Modulational Instability and Location Controllable Lump Solutions with Mixed Interaction Phenomena for the (2+1)-Dimensional Myrzakulov-Lakshmanan-IV Equation

Under consideration is a (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document}-dimensional spin nonisospectral Myrzakulov-Lakshmanan-IV (ML-IV) equation, which has close relation with the celebrated nonlinear Schrödinger equation. In the first place we study the modulational instability (MI) for this equation and deduce its formation mechanism for diverse localized waves from the plane wave background. Secondly, in view of the known nonisospectral Lax pair of this equation, we first successfully extend the generalized (n,N-n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n, N-n)$$\end{document}-fold Darboux Transformation (DT) from (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+1)$$\end{document}-dimensional equation to this (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document}-dimensional equation. As an application of the resulting DT, we derive some location controllable dark and bright lump, periodic wave and mixed breather-lump solutions, it is shown that the position of these localized waves can be controlled by some special parameters, so that we can move them to the positions we want. Especially, the dynamical behaviors of these localized waves are illustrated graphically. These results may have potential value for the study of applied ferromagnetism and nano magnetism.

This article will be organized as below: In Sect. 2, we will not only study the MI of Eq. (2), but also explain its generation principle from the plane wave background. In Sect. 3, generalized (n, N − n)-fold DT will be established for the (2 + 1)-dimensional Eq. (2). In Sect. 4, in terms of the generalized (1, N − 1)-fold DT, we will establish location controllable multi-lump and PW solutions for Eq. (2). In Sect. 5, we will exhibit the mixed breather-lump and PW-lump interaction structures by using the generalized (2, N − 2)-fold DT. Our summary will be discussed in the last section.

MI Analysis of ML-IV Equation
MI is a basic physical phenomenon in nonlinear science. It is a nonlinear process in which continuous wave or quasi continuous wave generates self-modulation of amplitude and frequency through nonlinear dispersive media, making the perturbation superimposed on continuous wave or quasi continuous wave grow exponentially. In the development and research of nonlinear equations, MI can determine the generation mechanism of breather and lump by numerical analysis. By analyzing and labeling the modulation stability (MS) and MI areas, we can easily explore the conditions for exciting soliton, plane wave, breather and lump. Subsequently, we will analyze the MI of Eq. (2) on plane wave background. First, we consider the plane wave seed solutions: where a, b, k, m, and w 1 are arbitrary real constants. For convenience of calculation, we take these parameters w 1 = 2, = 1 = 2 = 1 in advance, then we rewrite the seed solutions (4) after adding perturbations: where f 1± , f 2± , f 3± , f 4± and f 5± are small perturbations of the Fourier modes, Ω is the modulational frequency, and are called propagation parameters, and they are complex numbers. we can substitute Eq. (5) into Eq. (2) to obtain the following ten linear algebraic equations Linear algebraic equations (6) have nontrivial solutions only when , , Ω, a, b, k, m satisfies the following dispersion relations, namely (4) To facilitate analysis, when Then we define the MI gains G = |ℑ(Ω )| ( = 1, 2) , and the symbol ℑ is called this expression's imaginary part. Thus, we can sum up the generation factors of MI gains as follows: In Fig. 1a1 and b1, for MI gain G 1 , we clearly show the detailed distribution states on the coordinate axis Ω and a; In Fig. 1a2 and b2, for MI gain G 2 , we also clearly show the detailed distribution states on the coordinate axis Ω and k. In these figures, we mark the MI and MS areas in detail. Notably, on the line Ω = 0 , the perturbations are unstable and can excite lump in Fig. 1b1 and b2 except two points Ω = 0, a = 0 and Ω = 0, k = 0 . In Fig. 1b1 and b2, we can observe that the two red lines a = 0 and k = 1 2 are MS areas where solitons can be excited. The relevant proof of instability on the line Ω = 0 can refer to Ref. [22]. In the MI areas, we can obtain lump and breather, while in the MS areas, we can obtain soliton and plane wave.
It should be noted that we have given the values of the corresponding parameters in advance for the convenience of analysis, in fact, we can certainly determine other parameter values and analyze other MI gains, however, in order to save space, we will not analyze other parameters here.

Generalized (n, N − n)-Fold DT
The DT method is an effective way to seek for exact solutions of lax integrable nonlinear equations [23,24]. Near term, generalized (n, N − n)-fold DT has not only been came up, but also applied to solve some (1 + 1)-dimensional continuous and discrete nonlinear equations [25][26][27][28], there is no doubt that we can not only get the soliton solutions by this method, but also get the rational solutions and their mixed interaction structures by it. However, we have not studied how to extend this method to the nonisospectral (2 + 1)-dimensional nonlinear systems. Therefore in this section, we will try extending this generalized DT to the nonisospectral (2 + 1) -dimensional equation (2). Next, for Eq. (2), we will use the known Lax pair (3) to construct the generalized (n, N − n)-fold DT. Firstly, we know that there is a form about gauge transformation as below where = (x, y, t) = ( (x, y, t), (x, y, t)) T (T represents the transpose) satisfies Lax pair (3). We will determine the Darboux matrix T( ) later, which is a 2 × 2 order matrix, and ̃ =̃ (x, y, t) = (̃ (x, y, t),̃ (x, y, t)) T is required to satisfy the Lax pair in the same form as (3). Concurrently, ̃ must also subject to ̃ x =Ũ̃ ,̃ t = (2 1 + 4 2 2 )̃ y +Ṽ̃ , where Ũ ,Ṽ have the same forms as U, V, respectively, except that q, w, , , p are replaced by q,w,̃ ,̃ ,p . With the relevant knowledge of DT, we have the following expression With the help of symbolic computation, a classical DT matrix T will be established as below where N is a positive integer, A (j) and B (j) (j = 0, 1, ..., N − 1) are a class of unknown functions related to x, yand t that will be obtained later. According to (9), we can get the transformations between potential functions where For the sake of acquiring the novel solutions of Eq. (2), we must determine (11), and these unknown functions are also our next major goal.

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Journal of Nonlinear Mathematical Physics (2023) 30:600-627 According to Darboux matrix (10), we can clearly observe that detT( ) is a 2Nth order polynomial related to . Suppose that the spectral parameters T be n linearly independent solutions in the form of column vector for Lax pair (3) based on the = i (i = 1, 2, ..., n) . In addition, we need to ensure that T( i ) ( i ) = 0 , however, it only gives 2n algebraic equations so that 2N unknown functions A (N−1) and B (N−1) cannot be obtained when n < N . In order to get linear algebraic equations that can solve 2N unknown functions, for every i , we need to expand where Substituting the condition into Eq. (12), where the parameters i and r i are non-negative integers and r i must satisfy the relation N = ∑ n i=1 (n + r i ) . Subsequently, we will smoothly derive a linear algebraic system containing 2N equations as follows from which can adopt n suitable spectral parameters i and uniquely determine 2N (15). Through the description of the above process, once the original seed solution q 0 , w 0 , 0 , 0 and p 0 of Eq. (2) are given, then from Eq. (11) are as follows: will be clearly shown as follows: , and ΔB (N−3) will be uniquely determined by the determinant ΔN substituting its first, second, third, (N + 1)th, (N + 2) th and (N + 3) th columns via the vector (g (1) , ..., g (n) ) T with g (i) = (g i j ) 2(m i +1)×1 , its form is as follows In general, we call the transformations (8) and (11) as generalized (n, N − n)-fold DT, which adopts n spectral parameters, while conventional N-fold DT adopts N spectral parameters, which is the difference between them. We usually use n to represent the number of spectral parameters adopted in the generalized (n, N − n)-fold DT, while N − n denotes the sum of the derivative orders in the Taylor expansion for eigenfunction n . In this paper, we will use this method to obtain multi-lump, PW and abundant mixed interaction structures for Eq. (2).

Remark 1 (N, 0)-fold DT can be obtained by reducing generalized
, if we apply Taylor expansion at each i , we will get the lump and PW solutions for Eq. (2), conversely, we can get the soliton and breather solutions for Eq. (2). Generalized (1, N − 1)-fold DT can be obtained by reducing generalized (n, N − n)-fold DT, when n = 1 and r 1 = N − 1 , we can get lump solutions of Eq. (2) based on the plane wave background. In the same background, generalized(2, N − 2)-fold DT can also (16) be get by reducing generalized (n, N − n)-fold DT, when n = 2 and r 1 + r 2 = N − 2 , we can certainly get mixed interaction solutions of Eq. (2) for different localized waves. When 2 < n < N , abundant localized wave interaction solutions can also be obtained by using generalized (n, N − n)-fold DT, while it is not involved in this paper. Next, we can generate some multi-lump and mixed interaction solutions based on the plane wave background for Eq. (2) with the help of this method.

Location Controllable Multi-lump and PW Solutions via the Generalized (1, N − 1)-Fold DT
In order to obtain the PW and lump solutions for Eq. (2), in this section, in terms of the generalized (1, N − 1)-fold DT, we will only use a spectral parameter. Subsequently, we can substitute the plane wave seed solutions (4) into Lax pair (3), and obtain the fundamental solution of Eq. (2) as below: with where is an extremely small parameter, while g i and h i (i = 0, 1, 2, ..., N) are real value parameters, C 1 and C 2 are two arbitrary constants. The main objective of this article is to obtain some PW and lump solutions of Eq. (2), therefore, for different spectral parameters, we can adopt the two optional Taylor series expansions mentioned later. Subsequently, let the spectral parameter = 1 + 2 in (17), at the same time, the vector function is expanded by Taylor series around = 0 , in terms of the explicit calculation, we can get where ( ) = ( ( ) , ( ) ) T , ( = 0, 1, 2 ⋯) can be uniquely obtained through (13) and (17).

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To obtain the multi-lump solutions of Eq. (2), with the help of symbolic computation, we set M = − k 2 ± ai and think over two types Taylor series expansion conditions as follows: Type-I: Choosing C 1 = −C 2 = 1 , a = k = 1, = 1 + 2 with 1 = M and adopting Taylor series expansion at = 0 for (17). Then, these vectors ( ) = ( ( ) , ( ) ) T are the polynomial functions related to x, y, t, so that the rational bright-dark lump solutions can be easily established for Eq. (2). The relevant expansion expressions of the first type of (17) are as follows: Type-II: Choosing C 1 = −C 2 = a = k = 1, = 1 + 2 with 1 ≠ M (e.g., 1 = M + 1) and adopting Taylor series expansion at = 0 for (17). Then, these vectors ( ) = ( ( ) , ( ) ) T are the combinations of polynomial functions and exponential related to x, y, t, so that the PW solutions can be easily established for Eq. (2). Here, we only list an expansion expression of solution (17) as follows: Next, we only choose M = − k 2 − ai and primarily discuss three cases: N = 1, 2, 3 . For the convenience of calculation, we only discuss parameters a = b = = m = = k = 1 = 2 = w 1 = 1.

Location Controllable Bright-Dark First-order Lump and PW Solutions
When N = 1 , based on the transformations (11), in terms of the seed solutions (4) and generalized (1, 0)-fold DT, it will produce the following expressions Adopting Type-I, on the basis of (19) and (21), we can acquire the bright-dark firstorder lump solutions of Eq. (2), which can be rewritten as where (20) The above expressions contain two arbitrary parameters g 0 and h 0 , which can control the position of the first-order lumps. The relevant shapes of first-order lump are shown in Fig. 2. In terms of Fig. 2a1-e1, we can distinctly observe that the solution q owns bright lump shapes with single wave peak and double wave troughs, w owns the bright lump shapes with double wave peaks and double wave troughs, solution has the dark lump shapes with double wave peaks and double wave troughs, solution owns the dark lump shapes with double wave peaks and single wave trough, and p  Table 1 Position of the maximum and minimum of the lump solutions (22) Note: t is an arbitrary time and the corresponding coordinate points are as follows .

Lumps
Wave peak position Wave trough position Maximum Minimum has dark lump shapes with single wave peak and four wave troughs. The positions, maximum and minimum of the wave peaks and troughs of these lump solutions are listed in Table 1. In Fig. 2, we also show the lump structures with different parameters at different time. When the parameters g 0 = h 0 = 0 , we can clearly see that the lumps q, w, , and p propagate along the positive directions of x and y with the increase of time. In Fig. 2a2-e2, with the increase of h 0 , when t = g 0 = 0 , we can find that the lumps propagate along the negative direction of y, with the increase of g 0 , when t = h 0 = 0 , the lumps propagate along the negative direction of x. More importantly, we find that the characteristics of g 0 and h 0 for the control position are also applicable to the higher-order lumps. According to the above Type-II, we take 1 = M + 1 , on the basis of (20) and (21), the bright-dark first-order PW solutions can be easily obtained for Eq. (2), the related structures of the first-order PW are displayed in Fig. 3. Because of the complexity of the PW solutions, we here only give the structures of first-order PWs q 1 ,̃ 1 as shown in Fig. 3. When g 0 = h 0 = 0 , the PW solutions q and propagate along the positive direction of y with the increase of time in Fig. 3(a1)(a2)-(b1) (b2). With the increase of h 0 , when t = g 0 = 0 , the PW solutions q and propagate along the negative direction of y in Fig. 3(a3)-(b3). With the increase of g 0 , when t = h 0 = 0 , the PW solutions q and propagate along the negative direction of x in Fig. 3 (a4)-(b4).

Location Controllable Bright-Dark Second-Order Lump and PW Solutions
When N = 2 , based on the transformations (11), in terms of the seed solutions (4) and generalized (1, 1)-fold DT, it will produce the following expressions ΔA (1) , ΔA (0) , ΔB (1) and ΔB (0) can be uniquely determined by determinant Δ 2 substituting its first, second, third and fourth columns by the vector Owing to the complexity of the second-order lump and PW solutions, we will leave out the expressions of explicit analytical solutions here. From (23), adopting the expansion of Type-I, we can obtain the second-order bright-dark lump (23) solutions of Eq. (2), adopting the expansion of Type-II, we can also obtain the second-order bright-dark PW solutions of Eq. (2), which include four arbitrary parameters g i , h i , (i = 0, 1) , these parameters can not only control the position but also change the arrangement shape for second-order lump and PW.
However, because the propagation characteristics of second-order lumps q, w, , and p are similar to the first-order lumps, we merely give the relevant images of w and in Figs. 4 and 5 by selecting various parameter combinations. In Fig. 4, with the increase of time, when g i = h i = 0 , the second-order brightdark lumps w and not only propagate along the positive direction of x and y, but also separate into three first-order bright-dark lumps and arrange into triangles. In particular, if t = g i = h i = 0 , the strong interaction of second-order lump can be clearly displayed (see Fig. 4(a2)(b2)). When g 1 = 10 3 or h 1 = 10 3 and g 0 = h 0 = 0 , the second-order lumps can split into three first-order lumps and arrange into a triangle in Fig. 5(a1)(a2)-(b1)(b2). When the parameters g 0 = 8, g 1 = 0, h 0 = 10, h 1 = 10 3 , the second-order lumps are not only divide into three first-order lumps, but also arrange into triangles and propagate along the negative directions of x and y in Fig. 5(a3)(b3).
In Fig. 6, we show the relevant structures of second-order PW solutions q and w, respectively. The strong interaction of second-order PW can be clearly displayed in Fig. 6(a1)(b1) when t = g i = h i = 0 . We can also find that the second-order PW will split into two first-order PWs with the increase of time and propagate along the positive y direction when g i = h i = 0 in Fig. 6(a2)(b2). In Fig. 6(a3)(b3)-(a4)(b4), changing g 1 and h 1 will split the second-order PW into two first-order PWs, while changing g 0 and h 0 will control the position of the PW. Under the control of g 0 and h 0 , the position change of the second-order PW is consistent with the first-order PW.

Location Controllable Bright-Dark Third-Order Lump Solutions
If we use the expansion of Type-II, we can certainly obtain the third-order PW solutions of Eq. (2). However, the expressions of the third-order PW solutions are extremely complex, therefore, we merely discuss the third-order lump solutions in this section.
When N = 3 , based on the transformations (11), in terms of the seed solutions (4) and generalized (1, 2)-fold DT, it will produce the following expressions ΔA (2) , ΔA (1) , ΔB (2) and ΔB (1) can be uniquely determined by determinant Δ 3 substituting its first, second, fourth and fifth columns by the vector Journal of Nonlinear Mathematical Physics (2023) 30:600-627 Using the expansion of Type-I, according to (24), the third-order bright-dark lump solutions of Eq. (2) can be obtained, it include six arbitrary parameters g i , h i , (i = 0, 1, 2) , for the third-order lumps, these parameters can not only control the position but also change the arrangement shape.
The propagation characteristics of third-order lumps q, w, , and p are similar to the first and second order lumps, so we merely give the relevant structures of q and in Fig. 7 by selecting various parameter combinations. If t = g i = h i = 0 , the strong interactions of the third-order lump are shown in Fig. 7(a1)(b1). Especially, with the increase of time, when g i = h i = 0 , the third-order lumps can change their positions and shapes from the strong interactions as shown in Fig. 7(a1)(a2). When t = g i = h i = 0 except g 1 = 10 3 , we can see clearly that third-order lumps split into six first-order lumps and arrange into triangles as displayed in Fig. 7(a3)(b3). When t = g i = h i = 0 except g 2 = 10 6 or h 2 = 10 6 , the third-order lumps can split into six first-order lumps and arrange into pentagons in Fig. 7(b2)(a4)(b4). Since the propagation of the third-order lumps is similar to the first-order lumps after changing g 0 and h 0 , whose structures are omitted here. Subsequently, we will sum up the main mathematical characteristics of lump solutions related to order N in Table 2.

Mixed Breather-Lump Interaction Solutions via the Generalized
(2, N − 2)-Fold DT In this section, based on the two spectral parameters, we will exhibit the new mixed interaction solutions of lump and breather for Eq. (2) by using generalized (2, N − 2)-fold DT. In terms of the control of position parameters g 0 , h 0 , we can regulate the interaction between lump and breather. In the following contents, there are two cases: N = 2 (the generalized (2, 0)-fold DT) and N = 3 (the generalized (2, 1)-fold DT). For the convenience of calculation, we still discuss the parameters a = b = = m = = k = 1 = 2 = w 1 = 1.

Generalized (2,0)-Fold DT
Based on the case of N = 2 , the two spectral parameters 1 and 2 will be used, so there will be three cases: (i) Taylor series expansion is used for both spectral parameters at the same time; (ii) One of the spectral parameters is expanded by using Taylor series and the other is not used; (iii) Taylor series expansion is not used for both spectral parameters. From the above cases (i) and (ii), we can obtain new formal solutions. For case (i), if we use the expansion of Type-I at 1 and Type-II at 2 , we can get mixed interaction solutions for first-order bright-dark lump and first-order PW; If we use the expansion of Type-II at 1 and 2 , respectively, the mixed interaction solutions for two firstorder PWs will be obtained. For case (ii), if we use the expansion of Type-I at 1 or where A (1) , A (0) , B (1) and B (0) can be determined by: whereas ΔA (1) , ΔA (0) , ΔB (1) and ΔB (0) can be uniquely determined by determinant Δ 2 replacing its first, second, third and fourth columns by the vector According to the solutions (25), which contains two position parameters g 0 and h 0 . Subsequently, we can acquire the mixed interaction solutions of one-breather and first-order bright-dark lump, owing to the analytical expressions are extremely complex, which is leaved out here. In our research, it is found that the interaction characteristics of mixed solutions q, w, , and p are consistent, so we only give the related images of q and as shown in Fig. 8 by selecting various parameter combinations. If t = g i = j i = 0 , the strong interaction between the first-order lump and one-breather can be given in Fig. 8(a1) (b1). In particular, as time increases, when g 0 = h 0 = 0 , the first-order bright-dark lumps q and not only propagate along the positive x and y directions, but also separate from the breather, and the breather propagate along the negative x and positive y directions in Fig. 8(a1)(b1)-(a2)(b2).When the parameters g 0 = 8, h 0 = 0 or g 0 = 0, h 0 = 6 , the first-order lump propagates in the negative direction of x or negative direction of y, while the position of breather remains unchanged in Fig. 8(a3)(b3)-(a4)(b4). In terms of the generalized (2, 0)-fold DT, we will summarize the interaction structures between localized wave in Table 3.

Generalized (2,1)-Fold DT
Based on the case of N = 3 , the two spectral parameters 1 and 2 will be used, it will product two cases: (i) Taylor series expansion is used for both spectral parameters simultaneously; (ii) One of the spectral parameters is expanded by using Taylor series and the other is not used.
From the above two cases, we can obtain new formal solutions. For case (i), there are three mixed interaction solutions in the following forms: The first mixed type is first-order bright-dark lump and second-order PW; The second mixed type is the first-order PW and second-order bright-dark lump; The third mixed interaction type is second-order PW and first-order PW. For case (ii), there are two mixed interaction solutions in the following forms: The first mixed mold is one-breather and secondorder bright-dark lump; The second mixed mold is one-breather and second-order PW.
Next, we will ignore other cases and only discuss the following two cases: Case (a) One-breather and second-order bright-dark lump in the case (ii). Case (b) Second-order PW and first-order lump in the case (i).
In Case (a) , we use the expansion of Type-I at 1 and choose 2 = −i . According to the seed solutions (4) and generalized (2, 1)-fold DT, the mixed interaction solutions for one-breather and second-order bright-dark lump can be given as below images of w and as shown in Figs. 9 and 10. In Fig. 9, As time increases, when g i = j i = 0 , the second-order bright lump w not only propagates along the positive x and y directions, but also separates from the breather and splits into three first-order lumps that arrange into a triangle, at the same time, the breather propagate along the positive x and y directions as displayed in Fig. 9(a1)-(a3). In particular, if g i = j i = 0 , the strong interaction between the second-order lump and one-breather can be given in Fig. 9(a2) (b2). When the parameters g i = j i = 0 except g 1 = 10 3 , the secondorder lump splits into three first-order lumps clearly as shown in Fig. 9(b1)-(b3), whose propagation characteristics are consistent with Fig. 9(a1)-(a3). Figure 10 shows the mixed interactions between the second-order dark lump and one-breather, whose related properties are the same as Fig. 9.
Based on (27), In Case (b) , we use the Type-II at 1 and Type-I at 2 , the mixed solutions for second-order PW and first-order lump can be given, where A (2) , A (1) , B (2) and B (1) can be determined by: and first-order PW, respectively. In terms of the generalized (2, 1)-fold DT, we will summarize the interaction structures between localized wave in Table 4.

Conclusions
In this paper, we have investigated the nonisospectral (2 + 1)-dimensional Myrzakulov-Lakshmanan-IV equation (2), which may have potential value in applied ferromagnetism and nano magnetism. Compared with the published references [1,5], the main contributions in this paper are summarized as below: Table 4 Interaction structures between localized wave via the generalized (2, 1)-fold DT FL, SL, FPW, SPW and OB indicate first-order lump, second-order lump, first-order PW, second-order PW and one-breather, respectively. 1 , 2 are nonzero real constants, and 3 is an arbitrary complex number. The superscript (0) is the same as Table 3. The superscript (1)   (v) It should be specially noted that the positions of these localized waves can be controlled by some special parameters, which allows us to change these control parameters to make them appear at the desired positions. We believe that these results have certain theoretical significance for the parameter control of different localized waves.
These above-mentioned results are first reported. We expect that these achievements and new phenomena in this paper are helpful to understand some physical phenomena in applied ferromagnetism and nano magnetism.
Acknowledgements The authors would like to express their sincere thanks to other members of the discussion group for their valuable comments.
Author Contributions CXQ carried out the calculation and analysis studies of equation, participated in the programming design, drawing figures and dynamical analysis, and drafted the manuscript. ZBJ