Dynamic Behavior and Stability Analysis of Nonlinear Modes in the Fourth Order Generalized Ginzburg-Landau Model with Near PT-symmetric Potentials

We investigate the fourth order generalized Ginzburg-Landau equation (GGLE) which can be widely used in hydrodynamics and nonlinear optics. There are three major ingredients. Firstly, bilinear form of the fourth order GGLE is obtained by means of Hirota method. Then we get the analytical soliton solution with di ﬀ erent dynamic behaviors in two cases, corresponding to the constant and variable coe ﬃ cients respectively. Secondly, some numerical simulations with near PT -symmetric potentials are carried out. When the potential is closer to conventional PT symmetric potential, the nonlinear modes tend to unstable. But by increasing the value of imaginary part of potentials, It can be found that the amplitude of nonlinear mode is periodically oscillating. Thirdly, we consider the excitations of nonlinear modes and get some stable cases that have not been acquired in second part.

types of soliton solutions of GLE are analyzed in detail, including multi-peak solitons [11], exploding solitons [12], two-dimensional vortical solitons [13], lattice solitons [14] and peakons. Although the higher order GLEs have been investigated, nonlinear modes of them have rarely been analyzed so far. Therefore, the main purpose of this paper is to study the generalization of GLE with fourth order nonlinear dispersion by means of analytical and numerical methods.
Eq. (2) can be applied to describing propagation of light in an active dispersive medium [15]. In this case, u is the the complex envelope of the electric field, x denotes the propagation distance, and t is the retarded time. ↵ 1 is the the group velocity dispersion coefficient, and ↵ 2 describes the spectral filtering or linear parabolic gain [12]. 1 is Kerr-nonlinearity coefficient and 2 accounts for the nonlinear gain-loss processes [8]. ⇢ 1 is the parameter of the quintic nonlinearity and represents a higher order correction to the nonlinear amplification/absorption, and ⇢ 2 characterizes the saturation of the nonlinear gain and it is a possible higher order correction term to the intensity-dependent refractive index [15]. represents the effect due to discreteness and higher order magnetic interactions [16]. V is related to the refractive index waveguide, and W characterizes the amplification (gain) or absorption (loss) of light beam in the optical material [8,17].
There are three special cases of Eq. (2) that should be mentioned: (1)When ↵, , are constants and the other coefficients are zero, Eq. (2) is degenerated to Eq. (1). The stable optical soliton can be gotten by regulating the group velocity dispersion and nonlinear gain-loss coefficient [38]. Moreover, many kinds of analytical coherent structure solutions of this equation have been studied [39].
(2) When ↵ = =1 , (x, t)=V (x)+iW (x) and the other coefficients are zero, Eq. (2) can be reduced to in which the beam evolution is governed by the normalized nonlinear Schrödinger-like equation. It can be used to describe the propagation of the optical soliton in a self-focusing Kerr nonlinear PT -symmetric potential [40].
(3)When ↵, are complex constants, (x, t)=V (x)+iW (x) and the other coefficients are zero, Eq. (2) can be reduced to which can describe the spatial beam transmission in a cubic-nonlinear optical medium described by the complex Ginzburg-Landau equation with complex potentials. Moreover, the stability of soliton has been analyzed via numerical simulation [8].
In Sect. 2, under some constraints of the constant and variable coefficients that will be derived, Eq. (2) will be bilinearized. Based on the bilinear forms, one-soliton solutions will be derived. In Sect. 3, we will perform numerical simulation of Eq. (2) with near PT -symmetric potentials in two cases, corresponding to the near PT -symmetric Scarf-II and -signum potentials, respectively. Then we consider excitations of solitons via making the parameters rely on the propagation distance t. In, Sect. 4, results will be summarized. To get the bilinear forms of Eq. (2), we will present the following constraints on the variable coefficients: Via the dependent-variable transformation with the real f and complex g, we can derive the bilinear forms of Eq. (2): Through Hirota bilinear method, we introduce the bilinear operator D defined as [41] D m where m, n are positive integers, a, b are functions of x and t,andy stands for the small increment. Then we expand g and f in power series of a small parameter ✏ as where g m (m =1, 3, 5, ···), f n (n =2, 4, 6, ···) are functions of x and t to be determined.

The case of constant coefficients
In this section, we assume the coefficients of Eq. (2) are constant and set where ,a n dw e can derive the constraint relation of other parameters: Figs. 1 Parameters are chosen as: Without loss of generality, we set ✏ = 1. Thus, the expression of single soliton solution can be written as In Figs. 1, the propagation of the soliton along the distance x is illustrated. The velocity and direction of solitonic propagation phase will be changed if we choose different values of ⌘ 1 . Moreover, The amplitude of the soliton is determined by k 1 .

The case of various coefficients
Next, we will consider the situation that the coefficients of Eq. (2) are functions of x, t and assume Then we separate W (x, t) into two parts by W (x, t)=W 1 (x)+W 2 (t), and take 1 ( The expressions of variable coefficients can be obtained where . The analytical solution of Eq. (2) with variable coefficients can be likewise expressed as Eq. (12). In Figs. 2, we take the variable coefficient ⌘ 1 (t) as sine function. It is obvious that the soliton solution is periodic and 2 (t) is related to the amplitude. When

3A N A L Y S I SO FN U M E R I C A LS O L U T I O N S
To study the effect of fourth order nonlinear dispersion numerically, we consider the generalized case with constant coefficients and complex potentials (15) where ↵ i , i , i and ⇢ i (i =1 , 2) are all real parameters. We focus on the stationary solutions of Eq. (15) in the form: where µ is a real propagation constant. Substituting it into Eq. (15), we can obtain the complex localized field-amplitude function (x) that satisfies the ordinary differential equation: (17) In our numerical simulations, spatial differential and the integration in time are carried out by modified squared-operator method and the pseudospectral method, respectively [42]. In the following, we study the Eq. (15) under the role of the near PT -symmetric Scarf-II and -signum potentials, and find the stationary nonlinear modes of Eq. (15).

Nonlinear modes with the near PT -symmetric Scarf-II potential
The PT -symmetric complex potential V (x)+iW (x) has the features that V (x)=V (−x)a n d W (−x)=−W (x) [3]. Because of the appearance of complex coefficients, Eq. (2) is not PTsymmetric. The near PT -symmetric potentials are considered [8]. We initiate our analysis by introducing the following near PT -symmetric Scarf-II potential After setting some coefficients, we study the effects of other variables on the iterative image. The power of nonlinear mode is defined as P = R +∞ −∞ | (x, t)| 2 dx. The relationship between ⇢ 1 , ⇢ 2 , ↵ 2 , V 0 , W 1 and P is shown in the Figs. 4(a) and 4(b), in which the horizontal coordinate is W 1 ∈ (0, 6). When other parameters are specified, W 1 and P are positively correlated while ⇢ 1 , ⇢ 2 and P are negatively correlated [see Fig. 4(a)]. What's more, when ↵ 2 is different in two cases, we can make them have same power by adjusting W 1 [see Fig. 4(b)].
In addition, when W 1 =0,thenearPT -symmetric potential (18) is just the conventional PTsymmetric Scarf-II potential. With the value of W 1 decreasing, the solution will be unstable and become attenuation by the propagation of soliton. Now we consider the evolution of soliton solutions via Eq. (16). In our numerical simulations, the 5% initial random noise is added to simulate the wave transmission. With the given parameters, Fig. 4(c) displays a stable nonlinear mode. If the value of W 1 is decreased, the mode will become unstable. This is to say a little change of the gain-loss distributions can make the nonlinear mode unstable when W 1 is sufficiently small [see Fig. 4 Furthermore, It will get close to PT -symmetric Scarf-II potential by increasing the value of W 0 . In this case, the relationship between P and W 0 is illustrated in Fig. 5(a). If the initial value of W 0 =20W 1 , then the amplitude of nonlinear mode is periodically oscillating and it experiences more than 2 periods within 1450 ≤ t ≤ 1500 [see Fig. 5(b)]. This means the growth of W 0 can also change the stability of soliton with near PT -symmetric Scarf-II potential.
When other coefficients are fixed, we change W 0 to get different potentials. Let W 0 =0.1, and W approximates to an even function of x. Moreover, the soliton is stable and symmetric approximately [see Figs. 5(e) and 5(f)]. If we further increase W 0 to 20, then W is close to an odd function of x. What's more, the soliton is stable and asymmetric [see Figs. 5(b) and 5(c)], which indicates that we can get stable soliton with near PT -symmetric Scarf-II potential by increasing the value of W 0 .
In particular, for some small values of W 0 ,t h es o l i t o ni nc u b i cG L Ei su s u a l l ys t a b l ew i t h ↵ 2 ≤ 0a n d 2 ≥ 0, beyond which the soliton immediately becomes extremely unstable [8]. So we investigate the relationship between P and ↵ 2 with the fourth order magnetic interactions 2 [see Fig. 6]. When ↵ 2 > 0a n d 2 =0 ,P of the nonlinear mode changes suddenly near t =0 .Ifw efix 2 =0 .03, the curve about P and ↵ 2 of nonlinear mode will become smooth. Thus, power of the nonlinear mode can be transformed by changing 2 .

Nonlinear modes with the near PT -symmetric -signum potential
Next, we introduce the near PT -symmetric -signum potential to carry on our analysis, which plays a significant role in such fields as quantum physics, optics, and Bose-Einstein condensates [43,44].
The limit of the following Gaussian function can be used to express the function (x)= lim a→0 + g(x; a), thus we can use the Gaussian function g(x; a) with very small parameter a to approximate the function [43]. In order to facilitate calculations, without loss of generality, we choose a =0 .01 in this paper. As shown in the Figs. 7(a) and 7(b), the horizontal coordinates is W 1 ∈ (0, 6). W 1 is directly proportional to P . 2 , ⇢ 1 , ⇢ 2 are inversely proportional to P . And when W 1 is sufficiently small, the peakon solution is becoming unstable. Results of these numerical simulation are given in Figs. 7(c) and 7(e).
When the horizontal coordinates is W 0 ∈ (−5, 20), Fig. 8(a) shows the power of peakon solutions. When W 0 =0 ,P takes the minimum value. Considering W 0 as large as possible, the evolution of peakon is similar to the case under the near PT -symmetric Scarf-II potential and it experiences more than 5 periods within 1450 ≤ t ≤ 1500 [see Fig. 8(b)].
Based on the above results, we consider the effect of 2 and ⇢ 2 .I ti se a s yt os h o wt h a t 2 and ⇢ 2 are inversely proportional to P . The two stable peakon solutions are illustrated in Figs. 9(c)  Figs. 6 Parameters are chosen as: and 9(e). It can be seen that 2 are related to the morphology of peakons by changing the width of nonlinear mode.

Generalized model and excitations of solitons
In this section we consider excitations of above-mentioned solitons in Eq. (15) via adiabatical change of system parameters. We restrict our interests in the following nonlinear modes with the near PT -symmetric potentials (18) (19) and complex coefficients of Eq. (15) In order to modulate the system parameters smoothly, we consider the following "switch-on" function: where ✏ 1,2 respectively represent the real initial-state and final-state parameters and generate ↵ 2 (t), W 0 (t), W 1 (t)by✏(t). Under the synchronous modulation of system parameters, the excitation stage and the propagation stage can be described by Eq. (21), Eq. (22). During the excitation stage (0 <t<500), system parameters change slowly from ✏ 1 to ✏ 2 , and the initial state corresponding to ✏ 1 will be adiabatically driven to the new state corresponding to ✏ 2 ; during the propagation stage (500 ≤ t ≤ 1500), system parameters are maintained at ✏ 2 ,a n dt h ee x c i t e dn o n l i n e a rm o d ew i l l propagate in the final system. It should be noted that some system parameters can be constants [e.g., W 0 (x, t)=✏(t) = const.], if we set the "switch-on" function ✏(t) for ✏ 1 = ✏ 2 [45]. 14.

Figs. 11
Parameters are chosen as: We first execute a two-parameters excitation of the soliton with the near PT -symmetric Scarf-II potential. Figs. 10 shows that the the excitation of the nonlinear mode is stable with a lower amplitude with the change of W 0 and W 1 , due to both the final state and initial state are stable.
Because the solutions in Fig. 6 can not be gotten directly when ↵ 2 ≥ 0, we consider the case in this section. As can be seen, Figs. 11 shows a stable soliton solution.
Finally, we consider the near PT -symmetric -signum potential. We execute a single-parameter excitation of the peakon controlled by Eq. (21) via the initial condition determined. In Figs. 12, it can be seen that if W 01 =1andW 11 = 20, then the morphology of stable mode will change.

4C o n c l u s i o n s
In conclusion, we study the dynamic behavior and stability of nonlinear modes in the fourth order generalized GLE with near PT -symmetric potentials.
Firstly, we get the bilinear form of Eq. (2) by Hirota method. When the coefficients are constant and variable, analytical solutions and images of solitary wave solutions are obtained respectively. In constant coefficients, the velocity and direction of solitonic propagation phase will be changed if we choose different values of ⌘ 1 [see Figs. 1]. In various coefficients, the intensity of soliton is related to 2 (t) and the soliton solution is periodic when the variable coefficient ⌘ 1 (t) as sine function [see Figs. 2].
Secondly, we separately study the model of Eq. (15) with two novel categories of near PTsymmetric Scarf-II and -signum potentials, and get several stable soliton and peakon solutions. Besides, the relationships between system coefficients and the power of nonlinear modes are investigated. Roughly speaking, W 1 and P are positively correlated while ⇢ 1 , ⇢ 2 and P are negatively Finally, we analyse the excitations of nonlinear modes and get some stable cases that have not been acquired in second part. It can be seen that the morphology of stable mode is changed when we execute a single-parameter excitation of the peakon [see Figs. 11 and 12].