1. Introduction
A considerable amount of research work often gives rise to the nonlinear Schrödinger (NLS) equations with a variety of nonlinearities. Several methods, numerical and analytical, have been effectively used to handle these problems. The extended Jacobian elliptic function method [1], extended tanh-function method [2], Hirota method [3], bifunction method [4] provide examples of the methods used to analyze these problems.
Nowadays, it is well known that Hamiltonian systems can have entirely real spectrums although their Hamiltonians are non-Hermitian [5, 6]. They just need to admit ${ \mathcal P }{ \mathcal T }$-symmetry. Here, the linear space-reflection operator ${ \mathcal P }$ is defined by the relations ${ \mathcal P }$: $\left({\rm{i}},x,p\right)\to \left({\rm{i}},-x,-p\right)$ whereas that of the time-reversal operator ${ \mathcal T }$ by ${ \mathcal T }$: $\left({\rm{i}},x,p\right)\to \left(-{\rm{i}},x,-p\right)$, where x and p represent position and momentum operators, respectively and i is the imaginary unit. In that way, the ${ \mathcal P }{ \mathcal T }$ symmetricity may be regarded as a good property easily used to seek for the parameter domains of a unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase for the non-Hermitian Hamiltonians. We emphasize that the potential V(x) must satisfy V(x) = V*(−x), which implies that the real part of the potential is an even function, whereas the imaginary part is odd [7]. ${ \mathcal P }{ \mathcal T }$-symmetric structure has been easily realized in optics by placing symmetrically gain and loss elements [8], which was immediately followed by the experimental implementation in optical and atomic contexts [9], including synthetic photonic lattices, metamaterials, microring lasers, microresonators, and Bose–Einstein condensates [10–14]. Various ${ \mathcal P }{ \mathcal T }$-symmetric potentials have been introduced in optical systems, describe by a class of NLS equations [15–27]. For instance, considering the Kerr nonlinear media, one can find bright solitons [18], dark solitons and vortices [28], gray solitons [29], nonlocal Gap solitons [30]. Some solutions are also available in multi-dimensional ${ \mathcal P }{ \mathcal T }$-symmetric potentials [31–36].
Besides the cubic nonlinearities, power-law nonlinearity has been also proposed to help the stable evolution of spatial solitons [36–38]. In this way, stable localized spatial solitons in the power-law nonlinear media with ${ \mathcal P }{ \mathcal T }$-symmetric Scarf potential [36], Rosen–Morse potential [36], and Gaussian potential [37] have been discussed. The generalized (n+1)-dimensional NLS equation with power-law nonlinearity [39], cubic and power-law nonlinearities in ${ \mathcal P }{ \mathcal T }$-symmetric potentials [40] were also studied. It is well established that the localized mode excitations are always unstable in the case of Rosen–Morse potentials. However, in the case of Scarf and Gaussian potentials, the sech-type and Gaussian-type solutions are both stable below some thresholds for the imaginary part of ${ \mathcal P }{ \mathcal T }$-symmetric potentials in the focusing power-law nonlinear medium, while they are always unstable in the defocusing power-law nonlinear medium.
Most early works concentrated on the formation of solitonic solution under the exact balance between ${ \mathcal P }{ \mathcal T }$-symmetric potentials with a second order dispersion term and different order nonlinearities. However, higher-order dispersion/diffraction must be considered when the width of an optical beam is narrow [41–43]. Because of the higher-order diffraction, it is difficult to find exact analytical solutions of NLS equations and they require some kinds of techniques. Considering the effect of gain and loss, ${ \mathcal P }{ \mathcal T }$-symmetric configurations with higher-order diffractions have also been reported in [44–46], and the localized solutions are generally obtained with the aid of numerical techniques. Although the fourth order dispersion (FOD) was introduced to discuss the formation of soliton under ${ \mathcal P }{ \mathcal T }$-symmetric potentials in different nonlinear media [47–52], exact soliton solutions in ${ \mathcal P }{ \mathcal T }$-symmetric structures with the FOD and power-law nonlinearities are hardly reported.
In this paper, we investigate analytically and numerically the existence and the stability of 1D and 2D Gaussian-type soliton in power-law media with fourth-order diffraction and quartic Gaussian-type ${ \mathcal P }{ \mathcal T }$-symmetric potential. It emerges that the increase of the order of the nonlinearity parameter m leads to expansion of the stability region in the case of defocusing nonlinearity and reduction of the stability region in the other case. Therefore, one can just turn stable solitonic solutions into unstable ones, and vice versa. This assertion is confirmed by numerics.
2. Localized mode excitations IN 1D ${ \mathcal P }{ \mathcal T }$-symmetric complex potential
2.1. Model and general theory
Dynamic of the dimensionless light field envelope $\Psi$(z, x) propagating along the z-direction in a power-law nonlinear medium with FOD and quartic Gaussian-type ${ \mathcal P }{ \mathcal T }$-symmetric potential can be governed by the following partial differential equation:1 ) with a periodic potential, Kerr nonlinearity and FOD coefficient [59]. When β = 0 and m = 1, analytical stable Gaussian soliton supported by a ${ \mathcal P }{ \mathcal T }$-symmetric potential with power-law nonlinearity has been obtained in [37]. Sech-type localized mode solutions of equation (1 ) in the presence of complex hyperbolic ${ \mathcal P }{ \mathcal T }$-symmetric potentials have also been reported [60]. It is worth noting that three-dimensional optical solitons formed by the balance between different-order nonlinearities such as cubic, quintic, septimal nonlinearities, and FOD in ${ \mathcal P }{ \mathcal T }$-symmetric potentials have been also investigated [61].
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial \psi }{\partial z}+\displaystyle \frac{{\partial }^{2}\psi }{\partial {x}^{2}}-\beta \displaystyle \frac{{\partial }^{4}\psi }{\partial {x}^{4}}\\ \quad +\,[V(x)+{\rm{i}}W(x)]\psi +g| \psi {| }^{2m}\psi =0,\end{array}\end{eqnarray}$
where x is the transverse coordinate [37, 47]. The coefficient β is the coupling constant of the FOD, which can appear in a waveguide carrying a photonic crystal structure to modify the simple paraxial form of the diffraction [53]. The coefficients of the cubic term g = ±1 represent self-focusing and self-defocusing nonlinearities, respectively. m is an arbitrary non-zero constant. For m = 1, 2, and 3, one has the Kerr, quintic, and septic nonlinearity, respectively. On the other hand there are other models which admit non-integer nonlinearity. In fact, power-law with non-integer exponents are used for instance to quantify stress–strain relations of materials with pronouncedly non-Hookean behavior [54–56]. From a circuit realization standpoint, a fractional-order circuit is obtained using fractional capacitors [57, 58]. V(x) and W(x) are the real and imaginary parts of the complex ${ \mathcal P }{ \mathcal T }$-symmetric potential such that V(−x) = V(x) and W(−x) = −W(x). Physically, V(x) is responsible for the bending and slowing down of light, and W(x) can lead to either amplification (gain) or absorption (loss) of light within an optical material. In the absence of gain- and loss-distribution W(x), band gaps and lattice solitons are investigated in equation (In the following, we consider ${ \mathcal P }{ \mathcal T }$-symmetric potential, V(x) and W(x), as the quartic anharmonic potential [48, 62, 63] and Gaussian gain-or-loss distribution in the forms:1 ) in the form $\Psi$(z, x) = φ(x)eiμz, where μ is the real propagation constant and the complex function φ(x) satisfies the stationary power-law NLS equation:2 ), the localized boundary conditions $\phi \left(x\to \pm \infty \right)=0$ leads to the solution of equation (3 ) in the form
$\begin{eqnarray}\begin{array}{rcl}V(x) & = & {V}_{0}{x}^{2}+{V}_{1}\exp (-2{a}^{2}{x}^{2})\\ & & +\,{V}_{2}{x}^{4}+{V}_{3}{x}^{2}\exp (-2{a}^{2}{x}^{2})\\ & & +{V}_{4}\exp (-4{a}^{2}{x}^{2}),\\ W(x) & = & {W}_{0}x\exp (-{a}^{2}{x}^{2})+{W}_{1}x\exp (-3{a}^{2}{x}^{2})\\ & & +\,{W}_{2}{x}^{3}\exp (-{a}^{2}{x}^{2}),\end{array}\end{eqnarray}$
where the real parameters V0, V1, V2, V3, V4, W0, W1, and W2 determine the depth of the potential. The real constant a represents the width of the potential. We here deal with the problem of determining the stationary solutions of equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}}^{2}\phi (x)}{{\rm{d}}{x}^{2}}-\beta \displaystyle \frac{{{\rm{d}}}^{4}\phi (x)}{{\rm{d}}{x}^{4}}\\ \ \ \ +\,[V(x)+{\rm{i}}W(x)]\phi (x)+g| \phi {| }^{2m}\phi (x)=\mu \phi (x).\end{array}\end{eqnarray}$
For the quartic anharmonic Gaussian ${ \mathcal P }{ \mathcal T }$-symmetric potential ( $\begin{eqnarray}\phi (x)={\phi }_{0}\exp \left(-\displaystyle \frac{{a}^{2}{x}^{2}}{m}\right)\exp \left[{\rm{i}}\displaystyle \frac{A\sqrt{\pi }}{2a}\mathrm{erf}({ax})\right],\end{eqnarray}$
where erf(x) denotes the error function, and
$\begin{eqnarray}\begin{array}{rcl}{V}_{0} & = & -\displaystyle \frac{4{a}^{4}}{{m}^{2}}-\displaystyle \frac{48{a}^{6}}{{m}^{3}}\beta ,\quad {V}_{2}=\displaystyle \frac{16{a}^{8}}{{m}^{4}}\beta ,\\ {V}_{3} & = & -28{A}^{2}{a}^{4}\beta \left(\displaystyle \frac{12}{7m}+\displaystyle \frac{6}{7{m}^{2}}+1\right),\\ {V}_{4} & = & {A}^{4}\beta ,\quad {W}_{1}=12{A}^{3}{a}^{2}\beta \left(1+\displaystyle \frac{2}{3m}\right),\\ {W}_{2} & = & -\displaystyle \frac{12{{Aa}}^{6}}{{m}^{2}}\beta \left(\displaystyle \frac{2{m}^{2}}{3}+\displaystyle \frac{8m}{3}+\displaystyle \frac{8}{3m}+4\right).\end{array}\end{eqnarray}$
The propagation constant and the wave amplitude read $\begin{eqnarray}\begin{array}{rcl}\mu & = & -\displaystyle \frac{2{a}^{2}}{m}-12{a}^{4}{m}^{2}\beta ,\\ {\phi }_{0} & = & {\left(\displaystyle \frac{{A}^{2}\left[1+28\left(\tfrac{2}{7}+\tfrac{3}{7m}\right)\beta \right]-{V}_{1}}{g}\right)}^{\tfrac{1}{2m}},\end{array}\end{eqnarray}$
respectively. We notice the existence of the following condition $\begin{eqnarray*}g\left[{A}^{2}\left[1+28\left(\displaystyle \frac{2}{7}+\displaystyle \frac{3}{7m}\right)\beta \right]-{V}_{1}\right]\gt 0,\end{eqnarray*}$
where $\begin{eqnarray}\begin{array}{rcl}A & = & \displaystyle \frac{{{mW}}_{0}}{2{a}^{2}\left[6{a}^{2}\beta \left(m+\tfrac{10}{3}+\tfrac{4}{m}\right)+\left(m+2\right)\right]},\\ \mathrm{and}\ m & \ne & \displaystyle \frac{-\left(2+20{a}^{2}\beta \right)\pm \sqrt{{\left(20{a}^{2}\beta +2\right)}^{2}-4\left(6{a}^{2}\beta +1\right)\left(24{a}^{2}\beta \right)}}{2\left(6{a}^{2}\beta +1\right)}.\end{array}\end{eqnarray}$
which restricted the FOD parameter to the interval $\begin{eqnarray*}\left[\displaystyle \frac{16{a}^{2}-\sqrt{{\left(16{a}^{2}\right)}^{2}+2816{a}^{4}}}{352{a}^{4}}\quad \displaystyle \frac{16{a}^{2}+\sqrt{{\left(16{a}^{2}\right)}^{2}+2816{a}^{4}}}{352{a}^{4}}\right].\end{eqnarray*}$
We found from equations (5 ) and (6 ) that the order of the nonlinearity coefficient influences the potential parameters and consequently the soliton parameters of the solution. For β = 0, we recover the well known harmonic Gaussian potential [37]. Anharmonic potentials are linked to a perturbation [64, 65]. Moreover, anharmonics potential for the magneto-optical traps can be characterized by periodic potential.
2.2. Nonlinear modes, analytical and numerical Gaussian soliton: dynamic and stability
In order to study the stability of Gauss-type soliton solutions and investigate the effect of the the order of the nonlinearity parameter, we first use the eigenvalue method, to give the linear stability analysis of the solution (4 ), and with the help of numerical simulations namely the split step Fourier method, we check the stable and unstable regions. We then consider small perturbation to the solution $\Psi$(x, z), in the form [20–22]8 ) into equation (1 ), and linearizing with respect to ε, we obtain the following linear eigenvalue problem:9 ) can be solved numerically with the help of Fourier collocation method [66].
$\begin{eqnarray}\psi (x,z)=\left\{\phi (x)+\epsilon \left[F(x){{\rm{e}}}^{{\rm{i}}\delta z}+{G}^{* }(x){{\rm{e}}}^{-{\rm{i}}{\delta }^{* }z}\right]\right\}{{\rm{e}}}^{{\rm{i}}\mu z},\end{eqnarray}$
where ε ≪ 1, F(x) and G(x) are the perturbation eigenfunctions of the linearized eigenvalue problem and δ measures the growth rate of the perturbation instability. Substituting equation ( $\begin{eqnarray}\left[\begin{array}{cc}{L}_{1} & {L}_{2}\\ -{L}_{2}^{* } & -{L}_{1}^{* }\end{array}\right]\left[\begin{array}{c}F\\ G\end{array}\right]=\delta \left[\begin{array}{c}F\\ G\end{array}\right],\end{eqnarray}$
where L1 = −μ + ∂xx−β∂xxxx + V(x) + iW(x) + $(m+1)g| \phi {| }^{2m}$, ${L}_{2}={mg}| \phi {| }^{2m-2}{\phi }^{2}$. The ${ \mathcal P }{ \mathcal T }$-symmetric nonlinear modes are linearly stable if δ has no imaginary part, otherwise they are linearly unstable. The eigenvalue problem (In figure 1, we present stable and unstable regions of some peculiar parameters m for self-focusing $\left(g=1,{V}_{1}=-0.02\right)$ and self-defocusing $\left(g=-1,{V}_{1}=0.02\right)$ nonlinearities when other parameters are chosen as W0 = 0.01 and a = 0.5. In the case of self-focusing nonlinearity (see figure 1(a)), the instability region increased with m for β < 0 than the case of β > 0 and the solution exists for all the values of β and m. For the case of self-defocusing nonlinearity illustrated in figure 1(b) the stability region increased with m for β < 0 than the case of β > 0 and the solution does not exist for all the values of β and m. The linear stability eigenspectra for different values of m when the FOD parameter is set as β = 0.08 is depicted in figures 3(a1), (b1) for the focusing nonlinearity and figures 2(a1), (b1) for the defocusing nonlinearity, respectively. These figures implies that for the order of nonlinearity m chosen in the region of stability, the localized mode solution (4 ) is stable, while for the one chosen in the region of instability, it remains unstable.
Figure 1. Stable (black) and unstable (red) regions of solution ( |
Figure 2. (a1), (b1) Eigenvalues of solution ( |
Based on the linear stability analysis, we know the stable domains of analytical solution (4 ) with order of the nonlinearity, FOD parameter and extended ${ \mathcal P }{ \mathcal T }$-symmetric potential (2 ). To extend the stability analysis beyond the linear regime we resort to direct numerical simulations. To this end, we discuss numerically the stability of the solution (4 ) against a perturbation of 1% white noise by using a split-step Fourier beam technique, and monitor its evolution using equation (1 ). The evolution of the stable and unstable 1D Gaussian-type soliton solutions (4 ) with ${ \mathcal P }{ \mathcal T }$-symmetric potential (2 ) are shown in figures 2(a2), (b2) (defocusing nonlinearity) and figures 3(a2), (b2) (focusing nonlinearity). From figures 2(a2) and 3(a2), the noise perturbation seems to be completely washed out and the input beam persists for a relatively long distance (z = 1000). Thus, one could conclude that the order of nonlinearity m does not seem to cause any drastic deviation regarding the stability properties of the Gaussian solitons. In the case of figures 2(b2) and 3(b2), the white noise influences the background of soliton and a break up of the Gaussian soliton occurs at an early stage of its evolution. These results are different with the β = 0 case where Gaussian solitons were found to be linearly and nonlinearly stable only in the case of the defocusing nonlinearity [37].
Figure 3. (a1), (b1) Eigenvalues of solution ( |
We now deal with the case where the propagation constant does match with the one obtain by the analytical treatment. We numerically solve equation (3 ) and address the dependence of the propagation constant μ, of fundamental solitons on the integral power P, and the largest instability growth rate max [Im(δ)], for different values of the order of nonlinearity m. From figures 4(a1), (b1), we can see that the μ(P) curves originate at P = 0, and the slope of the dependence is positive and negative respectively for the focusing (see figure 4(a1)) and defocusing (see figure 4(b1)) case of nonlinearity. The order of nonlinearity slows the variation of μ with the growth of P, for both the focusing and defocusing signs of the nonlinearity. From figures 4(a2) one can see that the order of nonlinearity enhance the growth rate. However, in the case of defocusing nonlinearity shown in figure 4(b2), the instability appears with m = 1 and m = 2 only in tiny intervals of the propagation constant μ, then disappears with other values of m. We thus conclude that the order of nonlinearity m plays an important role in controlling both the growth rate and stability region of the fundamental solitons. To confirm the results of the linear stability analysis, we performed the propagation of stationary solutions by numerically simulating equation (3 ), in which the fundamental solitons are perturbed by a 1% random noise. The corresponding soliton profiles are summarized in figures 4(a3), for the focusing case, and figures 4(b3), for the defocusing case where we can find that the fundamental solitons can propagate robustly by the help of accelerated imaginary-time evolution methods [66].
Figure 4. The power P and linear stability of the numerical soliton versus the propagation constant μ for different values of the order of nonlinearity m in the case of focusing (a1), (a2) with V1 = −0.01 and defocusing (b1), (b2) with V1 = 0.01. The propagation dynamic of fundamental soliton with m = 3, μ = −4.85 for the focusing (a3) case and m = 3, μ = −5.11 for the defocusing (b3) case. The other parameters are V0 = −8, V2 = −1, V3 = 5, V4 = −4, W0 = 0.01, W1 = 1, W2 = 0.1, a = 0.5, β = −0.2. |
3. Localized modes in 2D ${ \mathcal P }{ \mathcal T }$-symmetric complex potential
We now investigate the 2D power-law NLS equation with a quartic Gaussian ${ \mathcal P }{ \mathcal T }$-symmetric potential and a FOD coefficient [67]:5 ), ${V}_{5}=-{A}^{2}\left[1+28\left(\tfrac{2}{7}+\tfrac{3}{7m}\right)\beta \right]$ and ${V}_{6}=\beta {A}^{4}{m}^{4}$.
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\displaystyle \frac{\partial \psi }{\partial z}+{\rm{\Delta }}\psi -\beta {{\rm{\Delta }}}^{2}\psi \\ \quad +\,[V(x,y)+{\rm{i}}W(x,y)]\psi +g| \psi {| }^{2m}\psi =0,\end{array}\end{eqnarray}$
where $\Psi$ = $\Psi$(x, y, z) is a complex field envelope function, Δ is the two-dimensional Laplacian in the (x, y) plane, and Δ2 is the so-called bi-Laplacian. The two-dimensional complex potential, which obey the ${ \mathcal P }{ \mathcal T }$-symmetric requirements V(−x, −y) = V(x, y) and W(−x, −y) = −W(x, y) are considered as $\begin{eqnarray}\begin{array}{rcl}V & = & {V}_{0}({x}^{2}+{y}^{2})-{V}_{1}{{\rm{e}}}^{-2{a}^{2}({x}^{2}+{y}^{2})}\\ & & +\,{V}_{2}({x}^{4}+{y}^{4})+{V}_{3}({x}^{2}{{\rm{e}}}^{-2{a}^{2}{x}^{2}}+{y}^{2}{{\rm{e}}}^{-2{a}^{2}{y}^{2}})\\ & & +{V}_{4}{{\rm{e}}}^{-4{a}^{2}({x}^{2}+{y}^{2})}-{V}_{5}({{\rm{e}}}^{-2{a}^{2}{x}^{2}}+{{\rm{e}}}^{-2{a}^{2}{y}^{2}})\\ & & -\,{V}_{6}({{\rm{e}}}^{-4{a}^{2}{x}^{2}}+{{\rm{e}}}^{-4{a}^{2}{y}^{2}})\\ W & = & {W}_{0}(x{{\rm{e}}}^{-{a}^{2}{x}^{2}}+y{{\rm{e}}}^{-{a}^{2}{y}^{2}})\\ & & +\,{W}_{1}(x{{\rm{e}}}^{-3{a}^{2}{x}^{2}}+y{{\rm{e}}}^{-3{a}^{2}{y}^{2}})+{W}_{2}({x}^{3}{{\rm{e}}}^{-{a}^{2}{x}^{2}}+{y}^{3}{{\rm{e}}}^{-{a}^{2}{y}^{2}}),\end{array}\end{eqnarray}$
where the constants V0, V1, V2, V3, V4, W0, W1, and W2 are given by equation (We find the exact analytical solutions of equation (10 ) with the order of nonlinearity m, the FOD coefficient β and the ${ \mathcal P }{ \mathcal T }$-symmetric potential (11 ) in the form
$\begin{eqnarray}\psi (x,y,z)=\phi (x,y)\exp \left({\rm{i}}\mu z+{\rm{i}}\theta (x,y\right),\end{eqnarray}$
where $\mu =2\left(-\tfrac{2{a}^{2}}{m}-12{a}^{4}{m}^{2}\beta \right)$, the phase θ(x, y) and soliton φ(x, y) are obtained as $\begin{eqnarray}\begin{array}{rcl}\phi (x,y) & = & {\left(\displaystyle \frac{{V}_{1}}{g}\right)}^{\tfrac{1}{2m}}{{\rm{e}}}^{-\tfrac{{a}^{2}({x}^{2}+{y}^{2})}{m}},\\ \theta (x,y) & = & \displaystyle \frac{A\sqrt{\pi }}{2a}\left[\mathrm{erf}({ax})+\mathrm{erf}({ay})\right].\end{array}\end{eqnarray}$
The linear stability analysis and direct numerical simulation of the localized modes, can be jointly used to discuss the stability of analytical solution (13 ) of equation (10 ). Therefore, the 2D generalization of the eigenvalue problem given in (9 ) is now considered and the results for some peculiar case of nonlinearity are shown in figures 5(a1), (b1) and figures 6(a1), (b1). From figures 5(a1) and 6(a1) obtained with m = 2, the spectrum is located entirely on the real axis and the 2D Gaussian soliton is stable. From the case presented in figures 5(b1) and 6(b1) with m = 5, non-zero imaginary parts of δ is found and the 2D Gaussian soliton is unstable.
Figure 5. (a1), (b1) Eigenvalues of solution ( |
Figure 6. (a1), (b1) Eigenvalues of solution ( |
To shed more light on the dynamics of the stability we again resort to direct numerical simulations of the solution (13 ) against a perturbation of 1% white noise. The results of numerical simulations are shown in figures 5(a2) and 6(a2) for the stable propagation, and figures 5(b2) and 6(b2) for the unstable propagation, respectively. From figures 5(a2) and 6(a2), one notices that the order of nonlinearity parameter does not alter the stability properties of the Gaussian soliton and the solution maintains a coherent structure for relatively long distances (z = 1000) and does not break up. However, in the case presented in figures 5(b2) and 6(b2), the Gaussian soliton does not resist to the white noise perturbation and one observes a break up of the solution. As in the 1D case, we would like to point out that our results are different to those obtained in the classical 2D NLS equation where it was shown that for β = 0, the 2D solitons are always unstable for self-focusing nonlinearity [37]. As in the case of one-dimensional systems, We solve numerically equation (10 ) and address the dependence of the propagation constant, μ, of fundamental solitons on the integral power, P and max[Im(δ)], for different values of the order of nonlinearity m. In figure 7(a1), we can see that in the case of focusing nonlinearity, the propagation constant growths with the soliton power and the slope of the dependence is in that case positive. The results of figure 7(a2) show that the fundamental soliton is stable in the case of lower order of nonlinearity. However, as the order of the nonlinearity increases, the instability appears and becomes more pronounced in the case of higher order of nonlinearity. In the case of defocusing nonlinearity, the results of figure 7(b1) show that the propagation constant decreases with the increase of soliton power and the slope of the dependence is in that case negative. Moreover, we see from the numerical values of max[Im(δ)] shown in figure 7(b2), that the instability is suppressed with the increasing of the nonlinearity. Based on these results, the dynamical behavior of the corresponding numerically found soliton solutions predicted to be stable in the case of focusing and defocusing power-law nonlinearities, under the action of random perturbations, with relative amplitudes 1% is depicted in figures 7(a3) and (b3), respectively.
Figure 7. The power P and linear stability of the numerical soliton versus the propagation constant μ for different values of the order of nonlinearity m in the case of focusing (a1), (a2) with V1 = 0.1 and defocusing (b1), (b2) with V1 = −0.1. The propagation dynamic of the humped numerical soliton with m = 2, μ = 12.7 for the focusing (a3), (a4) case and m = 2, μ = 12.7 for the defocusing (b3), (b4) case. The other parameters are V0 = 0.01, V2 = −0.001, V3 = 0.1, V4 = 0.1, V5 = −4, V6 = −4, W0 = 0.01, W1 = 0.01, W2 = −0.01, a = 0.5, β = −0.2. |
4. Conclusion
We have analytically and numerically studied Gaussian-type solitonic solutions in the power-law NLS equation with fourth-order diffraction characterized by quartic Gaussian-type ${ \mathcal P }{ \mathcal T }$-symmetric potential. The closed form expressions for the localized modes in such 1D and 2D self-focusing and self-defocusing nonlinear media are reported. The most essential problem is the stability of the solitons solutions. Numericals results for the stability of the solution were obtains by means of the numerical solution of the linearized equation of small perturbations, and verified by direct simulations. The increase of the order of nonlinearity parameter m leads to expansion of the stability region in the case of defocusing nonlinearity and reduction of the stability region in the other case. The results we obtained in this work provide a new way of control over soliton stability by using generalized ${ \mathcal P }{ \mathcal T }$-symmetric Gaussian potentials and FOD in different parameter domains. This may suggest new experiments for ${ \mathcal P }{ \mathcal T }$-symmetric nonlinear waves in optical media with higher-order nonlinearities and diffractions, and provide useful theoretical guidance for studies in related fields.
Eventhough we have found bright Gaussian-type soliton in our model, we emphasize that others type of soliton can be found. For instance, one can get dark soliton in 1D, and vortices in 2D. These issues will be subject of forthcoming studies.