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Effect of near ${ \mathcal P }{ \mathcal T }$-symmetric potentials on nonlinear modes for higher-order generalized Ginzburg–Landau model

  • Da Lin ,
  • Kai-Ru Dong ,
  • Jia-Rui Zhang ,
  • Yu-Jia Shen , *
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  • College of Science, China Agricultural University, Beijing 100083, China

*Author to whom any correspondence should be addressed.

Received date: 2022-07-10

  Revised date: 2022-08-08

  Accepted date: 2022-08-19

  Online published: 2022-11-21

Copyright

© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this paper, we study the higher-order generalized Ginzburg–Landau model which contributes to describing the propagation of optical solitons in fibers. By means of the Hirota bilinear method, the analytical solutions are obtained and the effect of relevant parameters is analyzed. Modulated by the near parity-time-symmetric potentials, the nonlinear modes with 5% initial random noise are numerically simulated to possess stable evolution. Furthermore, the evolution of nonlinear modes is displayed through the adiabatical change of some parameters. The investigation of the present work is intended as a contribution to the work for the higher-order generalized Ginzburg–Landau model.

Cite this article

Da Lin , Kai-Ru Dong , Jia-Rui Zhang , Yu-Jia Shen . Effect of near ${ \mathcal P }{ \mathcal T }$-symmetric potentials on nonlinear modes for higher-order generalized Ginzburg–Landau model[J]. Communications in Theoretical Physics, 2022 , 74(12) : 125001 . DOI: 10.1088/1572-9494/ac8afd

1. Introduction

Optical solitons, which have the unique characteristic that waveform and velocity remain unchanged over long distant propagation, have been paid increasing attention in recent years [15]. It is found that the formation mechanism of optical solitons during the propagation process is the balance between group velocity dispersion and the self-phase modulation effect in the anomalous dispersion region [1]. To describe the propagation of optical solitons in optical fibers, the nonlinear Schrödinger equation (NLSE) known as an important and universal model has been developed with some generalizations and soliton solutions presented [613]. Nevertheless, the generalized Ginzburg–Landau equation (GGLE), which is widely applied in such fields as superconductivity, liquid crystal, Bose–Einstein condensate, can be considered as a dissipative generalization of NLSE [1417]. Different analytical and numerical methods have been applied to the GGLE, while various novel solutions including the pulsating, erupting and creeping solitons have been obtained [1823]. By means of numerical simulations, the stability of various solutions has been proved [24, 25]. For a wider application prospect, the model has been extended to higher-dimension and higher-order cases [2630]. Moreover, parity-time (${ \mathcal P }{ \mathcal T }$) symmetric potentials have been introduced to the GGLE with several interesting results [25, 31, 32]. Though different ${ \mathcal P }{ \mathcal T }$-symmetric behaviors have been studied theoretically or observed in experiments [3338], limited research has been done which is relevant to the higher-order GGLE. In previous work, we have investigated the fourth-order GGLE with quintic nonlinearities and near ${ \mathcal P }{ \mathcal T }$-symmetric structures [39].
In this paper, we will study the GGLE with third-order dispersion and nonlinear gradient:
$\begin{eqnarray}\begin{array}{l}\,{\rm{i}}{u}_{t}+\alpha (x){u}_{{xx}}+U(x,t)u+\beta (x,t)| u{| }^{2}u\\ \quad +\,{\rm{i}}\sigma (t){u}_{{xxx}}+\,{\rm{i}}\gamma (t)| u{| }^{2}{u}_{x}+\,{\rm{i}}\rho (t){u}_{x}=0,\end{array}\end{eqnarray}$
where u(x, t) represents a complex wave envelope, x denotes the propagation distance and t is the time. The subscripts denote the partial derivative with respect to x or t and i represents the imaginary unit. α(x), U(x, t) and β(x, t) are complex functions that can be assumed as α = α1 + iα2, U = V + iW, β = β1 + iβ2. α(x), U(x, t), β(x, t), σ(t) and γ(t) can describe the variable effect of group velocity dispersion, gain or loss, self-phase modulation, third-order dispersion and nonlinear gradient terms, respectively [18, 25, 30, 40].
There are three special cases that can be reduced by equation (1).

(1)When α2(x) = W(x, t) = β2(x, t) = 0, equation (1) turns into the third-order NLSE. It has been used to describe the propagation of ultra-short pulses and optical solitons in fibers in [6, 7, 41]. Some exact solutions and the corresponding abundant structures have been obtained [7], and the linear stability of solitons has been studied [6].

(2)When σ(t) = γ(t) = ρ(t) = 0, equation (1) can be reduced to the second-order GGLE. The analytical solutions have been derived by means of the Hirota bilinear method [21, 25, 42, 43]. The stability of soliton has been analyzed via numerical simulations in [25, 42].

(3)When σ(t) = γ(t) = ρ(t) = 0 and U(x, t) is ${ \mathcal P }{ \mathcal T }$-symmetric. Equation (1) is changed into the GGLE with ${ \mathcal P }{ \mathcal T }$-symmetric potential, which has been less investigated so far except [25]. The effect of near ${ \mathcal P }{ \mathcal T }$-symmetric potentials on nonlinear modes has been reported [32].

The rest of this paper is arranged as follows. In section 2, the bilinear form of equation (1) is derived under some constraints. In addition, soliton solutions of equation (1) with constant and variable coefficients are obtained respectively. In section 3, the stable transmission of nonlinear modes is verified through numerical simulations with 5% perturbations. The effect of near ${ \mathcal P }{ \mathcal T }$-symmetric potentials is discussed with relevant figures illustrated, and the adiabatic change of some parameters is considered. Finally, the conclusions are given in section 4.

2. Analytical solutions of equation (1)

The analytical solutions of equation (1) are derived by the Hirota bilinear method. Through variable transformation
$\begin{eqnarray}u(x,t)=\displaystyle \frac{G(x,t)}{F(x,t)},\end{eqnarray}$
with the real function F and complex G, and the constraint αγ = 3σβ, the bilinear equations of equation (1) are written as
$\begin{eqnarray}\begin{array}{l}\left[\,{\rm{i}}{D}_{t}+\alpha {D}_{x}^{2}+U+\,{\rm{i}}\sigma {D}_{x}^{3}+\,{\rm{i}}\rho {D}_{x}\right]G\cdot F\,=\,0,\\ \quad \alpha {D}_{x}^{2}F\cdot F-\beta | G{| }^{2}\,=\,0.\end{array}\end{eqnarray}$
The Hirota operator is defined by [44]
$\begin{eqnarray}\begin{array}{l}{D}_{x}^{m}{D}_{t}^{n}G(x,t)\cdot F(x,t)=\displaystyle \frac{{\partial }^{m}}{\partial {y}^{m}}\\ \quad \times \displaystyle \frac{{\partial }^{n}}{\partial {s}^{n}}G(x+y,t+s)F(x-y,t-s){| }_{y\,=\,0,s\,=\,0}.\end{array}\end{eqnarray}$
We expand G and F in power series of ε as
$\begin{eqnarray}\begin{array}{rcl}G & = & \epsilon {G}_{1}+{\epsilon }^{3}{G}_{3}+{\epsilon }^{5}{G}_{5}+\cdot \cdot \cdot ,\\ F & = & 1+{\epsilon }^{2}{F}_{2}+{\epsilon }^{4}{F}_{4}+{\epsilon }^{6}{F}_{6}+\cdot \cdot \cdot ,\end{array}\end{eqnarray}$
where ε is a small parameter, Gi(i = 1, 3, 5, …) and Fj(j = 2, 4, 6, …) are functions of x and t to be determined.
In this section, we study two cases of constant and variable coefficients. For the sake of calculation, we set ε = 1. The analytical expression of a single soliton solution for equation (1) is
$\begin{eqnarray}u(x,t)=\displaystyle \frac{{G}_{1}}{1+{F}_{2}}.\end{eqnarray}$
Case 1:
Under the constraints β1 = c0α1, β2 = c0α2, $W=-{k}_{1}^{2}{\alpha }_{2}$, where αi(i = 1, 2), c0, k1 are real constants, we substitute equation (5) into equation (3) and collect the coefficients of ε with the same power. Then we can get
$\begin{eqnarray}\begin{array}{rcl}{G}_{1}(x,t) & = & ({m}_{1}+\,{\rm{i}}{m}_{2}){{\rm{e}}}^{{k}_{1}x+({w}_{1}+\,{\rm{i}}{w}_{2})t},\\ {F}_{2}(x,t) & = & {A}_{1}{{\rm{e}}}^{2{k}_{1}x\,+\,2{w}_{1}t},\\ {w}_{1} & = & -{k}_{1}\rho -{k}_{1}^{3}\sigma ,\\ {w}_{2} & = & V+{k}_{1}^{2}{\alpha }_{1},\\ {A}_{1} & = & \displaystyle \frac{{c}_{0}({m}_{1}^{2}+{m}_{2}^{2})}{8{k}_{1}^{2}},\end{array}\end{eqnarray}$
i.e. the soliton solution of equation (1) with constant coefficients, where mi, wi(i = 1, 2), ρ, σ, V, W are real constants.
Case 2:
Similarly, we set β1(x, t) = 2c0(t)α1(x), β2(x, t) = 2c0(t)α2(x), ${c}_{0}(t)={c}_{1}{{\rm{e}}}^{\int 2{W}_{1}(t)\,{\rm{d}}t}$, $V(x,t)=-{k}_{1}^{2}{\alpha }_{1}(x)+{V}_{1}(t)$, $W(x,t)\,=-{k}_{1}^{2}{\alpha }_{2}(x)+{W}_{1}(t)$ and derive
$\begin{eqnarray}\begin{array}{rcl}{G}_{1}(x,t) & = & ({m}_{1}+\,{\rm{i}}{m}_{2}){A}_{1}(t){{\rm{e}}}^{{k}_{1}x+{w}_{1}(t)+\,{\rm{i}}{w}_{2}(t)},\\ {F}_{2}(x,t) & = & \displaystyle \frac{{m}_{1}^{2}+{m}_{2}^{2}}{4{k}_{1}^{2}}{{\rm{e}}}^{2{k}_{1}x\,+\,2{w}_{1}(t)},\\ {A}_{1}(t) & = & {{\rm{e}}}^{\displaystyle \int -{W}_{1}(t)\,{\rm{d}}t},\\ {w}_{1}(t) & = & \displaystyle \int [-{k}_{1}\rho (t)-{k}_{1}^{3}\sigma (t)]\,{\rm{d}}t,\\ {w}_{2}(t) & = & \displaystyle \int {V}_{1}(t)\,{\rm{d}}t,\end{array}\end{eqnarray}$
where mi (i = 1, 2), k1 are real constants. Substituting Expressions (8) into (6), we get the analytical soliton solution likewise.
By modulating dispersion and gain or loss terms, we illustrate their effect of them on the structure and propagation of soliton in figure 1. In figure 1(a), when σ, ρ and W1(t) are chosen as sine functions, the amplitude of the soliton varies with time periodically. Once the dispersion terms are taken as aperiodic functions like exponential functions, the amplitude is still periodic except for a phase shift around t = 0. Obviously, the periodicity of amplitude is only related to the gain or loss term and the dispersion terms affect the structures. As shown in figure 1(c), the value of k1 is adjusted. When the value of k1 reduces to 0.5, the maximum amplitude decreases and the structure of soliton has changed.
Figure 1. Structures of soliton solution with variable coefficients. Parameters are chosen as: (a) ${W}_{1}(t)=0.2\sin (0.5t),\rho (t)=\sigma (t)=\sin t$, (b) ${W}_{1}(t)=0.2\sin (0.5t),\rho (t)=\sigma (t)={{\rm{e}}}^{-{t}^{2}}$, (c) ${W}_{1}(t)=0.2\sin (0.5t),\rho (t)=\sigma (t)=\sin t,{k}_{1}\,=\,0.5$ and other parameters are fixed as 1.

3. Numerical simulations of equation (1)

The stability of solitary wave solutions plays a crucial role in practical applications. Due to the non-integrability of equation (1), the stability of solitary waves propagating in non-Kerr nonlinear media can not be guaranteed [9]. So the stability will be tested in this section via numerical simulations with a perturbation of 5% initial random noise. Furthermore, the modified squared-operator and pseudospectral methods are used in the numerical simulations [45]. Under near ${ \mathcal P }{ \mathcal T }$-symmetric potential, the equation (1) can be rewritten as
$\begin{eqnarray}\begin{array}{l}\,{\rm{i}}{u}_{t}+({\alpha }_{1}+{\rm{i}}{\alpha }_{2}){u}_{{xx}}+[V(x)+\,{\rm{i}}W(x)]u\\ \quad +({\beta }_{1}+\,{\rm{i}}{\beta }_{2})| u{| }^{2}u+\,{\rm{i}}\sigma {u}_{{xxx}}\\ \quad +\,{\rm{i}}\gamma | u{| }^{2}{u}_{x}+\,{\rm{i}}\rho {u}_{x}=0,\end{array}\end{eqnarray}$
where σ, γ, ρ, αi, βi (i = 1, 2) are real constants and V + iW denotes the near ${ \mathcal P }{ \mathcal T }$-symmetric potential.
The nonlinear mode of equation (9) can be defined as
$\begin{eqnarray}u(x,t)=\phi (x){{\rm{e}}}^{-\,{\rm{i}}\mu t},\end{eqnarray}$
where μ is a real propagation constant. We will first study the stability of nonlinear modes with the effect of the last three terms ignored, and reduce equation (9) to the second-order GGLE:
$\begin{eqnarray}\begin{array}{l}\,{\rm{i}}{u}_{t}+({\alpha }_{1}+\,{\rm{i}}{\alpha }_{2}){u}_{{xx}}+[V(x)\\ \quad +\,{\rm{i}}W(x)]u+({\beta }_{1}+\,{\rm{i}}{\beta }_{2})| u{| }^{2}u\,=\,0.\end{array}\end{eqnarray}$
In the last subsection, these parameters will be considered again through adiabatical excitation of them.

3.1. Nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential

We introduce the near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential [32]
$\begin{eqnarray}\begin{array}{rcl}V(x) & = & {V}_{0}{{\rm{sech}} }^{2}(x),\\ W(x) & = & {W}_{0}{\rm{sech}} (x)\tanh (x)-{W}_{1}{{\rm{sech}} }^{2}(x),\end{array}\end{eqnarray}$
where the value of real constants V0, W0 and W1 can be modulated to obtain stable nonlinear modes.
The power of nonlinear mode is defined as $P\,={\int }_{-\infty }^{+\infty }| \phi (x,t){| }^{2}\,{\rm{d}}x$. Figure 2(a) shows the result that stable evolution does not exist when the value of W1 approaches zero because the potential turns into ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. The power decreases obviously with increasing the value of β2, but α2 has less effect on the power. The two curves with different values of α2 intersect at W1 = 2.8. Symmetric curves with respect to W0 = 0 are shown in figure 2(b). Moreover, they attain the lowest power at the point W0 = 0 simultaneously. By the change of V0 or W1, the lowest power can be adjusted.
Figure 2. Effect of parameters on the power of nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. (a) α1 = β1 = V0 = W0 = 1, (b) α1 = β1 = β2 = 1 and α2 = −1.
The stable nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential are shown in figures 3(a) and 3(d) with 5% initial perturbations. Increasing the value of W1 to 5, the amplitude becomes larger and the nonlinear mode has a narrower width. That is to say, the energy becomes more concentrated than before. At the same time, the imaginary part of the nonlinear mode takes up a larger proportion.
Figure 3. Stable evolution of nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. (a), (b), (c) W1 = 2, (d), (e), (f) W1 = 5. α2 = −4 and other parameters are fixed as 1.

3.2. Nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric δ-signum potential

Equation (11) with near ${ \mathcal P }{ \mathcal T }$-symmetric δ-signum potential is discussed as follows. The potential can be expressed as
$\begin{eqnarray}\begin{array}{rcl}V(x) & = & 2{V}_{0}\delta (x),\\ W(x) & = & {W}_{0}\mathrm{sign}(x){{\rm{e}}}^{-{V}_{0}| x| }-{W}_{1}\delta (x),\end{array}\end{eqnarray}$
where $\delta (x)={\mathrm{lim}}_{a\to {0}^{+}}g(x;a)$, $g(x;a)=\tfrac{{{\rm{e}}}^{-{x}^{2}/{a}^{2}}}{a\sqrt{\pi }}$, and a is set as 0.01 for calculation expediently [46]. It is found that the two curves with different values of α2 are nearly parallel to each other in figure 4(a). Figure 4(b) illustrates the effect of near ${ \mathcal P }{ \mathcal T }$-symmetric δ-signum potential. When V0 = 0.4, the nonlinear modes exist even if W1 = 0. At the points in blue solid and red dashed curves, the potential reduces to ${ \mathcal P }{ \mathcal T }$-symmetric δ-signum potential, which satisfies that V(x) = V(−x) and W(−x) = −W(x).
Figure 4. Effect of parameters on the power of nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric δ-signum potential. (a) α1 = β1 = V0 = W0 = 1, (b) α1 = β1 = β2 = 1 and α2 = −1.
Next, we consider the evolution of nonlinear modes with the potential. In the numerical simulations, 5% initial random noise is added likewise. Figures 5(a), (d) and (g) show the stable evolution of peakons, while W0 affects the amplitude and period of oscillation. In contrast to figure 5(a), the peakon maintains a certain value and does not oscillate when the value of V0 increases to 1 in figure 5(g).
Figure 5. Stable evolution of nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric δ-signum potential. (a), (b), (c) V0 = 0.4, W0 = 2, (d), (e), (f) V0 = 0.4, W0 = 1.5, (g), (h), (i) V0 = 1, W0 = 2. Other parameters are α1 = β1 = β2 = 1, α2 = −1, W1 = 5.

3.3. Adiabatic excitation and evolution of the nonlinear modes

The adiabatic change of parameters in equation (9) with near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential will be considered. The ‘switch-on' function in [47] is used so that the parameters can be smoothly adjusted:
$\begin{eqnarray}\xi (t)=\left\{\begin{array}{ll}{\xi }^{(\mathrm{ini})}, & t\,=\,0,\\ {\xi }^{(\mathrm{ini})}+\displaystyle \frac{{\xi }^{(\mathrm{end})}-{\xi }^{(\mathrm{ini})}}{2}\Space{0ex}{0.25ex}{0ex}[1+\sin \left(\displaystyle \frac{\pi }{500}t-\displaystyle \frac{\pi }{2}\right)\Space{0ex}{0.25ex}{0ex}], & 0\lt t\lt 500,\\ {\xi }^{(\mathrm{end})}, & 500\leqslant t\leqslant 1500.\end{array}\right.\end{eqnarray}$
The process can be divided into two stages: the excitation stage (0 < t < 500) and the propagation stage (500 ≤ t ≤1500). In the excitation stage, ξ changes constantly from ξ(ini) to ξ(end) and remains unchanged in the propagation stage. We generate the parameters σ, γ, ρ, V0, W0 and W1 by ξ(t). When ξ(ini) = ξ(end), the function ξ(t) turns into a constant. Figures 6(a), (c) and (e) display the stable excitation and evolution of the nonlinear modes. With σ, γ and ρ changing from 0 to 1, the amplitudes of nonlinear modes are all decreasing. Then we excite W0 and W1 simultaneously to meet ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in figure 6(e). In contrast with figure 6(a), the initial condition is the same and the final state changes greatly. In addition, the amplitude changes rapidly during the excitation stage for a short time.
Figure 6. Adiabatic excitation and evolution of the nonlinear modes under near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. (a, b) W0 = 0, W1 = 1, (c), (d) W0 = 0, W1 = 0.5, (e), (f) ${W}_{0}^{(\mathrm{ini})}=0$, ${W}_{0}^{(\mathrm{end})}=2$, ${W}_{1}^{(\mathrm{ini})}=1$, ${W}_{1}^{(\mathrm{end})}=0$ and other parameters are α2 = −1, α1 = β1 = β2 = V0 = 1, σ(ini) = γ(ini) = ρ(ini) = 0, σ(end) = γ(end) = ρ(end) = 1.

4. Conclusions

In this paper, we study the higher-order GGLE, i.e. equation (1), with variable parameters and near ${ \mathcal P }{ \mathcal T }$-symmetric potentials. Under some constraints, the analytical solutions of equation (1) have been derived by the Hirota bilinear method. And several structures of solitons have also been illustrated in figures by the modulation of corresponding parameters. With the near ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II and δ-signum potentials introduced, stability of the nonlinear modes is proved via numerical simulations. Through the process of adiabatic excitation, stable nonlinear modes are also displayed. The results obtained might advance further investigations on generalized Ginzburg–Landau models by means of analytical and numerical methods. These new findings of nonlinear modes in the generalized Ginzburg–Landau model might be potentially applied to hydrodynamics, optics and matter waves in Bose–Einstein condensates.

We express our sincere thanks to all the members of our discussion group for their valuable comments.

The authors declare no conflicts of interest.

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