\\ \quad \times \,\mathrm{arctanh}\left[\displaystyle \frac{-{f}_{4}-\sqrt{{f}_{4}^{2}-4{f}_{6}{f}_{2}}}{2\sqrt{{f}_{6}{f}_{2}}}\tanh (\sqrt{{f}_{2}}\xi )\right]\\ \quad -\,\displaystyle \frac{q}{d}\xi -\kappa Z+\displaystyle \frac{{c}_{0}}{2[1-{c}_{0}D(x)]}{t}^{2}\\ \quad +\,\displaystyle \frac{{b}_{0}}{1-{c}_{0}D(x)}t+\displaystyle \frac{{b}_{0}^{2}D(x)}{2[1-{c}_{0}D(x)]}\\ \quad +\,{\displaystyle \int }_{0}^{x}{V}_{1}(s){\rm{d}}s+{\phi }_{0},\end{array}\end{eqnarray}$
and the corresponding chirp is given by $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{b}=-\displaystyle \frac{\partial {{\rm{\Phi }}}_{b}}{\partial t}=\left[q-\displaystyle \frac{2{\alpha }_{4}{f}_{2}}{\sqrt{{f}_{4}^{2}-4{f}_{6}}\cosh (2\sqrt{{f}_{2}}\xi )-{f}_{4}}\right]\\ \quad \times \,\displaystyle \frac{1}{{dW}(x)}-\displaystyle \frac{{c}_{0}}{1-{c}_{0}D(x)}t-\displaystyle \frac{{b}_{0}}{1-{c}_{0}D(x)}.\end{array}\end{eqnarray}$
Notice that the bright soliton in [23] was presented with the parametric condition ${f}_{4}^{2}-4{f}_{6}{f}_{2}={f}_{2}^{2}$, which is sufficient but not necessary. In fact, the expressions in the following form are all the solutions of equation (3 ):
$\begin{eqnarray}\begin{array}{l}E[T(x,t),Z(x)]\\ \quad =\,\sqrt{\displaystyle \frac{2{f}_{2}}{\varsigma {{\rm{e}}}^{2\sqrt{{f}_{2}}\xi }+[({f}_{4}^{2}-4{f}_{6}{f}_{2}){{\rm{e}}}^{-2\sqrt{{f}_{2}}\xi }]/\varsigma -{f}_{4}}}\\ \quad \times \,\exp \{{\rm{i}}[{{\rm{\Phi }}}_{b}(x,t)-\phi (x,t)]\},\end{array}\end{eqnarray}$
where ς is a arbitrary none-zero real number.4.2. Chirped kink solitons
From equation (2 ) and [23], we get the chirped kink soliton solution of equation (1 )16 ), and f2 is given by equation (17a )
$\begin{eqnarray}\begin{array}{l}{u}_{k}(x,t)={\left[\displaystyle \frac{{\alpha }_{3}{\beta }_{2}(x)}{d{\gamma }_{3}(x)}\displaystyle \frac{1}{{W}^{2}(x)}\right]}^{1/6}{\left[\displaystyle \frac{2{\alpha }_{5}({q}^{2}-2d\kappa )}{{d}^{2}{\alpha }_{3}}\right]}^{1/4}\\ \quad \times \,\sqrt{1+\displaystyle \frac{\tanh (2\sqrt{{f}_{2}}\xi )}{1+{\rm{sech}} \left[2\sqrt{{f}_{2}}\xi \right]}}\exp [{\rm{i}}{{\rm{\Phi }}}_{k}(x,t)],\end{array}\end{eqnarray}$
where the traveling coordinate ξ is given by equation (The total phase of field Φk(x, t) is
$\begin{eqnarray}\begin{array}{l}{{\rm{\Phi }}}_{k}(x,t)=\displaystyle \frac{{\alpha }_{4}}{d}\sqrt{\displaystyle \frac{2{\alpha }_{5}({q}^{2}-2d\kappa )}{{d}^{2}{\alpha }_{3}}}\\ \quad \times \,\left\{\xi +\displaystyle \frac{\mathrm{ln}\left[1+{\rm{sech}} (2\sqrt{{f}_{2}}\xi )\right]-\mathrm{ln}\left[{\rm{sech}} (2\sqrt{{f}_{2}}\xi )\right]}{2\sqrt{{f}_{2}}}\right\}\\ \quad -\displaystyle \frac{q}{d}\xi -\kappa Z\\ \quad +\,\displaystyle \frac{{c}_{0}}{2[1-{c}_{0}D(x)]}{t}^{2}+\displaystyle \frac{{b}_{0}}{1-{c}_{0}D(x)}t\\ \quad +\,\displaystyle \frac{{b}_{0}^{2}D(x)}{2[1-{c}_{0}D(x)]}+{\displaystyle \int }_{0}^{x}{V}_{1}(s){\rm{d}}s+{\phi }_{0},\end{array}\end{eqnarray}$
and the accompanying chirp has the form $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{k}=-\displaystyle \frac{\partial {{\rm{\Phi }}}_{k}}{\partial t}\\ \quad =\left\{q-{\alpha }_{4}\sqrt{\displaystyle \frac{2{\alpha }_{5}({q}^{2}-2d\kappa )}{{d}^{2}{\alpha }_{3}}}\left(1+\displaystyle \frac{\tanh \left[2\sqrt{{f}_{2}}\xi \right]}{1+{\rm{sech}} (2\sqrt{{f}_{2}}\xi )}\right)\right\}\\ \quad \times \,\displaystyle \frac{1}{{dW}(x)}-\displaystyle \frac{{c}_{0}}{1-{c}_{0}D(x)}t-\displaystyle \frac{{b}_{0}}{1-{c}_{0}D(x)}.\end{array}\end{eqnarray}$
Except for the first constraint condition equation (18 ), the other new three parametric conditions (the excitation conditions) of the analytical kink soliton solution equation (23 ) are given by:
$\begin{eqnarray}2d\kappa -{q}^{2}\gt 0,\end{eqnarray}$
$\begin{eqnarray}{\alpha }_{3}{\alpha }_{5}\lt 0,\end{eqnarray}$
$\begin{eqnarray}4{\alpha }_{5}(2d\kappa -{q}^{2})+{d}^{2}{\alpha }_{1}=-{d}^{3}\sqrt{\displaystyle \frac{{\alpha }_{3}({q}^{2}-2d\kappa )}{2{d}^{2}{\alpha }_{5}}}.\end{eqnarray}$
5. Propagation evolution of chirped self-similar solitons
Now, we will analyze the dynamical behaviors of chirped bright and kink similariton solutions (15 ) and (23 ) by appropriately choosing the varying coefficient. As pulse amplitude A(x) is a real function, β2(x)/γ3(x) must be greater than or equal to zero (i.e. β2(x)/γ3(x) ≥ 0) when α3d > 0. And to ensure V2(x) an odd function, we choose β2(x) and γ3(x) to be odd functions. Specifically, we consider two types of periodic distributed amplification systems, for which the varying coefficients of GVD are both
$\begin{eqnarray}{\beta }_{2}(x)={\beta }_{0}\sin (\sigma x),\end{eqnarray}$
and the septic nonlinearity parameters are $\begin{eqnarray}{\rm{Type}}\,1:\,{\gamma }_{3}(x)={\gamma }_{0}\sin (\sigma x){\rm{{\rm{sech}} }}({hx}),\end{eqnarray}$
$\begin{eqnarray}{\rm{Type}}\,2:\,{\gamma }_{3}(x)={\gamma }_{0}\sin (\sigma x)\cosh ({hx}),\end{eqnarray}$
where σ and h control the rate of dispersion and septic nonlinearity inside the nonlinear optical fibers, while β0 and γ0 are arbitrary constants.For these two types of periodic distributed system parameters, the width of the pulse given by equation (5 ) takes the form
$\begin{eqnarray}W(x)={W}_{0}\left\{1+\displaystyle \frac{{c}_{0}{\beta }_{0}}{\sigma }[\cos (\sigma x)-1]\right\},\end{eqnarray}$
indicating that W depends on the initial chirp parameter c0 and varies periodically with the propagation distance x.Moreover, the gain (loss) coefficients are
$\begin{eqnarray}\begin{array}{c}{\rm{Type}}\,1:\,{V}_{2}(x)=-\displaystyle \frac{h\tanh ({hx})}{6}\\ \,+\,\displaystyle \frac{\sigma {c}_{0}{\beta }_{0}\sin (\sigma x)}{6\left\{\sigma +{c}_{0}{\beta }_{0}[\cos (\sigma x)-1]\right\}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{\rm{Type}}\,2:\,{V}_{2}(x)=\displaystyle \frac{h\tanh ({hx})}{6}\\ \,+\,\displaystyle \frac{\sigma {c}_{0}{\beta }_{0}\sin (\sigma x)}{6\left\{\sigma +{c}_{0}{\beta }_{0}[\cos (\sigma x)-1]\right\}}.\end{array}\end{eqnarray}$
And V1 is an arbitrary even function of x, which only influences the total phases of bright and kink solitons, and have no relationships with the amplitudes and chirps.Now we investigate the dynamical behaviors of the chirped bright and kink similariton solutions (15 ) and (23 ) for the control system described by equations (27 ), (28a ) and (28b ) with different initial chirp parameter c0 and septic nonlinearity parameter γ3(x). As seen from figures 1(a)–(c), the chirped similariton pulses show a snake-like behavior as they propagate through the nonlinear optical fibers. For pulse center positions, these six figures in figure 1 have the same variation with a period of 2π, as shown in figure 2(a). It coincides with equation (16 ), which indicates that if the initial center position t0 = 0, the center position (${t}_{c}(x)\,={t}_{0}+({b}_{0}+{c}_{0}{t}_{0}-\tfrac{q}{{{dW}}_{0}}){\int }_{0}^{x}{\beta }_{2}(s){\rm{d}}s$) is independent of c0 and γ3(x).
Figure 1. Evolution of chirped self-similar bright soliton ( |
Figure 2. The influence of initial chirp parameter c0 and septic nonlinearity parameter γ3(x) on the pulse center position, amplitude and FWHM of bright soliton. (a) The center position of bright solitons in figures 1(a)–(f), which vary with propagation distance x in an identical way. (b) The amplitude peak values of bright solitons ${\rm{Max}}(| {u}_{b}{| }^{2})$ in figures 1(a)–(f) change along x. The inset shows the corresponding gain/loss coefficient V2(c). (c) The pulse profiles of bright solitons at x = 10 with a different value of c0. Other parameters are the same as in figure 1(a). The FWHM values are marked. The inset shows the FWHM of pulses vary with x with c0 = 0 (black line), c0 = 0.5 (green dash line) and c0 = 1 (red dashed–dotted line). (d) The FWHM relation with c0 at different positions of propagation distance x. (n is an arbitrary natural number.) And the change of h in septic nonlinearity parameter γ3(x) of Type 1 and Type 2 do not influence the result of FWHM. Other parameters are the same as in figure 1(a) except c0 and γ3(x). |
To find the factors that affect the amplitude, the variation of amplitude peak values ${\rm{Max}}(| {u}_{b}{| }^{2})$ in figure 1 with respect to x is plotted in figure 2(b). And the inset shows the corresponding ${ \mathcal P }{ \mathcal T }$-symmetric potential imaginary part V2(x) = 0. For c0 = h = 0, which means V2(x) = 0, the pulse shape in figure 1(a) remains the same propagating along x, so ${\rm{Max}}(| {u}_{b}{| }^{2})$ is a constant function of x (see line ‘a' in figure 2(b)). With c0 = 0, and h = 0.1 in Type 1/Type 2 of septic nonlinearity parameter γ3(x), which causes V2(x) to be negative/positive, the amplitude peak value monotonic increase/decrease with the increasing of x (line ‘b'/‘c' in figure 2(b)). For c0 = 0.5 and h = 0, in which case the V2(x) is a periodic function, the amplitude peak value presents variation in the same periods (2π) as V2(x) (line ‘d'). For c0 = 0.5, and h = 0.1 in Type 1 / Type 2 of γ3(x), V2(x) can be seen as the V2(x) of ‘b'/‘c' adds that of ‘d', and the ${\rm{Max}}(| {u}_{b}{| }^{2})$ of Type 1 / Type 2 (line ‘e'/‘f' in figure 2(b)) is equal to the product of the ${\rm{Max}}(| {u}_{b}{| }^{2})$ of ‘b'/‘c' and ‘d'. Therefore, we can control the amplitude by adjusting the ${ \mathcal P }{ \mathcal T }$-symmetric potential imaginary part V2(x).
To quantitative analyze the influence of initial chirp parameter c0 on the width of the bright soliton pulse, we plot the pulse profiles at x = 10 with ${\gamma }_{3}(x)={\gamma }_{0}\sin (\sigma x)$ and c0 = 0, 0.5, 1 as shown in figure 2(c). The full width at half maxima (FWHM) for c0 = 0, 0.5 and 1 are approximately 3.75, 7.20 and 10.65, respectively. The inset of figure 2(c) shows that, for c0 = 0, the FWHM keeps constant. While for c0 = 0.5 or 1, the FWHM changes periodically with x (period: 2π), and the maximum and minimum values of FWHM are located at x = (2n + 1)π, (n = 0, 1, 2, …) and x = 2nπ, respectively. One can find from figure 2(d) that, the pulse width of self-similar soliton varies linearly with c0 at a certain position of x. These results conform to equation (29 ).
The results for the kink soliton (23 ) are shown in figure 3 with the same values of c0 and γ3(x) as those in figure 1. We can see that the kink solitons share the same features influenced by parameters c0 and h with the bright ones. The propagations of the chirped self-similar solutions (15 ) and (23 ) for the parameters b0 = 0, c0 = 0.5, and q = 0 are shown in figure 4. It can be seen that the initial chirp c0 leads to the ‘breathing character' [39] of similariton pulses. And in this situation, the centre positions of pluses keep constant at t = 0.
Figure 3. Evolution of chirped self-similar kink soliton ( |
In figures 1 and 3, the constant parameter values α2 and α4 are not given, because these two parameters do not affect the evolution of soliton pulse shapes, but they must satisfy the condition (18 ). While the chirp of solitons is related to α4. In order to understand the influence of nonlinear effect on the chirp of propagating self-similar waves, here the chirping profile of the bright and kink similiaritons given by (21 ) and (25 ) for different values of α4 at x = 10 are depicted in figures 5(a) and (b), respectively. The figures show that, for both kinds of solitons, the constant parameter α4 determines whether the chirp is linear or nonlinear, and the amplitude of the chirp increases continuously as the value of α4 increases. And it is found that the chirps have the same center positions with the corresponding soliton intensities ∣ub∣2 and ∣uk∣2.
6. Numerical results and the stability analysis
The numerical simulation of chirped bright and kink solitons is carried out based on the mixed method of split-step Fourier method (SSFM) and Runge–Kutta method [40]. The parameters of bright/kink soliton are the same as figures 1/figures 2(a), (b), (d), (e), and equations (15 )/(23 ) with x = 0 is used as the input pulse. The pulse profiles of bright/kink solitons at x = 5π (2.5 times of the period) of the SSFM simulation solutions and the self-similar exact ones are shown in figures 6/7(a)–(d). One can see that the simulation results agree extremely well with the analytical results. Figures 6 and 7(e)–(h) show that the simulated bright and kink solitons evolve in exactly the same way as the self-similar ones of figures 1 and 2(a), (b), (d), (e), respectively. When we change the form of ${ \mathcal P }{ \mathcal T }$-symmetric potential real component V1(x), the results show no difference from figures 6/7(a)–(h), confirming that the amplitudes of these solitons ∣ub∣2 and ∣uk∣2 are independent of V1(x). While the total phases (20 ) and (24 ) of bright and kink solitons can be affected by V1(x).
Figure 6. Numerical calculation results of Chirped bright solitons based on SSFM combined with Runge–Kutta methord. (a)–(d) The profiles of initial pulses (x = 0), and of the numerical simulation solutions (Num.) and the self-similar analytical ones (Anal.) at x = 5π with the same parameters of figures 1(a), (b), (d), (e). (e)–(h) The evolution of chirped bright soliton by numerical calculation with the same parameters of (a)–(d). (i)–(l) The evolution of chirped bright soliton by numerical calculation corresponding to (e)–(h) with 5% noise. (m)–(p) The pulse profiles at x = 5π of the numerical simulation solutions (Num.) with noise, and the inset in each figure shows the zoom-in profile near the peak. |
Figure 7. Numerical calculation results of Chirped kink solitons based on SSFM combined with Runge–Kutta methord. (a)–(d) The profiles of initial pulses (x = 0), and of the numerical simulation solutions (Num.) and the self-similar analytical ones (Anal.) at x = 5π with the same parameters of figures 3(a), (b), (d), (e). (e)–(h) The evolution of chirped kink soliton by numerical calculation with the same parameters of (a)–(d). (i)–(l) The evolution of chirped kink soliton by numerical calculation corresponding to (e)–(h) with 5% initial random noise. (m)–(p) The pulse profiles at x = 5π of the numerical simulation solutions (Num.) with noise. |
For soliton, the stability of any solution largely determines the practical application possibility. Thus, we perform a robustness analysis through numerical simulations by testing the ability of each soliton to propagate over an appreciable distance under an environment with a finite perturbation. For this aim, a photon initial noise (random noise) corresponding to 5% of each input profile is generated, which is significant enough to perturb and destroy any unstable structures. Figures 6 and 7(i)–(l) attribute to the propagations of bright and kink solitons under a noise environment for a length of 20 m. Figures 6 and 7(m)–(p) show the pulse profiles of the right and kink solitons with noise at x = 5π. The presented solutions preserve the basic features of soliton dynamical behaviors and ensure remarkable stability despite a noise perturbation. Therefore, it can be concluded that the obtained solutions of the above solitons are stable and robust.
7. Conclusion
The fs light propagation in a polynomial law nonlinear optical fiber has been studied based on the space-modulated higher-order NLSE (1 ) with distributed weak nonlocal nonlinearity and SSF. Through the similarity transformation, equation (1 ) is converted to the constant-coefficient NLSE (3 ), and then exact chirped self-similar bright and kink soliton solutions are obtained. Two types of periodic distributed amplification systems are considered to study the transmission evolution of chirped self-similar solitons. By choosing appropriate parameters, the soliton structures and propagation behaviors can be controlled. Specifically, the imaginary part of the ${ \mathcal P }{ \mathcal T }$-symmetric potential can control the amplitude of solitons, and the pulse width of the self-similar soliton is found to have a linear correlation with the initial chirp parameter through quantitative analysis. The numerical simulation of bright and kink solitons based on the SSFM and Runge–Kutta method shows excellent agreement with the analytical solution. The stability test against perturbation taken by numerical calculation demonstrates that the bright and kink solitons are remarkably stable. These results constitute the first analytical demonstration of the propagation of nonlinearly chirped solitons in a spatially inhomogeneous fiber system under the ${ \mathcal P }{ \mathcal T }$-symmetric potential, polynomial law nonlinearity, weak nonlocal nonlinearity, SSF and second-order dispersion.