Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

Solving new integrable systems and exploring their physical applications have been a hot topic. This paper gives the modulation instability in the continuous wave background of the new modified Gerdjikov–Ivanov equation with anomalous dispersion. Based on the extend Lax pair, the localized wave solutions of the new model are obtained via generalized Darboux transformation method, various types solutions including breather, rogue waves, and interaction solutions are presented and their dynamic properties are analyzed. The results obtained have certain application value for nonlinear optics and long-distance transmission.

all embody nonlinear physical phenomena. Among the major studies in nonlinear science, nonlinear mathematical physical equations are often used to describe various nonlinear physical phenomena in nature. After more than 100 years of development, a series of effective methods have emerged for the solving integrable systems, among which the well-known ones include the Hirota bilinear method, inverse scattering transformation method, Darboux transformation (DT) method, etc. [1][2][3][4][5][6][7][8][9][10]. The emergence of these well-known methods has given an extremely powerful step forward in the development of nonlinear equations, and more equations with physical backgrounds are discovered and their numerical or analytical solutions are constructed, it is built a basis for whether solitons are destroyed after collisions occur.
Rogue waves (RWs) with a variety of properties have been uncovered as a hot research topic in recent years. It first appeared in the deep ocean in the form of an extremely destructive wave, surprisingly, RWs have also been found in various physical systems, such as optical fiber systems, water fluids, plasma, Bose-Einstein condensates and even financial systems [11][12][13][14][15][16][17][18][19], which are currently more popular. There is a consensus in the current study of RWs that they are produced by tuning to modulation instability (MI) waves, and the breather solution involved in them usually comes from instabilities with small amplitude perturbations, the magnitude of which has the potential to affect the degree of catastrophe suffered. Mathemati-cally, RWs can be understood as the limit of a soliton approaching infinity in time period or space period [16]. Therefore, considering the realistic physical applications, it is necessary to do further research on different models of higher-order RWs. So far, the generalized Darboux transformation (GDT) method as the main research tool has been used for solving higher-order RWs [20][21][22][23][24].
The nonlinear Schrödinger equation (NLSE) has been widely studied as an important class of models for mathematical physics equations. In order to analyze the effects of higher-order perturbations, scholars have started to study the modified NLSE, which include the third derivative nonlinear Schrödinger equation (DNLSE), i.e., the Gerdjikov-Ivanov (GI) equation [24][25][26][27][28], where i 2 = −1, x, t represent the space and time coordinates, q ≡ q(x, t) is the transverse magnetic field perturbation, q * denotes the complex conjugate of q. Equation (1) has been considered as a plasmaphysics model for the Alfvén waves propagating parallel to the ambient magnetic field [28]. The generation of Alfvén waves results from hydromagnetic turbulence and ionized inhomogeneity [29], e.g., in the sun photosphere, solar atmosphere and solar wind originate. It has been observed in the laboratory, plasma machines, and fusion reactors [30]. In this context, the study of the extended GI equation is also of potential interest for plasma objects. Therefore, the generalized GI equation iq t − 2q x x + 1 4 i((48η + 12)|q| 2 + 8iα)q x + 4iηq 2 q * x + 1 4 (32η 2 + 24η + 3)|q| 4 q +4iαη|q| 2 q + iα|q| 2 is investigated in this paper. If the free parameters are taken as α = 0, β = 0, η = −1/4, Eq. (2) is converted to it is a generalized GI equation with anomalous dispersion. In order to simplify the constraint parameters, and there is an integral relation between α and β in the derivation, therefore we can assume that α = 2iη, β = iη 2 , Eq. (2) is converted to η is an arbitrary real number, it is noted that the 2iη(|q| 2 q) x is physically defined as a self-steepening term, it is unique in that it causes the optical pulse to become asymmetrical at the trailing edge and steepen upward.
As the name implies, two forms of DNLSE have existed before Eq. (1), namely the Kaup-Newell (KN) equation [31] iq t + q x x − i(|q| 2 q) x = 0 and the Chen-Lee-Liu equation [32] iq t +q x x +i|q| 2 q x = 0, respectively. All three types of equations come from the KN system, and the display solution of the KN equation has been obtained by DT in [33], while in the text, we mainly via the GDT method to research the higherorder RWs and interaction solutions of Eq. (4) based on the extended Lax pair. The excitation conditions of various waves are obtained by modulational instability, and the phenomenon of multiple peaks in the formation of RWs is found in the morphology of the analytical solution, while the interaction principle demonstrates that the transmission process does not change the waveform, which has potential applications to plasma physics and nonlinear optics.
The layout of the full text can be summarized as follows: First, the modulation instability of the basic solution and the excitation conditions of the localized waves of the modified GI equation are analyzed in Sect. 2. In Sects. 3 and 4, the Lax pair of the modified model is presented and the determinant representations of the complete N -fold DT and the GDT are given by feat of symbolic calculations, respectively. The higher-order RWs and interaction solutions of Eq. (4) are analyzed in Sects. 5 and 6, and the shapes of the solutions are shown graphically. The dynamic behavior, physical properties and the applications involved of the solutions are analyzed in Sect. 7. The last section concludes the whole text.

Modulational instability
The research shows that the necessary condition for the generation of breathers and RWs in localized waves is MI [34], the MI of Eq. (4) will be studied based on the nonlinear continuous wave (CW) background. First, by solving the basic solution to the equation we get b = (8c 4 η 2 + 6c 4 η − 8c 2 aη + 3 4 c 4 − 8c 2 η 2 − 3c 2 a − 2c 2 η + 2a 2 + 4aη + 2η 2 ), a small Fourier mode perturbation is added to the plane-wave solution q 0 = ce i(ax+bt) of Eq. (7), i.e., q where ω is the modulation frequency, a,c are arbitrary real numbers, ρ + and ρ − denotes the small amplitude of the Fourier mode. Substituting the above expression into Eq. (4) for linearization, a coupled linear system with respect to ρ + and ρ − of Eq. (4) can be obtained, i.e., where Solving for the determinant formed by the coefficients on ρ + and ρ − in the coupled system (5) leads to the following conclusions. When the system of Eqs. (13) has nontrivial solutions, P, a and c need to satisfy the following dispersion relation, Define the growth rate of MI as G = |Im(ωP)|, Im is the imaginary part. However, G does not prove the MI on the resonance line ω = 0 because it can eliminate the modulation instability growth on the line. Therefore, the perturbation on the plane wave can be rewritten q = q 0 (1 + σ π(t)), substituting it into Eq. (4), we can get π(t) = 1+i(32c 4 η 2 +24c 4 η−16c 2 aη+3c 4 − 16c 2 η 2 − 6c 2 a − 4c 2 η)t, which proves the MI on the resonance line ω = 0. Figure 1a shows the relationship between the modulation frequency and the CW background at a = 1 4 . In the MI plane, it can be clearly observed that the small amplitudes are exponentially amplified, which in turn reflects their instability. However, in the MS plane, all perturbations are stable and unchanged. In MI, the perturbation frequency on the resonance line is equal to the perturbation frequency of the plane wave, whereas in MS, the growth rate on the line does not describe the stability of the perturbation. For other forms of exact solutions, such as Akhmediev breathers, Kuznetsov-Ma breathers, bright solitons and RWs solutions, they can be expressed as a plane wave form plus a perturbation form, i.e., q = q 0 + ρ pert . Taking the Fourier transformation on their perturbation terms ρ pert , the frequency ω corresponding to the maximum spectral intensity can be obtained, which is the dominant frequency of the CW. Figure 1b shows the relationship between multiple localized wave excitation regions and MI. The different types of nonlinear continuous waves of Eq. (4) are clearly shown in Fig. 1b, especially RW and K-M are not accurately distinguished in the MI plane, but can be identified by their dominant propagation constants.

N-fold Darboux transformation
With the advent of the DT method, the solutions of a large number of nonlinear models in the integrable system are mined and studied based on the classical Lax pair. Next, to ensure a smooth research of the N -ford DT for the modified model with anomalous dispersion Eq. (4), we first give the Lax pair based on the matrix form: where the spectral matrix is expressed as T is a complex function to be determined, μ ∈ C is called spectral parameter. Obviously, deriving the zero-curvature equation (2) focused on in this paper can be obtained. The next goal of this section is to compute and derive the specific determinant in terms of the N -ford DT to Eq. (4). In order to calculate conveniently, we will defined the eigenfunction is k = (x, t, μ k ) = k,1 , k,2 T . First, the gauge transformation is con- where T is a 2 × 2 matrix with respect to μ. It is known from the basics of the DT that needs to satisfy the following conditions: Theoretically, it is easy to prove thatŪ andV with respect to q N have the same expression form as U and V with respect to q, respectively, and only need to replace q in U, V with q N . Therefore, with the help of compatibility condition¯ xt =¯ t x due to Eq. (10), by calculating the above two equations Eq. (9) and Eq. (10), it is known that T, U, V,Ū ,V must satisfyŪ The symbol "−1" indicates the inverse of the matrix. Under the transformation Eq. (11), the Lax pair (10) has the same form as the initial Lax pair (7), i.e.,Ū , V and U , V are guaranteed to have the same form. Therefore, the following expression can be obtained which in turn generates Eq. (4) with q → q N , that is, it shows that q N under the new system (9) and (10) is still a solution of Eq. (4). Next, a special Darboux matrix is constructed as follows: where A (2 j) , B (2 j+1) , j = 0, 1, . . . , N −1 denote complex functions and are obtained by solving the linear is a solution of the Lax pair Eqs. (7) and (8) under the given spectral parameters μ k . Obviously, according to the formula det T (μ) = 0, we can get the spectral parameters are the 4N roots of the T k = 0, i.e., it can be written as Substitute Eq. (12) into Eq. (11), the following N -fold DT theorem of Eq. (4) can be derived.

Generalized (n, N − n)-fold Darboux transformation
In Sect. 3, by means of symbolic calculation, a complete determinant expression of the N -ford DT to the studied model is given. For the diversity of multiple RW solution of Eq. (4), the small parameter ε is introduced to obtain the determinant of (n, N −n)-fold GDT method to the equation by using Taylor expansion and limit idea. Obviously, (μ i ) is a column vector solution of Eqs. (7) and (8) about μ k and q 0 , each component in According to mathematical theory it is known that , and the definition also applies to T . Therefore, as long as the appropriate spectral parameters μ i are selected, A 2 j , B 2 j+1 can be obtained with the aid of solving the following system where m i , (i = 0, 1, 2, ...) represent the highest order derivative term of the Taylor expansion of T and in the above system for n different spectral parameters μ i , in actual calculation, the relationship between m i and can be determined by solving the system of algebraic Eq. (13). If the appropriate spectral parameters μ i can be selected so that the determinant of the coefficient matrix in the system (13) is nonzero, then T can be uniquely determined by system (13). It also proves that the new Darboux matrix T , which consists of the coefficients of Taylor's expansion and the corresponding derivative terms, still satisfies Theorem. 1. Finally, the expression of the (n, N − n)-fold GDT of Eq. (4) is constructed and the following theorem can be derived.
T be a column vector solution of the Lax pair Eqs. (7) and (8), the obtaining of the solution depends on the initial solution q 0 and the given spectral parameter μ 1 of Eq. (4). With the help of the (n, N − n)-fold GDT, the relationship between the new solution of Eq. (4) and the original solution can be expressed as Obviously, with the help of Cramer's rule and solving the linear system (13), the determinant expression of . . , n) can be represented by the following formulas: It is worth noting that if the parameters are selected to be n = 1 and m 1 = N − 1, the (n, N − n)-fold GDT becomes the (1, N − 1)-fold GDT; if the parameters are selected to be n = N and m i = 0, 1 ≤ i ≤ n, Theorem 2 can be reduced to Theorem 1.

Higher-order rogue wave solutions
Starting from the plane-wave solution, with the help of Taylor expansion and limit principle, this chapter intends to obtain the determinant expression of the higher-order RW solution of Eq. (4) by using the (1, N − 1)-fold GDT method, so as to obtain the accurate expression of the analytical solution. First, by substituting the plane-wave solution q 0 = ce i(ax+bt) into Eq. (4) for simplification, the basic expression can be obtained as follows: where a,c = 0 and η are real number, physically a can be used to control the wavelength, while c can be used to control the amplitude of nonlinear continuous wave.
Substitute q 0 into linear system Eqs. (7) and (8) yields the expression for the solution containing the spectral parameters μ as follows: where where ε is a small parameter introduced. The spectral parameters μ = μ i + ε 2 are substituted into Eq. (17). The Taylor expansion of the functions 1,1 and 1,2 in Eq. (17) at ε = 0 yields As above, (k) is still defined by since the expanded expression of (ε 2 ) is too complex, it is omitted here. The next part of the work is to explore RW phenomena and the interaction solutions between various kinds of localized waves of Eq. (4), for convenience, we assume that R = −8c 2 η−2c 2 +2 √ 8c 4 η+c 4 −4ac 2 +4a 2 . The localized wave solutions with different parameters can be obtained more accurately by analyzing and comparing the relationship between the spectral parameters μ i and R. The follow-up study involved the following two forms of Taylor series expansions of Eq. (17), defined as Expansion I Let μ i = R and C 1 , C 2 are taken to be special values of C 1 = 1 ε , C 2 = − 1 ε , the selection is often used to construct the RW solution of Eq. (4). Expansion II Let μ i = R and C 1 , C 2 are taken to be special values of C 1 = 1, C 2 = −1, in this case, the RWs cannot be obtained, but it is generally used to obtain semi-rational soliton of Eq. (4).
In this paper, we focus on giving the discussion in two cases N = 2 and N = 3. Case I When N = 2, obviously, with the aid of recurrence formula of the Darboux matrix and linear equa-tions, the expression for the second-order RW solutions are written as: It has been found that in order to construct the secondorder RW solution, the spectral parameter needs to be chosen as μ 1 = R, in the calculation then it is necessary to substitute μ 1 = μ 1 + ε 2 into the system (17), and then the vector function in Eq. (17) is subjected to a Taylor series expansion at ε = 0 by taking the special parameters of the "Expansion I." The coefficients of the corresponding terms are selected to form the determinant, and then, the expression of B (3) (x, t) can be established by the following system: Case II Same as Case I, when N = 3, with the aid of recurrence formula of the Darboux matrix and linear equations, the expression for the third-order RW solution can be written as: (5) . Therefore, with the help of system (13), B (5) can be obtained by solving the following linear system: where B (5) (x, t) =¯ 5 6 with¯ 6 can be deduced by Eqs. (14) and (15). The expressions for the determinants involved are too long and are omitted here.¯ 6 can be determined by system (14), and¯ 5 is given by¯ 6

Interaction solutions
Scenario n = 1 was discussed and analyzed in the previous section, and this section will focus on the case n = 2, when the (n, N − n)-fold GDT becomes the (2, N − 2)-fold GDT, which contains two spectral parameters μ 1 and μ 2 conveniently leading to the collision various types of localized waves, mainly involving the collision phenomenon of RWs, breather and semirational soliton with each other. The final presentation of the collision phenomena of localized waves provides a reference for certain applications in the physical domain such as optical transmission.

Interactions of localized waves via the GDT for
When the value of N is taken as 2, it is known that the number of spectral parameters to be selected is 2 and assumed to be μ 1 and μ 2 , respectively. Substituting μ 1 and μ 2 into the system (13), the new solution expression of Eq. (4) can be derived with the help of the (2, 0)-fold GDT method in Theorem. (2) as It is well known that the choice of spectral parameters μ i determines the type of localized wave solutions. However, diverse free parameters a, c, η can also lead to unique collision phenomena between the two solutions. Combining with the above analysis, the interaction schemes are given for four cases in this paper. Case 1 The collision of the first-order RW and onebreather In order to successfully obtain the collision phenomenon of the first-order RW and one-breather, the spectral parameter is chosen as μ 1 = R,μ 2 = R and substitutes the spectral parameters μ 1 = μ 1 + ε 2 into Eq. (17), expand the vector function 1 in Eq. (17) about Taylor series at ε = 0 based on "Expansion I," therefore B (3) is constructed by solving the following linear equations: where ,¯ 3 ,¯ 4 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the first-order RW and one-breather to Eq. (4) with different parameters is shown in Fig. (3). Case 2 The collision of the first-order semi-rational soliton and one-breather Similarly, in order to successfully obtain collision phenomenon of the first-order semi-rational soliton and one-breather, the spectral parameters need to be chosen as μ 1 = R,μ 2 = R, μ 1 = μ 2 and substitute the spectral parameter μ 1 = μ 1 + ε 2 into Eq. (17), the vector solution 1 in Eq. (17) is expanded by Taylor series at ε = 0 based on "Expansion II," therefore B (3) is determined in terms of similar system (19), where B (3) =¯ 3 4 ,¯ 3 ,¯ 4 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the first-order semi-rational soliton and one-breather to Eq. (4) with different parameters is shown in Fig. (4).

Case 3 The collision of the first-order RW and firstorder semi-rational soliton
If the collision phenomenon of the first-order RW and first-order semi-rational soliton to Eq. (4) is obtained, the spectral parameter needs to be chosen as μ 1 = R,μ 2 = R, substitute the spectral parameter μ 1 = μ 1 + ε 2 , μ 2 = μ 2 + ε 2 into Eq. (17), the vector solution 1 in Eq. (17) is expanded by Taylor series at ε = 0 based on "Expansion I" and "Expansion II," finally, the expression of B (3) can be represented based on the linear system, where ,¯ 3 ,¯ 4 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of firstorder semi-rational soliton and first-order RW to Eq. (4) with different parameters is shown in Fig. (5).
To facilitate the study of the collision of the localized wave solutions of Eq. (4), with the help of the (2, 1)fold GDT, the localized wave interaction solutions with four different parameters are discussed as follows.

Case 5
The collision of the second-order RW and onebreather As in the previous subsection, in order to successfully obtain the collision of the second-order RW and one-breather, the spectral parameter is chosen as μ 1 = R,μ 2 = R and substitute the spectral parameter μ 1 = μ 1 + ε 2 into Eq. (17), expand the vector function 1 in Eq. (17) about Taylor series at ε = 0 based on "Expansion I," therefore B (5) is constructed by solving the following system: where B (5) =¯ 5 6 ,¯ 5 ,¯ 6 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the second-order RW and one-breather to Eq. (4) with different parameters is shown in Fig. (7). Case 6 The collision of the second-order semi-rational soliton and one-breather Similarly, in order to successfully obtain the collision of the second-order semi-rational soliton and onebreather, the spectral parameter needs to be chosen as μ 1 = R,μ 2 = R, substitute the spectral parameter μ 1 = μ 1 + ε 2 into Eq. (17), the vector solution 1 in Eq. (17) is expanded by Taylor series at ε = 0 based on "Expansion II," then the expression of B (5) is obtained by the similar system (19), where B (5) =¯ 5 6 , 5 ,¯ 6 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the second-order semi-rational soliton and one-breather to Eq. (4) with different parameters is shown in Fig. (8). Case 7 The collision of the first-order RW and secondorder semi-rational soliton If the collision phenomenon of one breather and second-order RW to Eq. (4) is obtained, the spectral parameter is chosen as μ 1 = R,μ 2 = R, substitute the spectral parameters μ 1 = μ 1 + ε 2 , μ 2 = μ 2 + ε 2 into Eq. (17), the vector solution 1 in Eq. (17) is expanded by Taylor series at ε = 0 based on "Expansion I" and "Expansion II," then B (5) is represented in terms of the linear system, where B (5) =¯ 5 6 ,¯ 5 ,¯ 6 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the first-order RW and second-order semi-rational soliton to Eq. (4) with different parameters is shown in Fig. (9). Case 8 The collision of the second-order RW and firstorder semi-rational soliton Similarly, if the collision of the second-order RW and first-order semi-rational soliton to Eq. (4) is obtained, the spectral parameter needs to satisfy μ 1 = R,μ 2 = R, substitute the spectral parameter μ 1 = μ 1 + ε 2 , μ 2 = μ 2 + ε 2 into Eq. (17), the vector solution 1 in Eq. (17) is expanded by Taylor series at ε = 0 based on "Expansion I" and "Expansion II," therefore B (5) is constructed based on the similar linear system (20), where B (5) =¯ 5 6 ,¯ 5 ,¯ 6 can also be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the second-order RW and first-order semi-rational soliton to Eq. (4) with different parameters is shown in Fig. (10).
In particular, we give the one kind localized wave interaction behavior for Eq. (4) at n = 3. Obviously, 3 spectral parameters which can be assumed the μ 1 ,μ 2 and μ 3 , and the spectral parameters need to satisfy the condition μ 1 = R,μ 2 = R, μ 3 = R, therefore the interactions of first-order RW, first-order semi-rational soliton, and one-breather can be obtained. Case 9 The collision of the first-order RW, first-order semi-rational soliton and one-breather First, substitute the spectral parameter μ 1 = μ 1 + ε 2 , μ 2 = μ 2 + ε 2 into Eq. (17), the vector solution 1 in Eq. (17) is expanded by Taylor series at ε = 0 based on "Expansion I" and "Expansion II," therefore B (5) is constructed based on the linear system, where B (5) =¯ 5 6 ,¯ 5 ,¯ 6 can be deduced by Eqs. (14) and (15). Therefore, the collision phenomenon of the first-order RW, first-order semi-rational soliton and one-breather to Eq. (4) with different parameters are shown in Fig. (11).
Of course, for the case of n = 3 there are various combinations, such as two one-breather and first-order RW and two one-breather and first-order semi-rational soliton, which are not shown in the text due to the complexity of the calculation, but you can continue to study them if you are interested.

Dynamics analysis and discussion
In this section, the dynamic properties of the solution to Eq. (4) and their physical application will be simply Fig. 11 The collision of the first-order RW, first-order semirational soliton with one-breather of Eq. (4) with (a-c). a a = 1/2, c = −1, η = 0, analyzed. Since the model Eq. (4) has some free parameters, the variation of the free parameters also affects the presentation of the final solution of the model. Due to the diversity and complexity of the solutions, this paper focuses on giving the higher-order RW solutions of the model and the collision phenomenon of nonlinear continuous solutions including breather, RWs and semi-rational soliton, etc.
First, observing the morphology of the RWs in Fig. 2a and c shows the special phenomenon from the conventional RWs due to the different selection of parameters, where multiple RWs collide, a splitting phenomenon occurs during the fusion process, and a sharp wave peak is accompanied by the appearance of small wave peaks, while the other solutions after the fission remain consistent with the existing RWs morphology.
Second, we observe the collision phenomenon of various solutions when N = 2. Figure 3a shows the strong collision structure of the first-order RW and the K-M breather. It is clearly observed that during the collision of the two waves, the original breather reaches a new amplitude with the appearance of the RW, then the RW disappears and the breather returns to the original amplitude and continues to move forward. Figure 3b and c shows the elastic action of the first-order RW with the spatiotemporal periodic breather, respectively. During the action process, the RW propagates in a short time and space that appears in the intersection with the double localized RWs in time and space. In practical physical applications, for the purpose of long distance transmission, the signal amplification can be achieved by using the RW to the signal amplitude. Fig-ure 4 shows the collision phenomenon of the first-order semi-rational soliton and one-breather, and three different parameters are chosen for μ 2 for observation. Both waves continue to propagate according to their original trajectories after colliding with each other and do not change their orbits as a result, except that the amplitude increases significantly at the intersection of them, which is more obvious in Fig. 4a and b. Figure 5 shows the collision of the first-order RW and first-order semirational soliton, and the physical forms they show are basically the same as Fig. 3, except that the amplitude heights after the collision are not the same due to the different free parameters and spectral parameters chosen, so we can get the desired amplitude heights by changing the free parameters and their spectral parameters, but this situation does not apply to most of the equations of the NLS class that can reach the ideal state by changing the spectral parameters. The collision phenomenon occurring in Fig. 6 is similar to Fig. 4, and this phenomenon is not described too much.
Finally, let us observe the phenomenon of mutual collisions of different waves when N = 3. For this case we still give four different discussions, and since the collision of the first-order semi-rational soliton and the second-order semi-rational soliton is similarly to Fig. 8, it is not discussed here. Observing Fig. 7, we can find that there is an influence on the fusion of solutions when η = 0 and η = 0. For example, there is a clear difference between Fig. 7a and c, where the former splits the RWs into different waveforms by crashing them apart, while the latter does gather to one waveform. After observing Fig. 8, it is found that when the second-order semi-rational soliton collides with one breather, all three waveforms reach a new height of amplitude at the collision, and then all continue according to their respective trajectories. The following conclusion can be obtained by changing the graph given by the parameter g 1 : as g 1 increases, the distance between the second-order semi-rational solitons then becomes larger and fission occurs, but the respective propagation direction and trajectory do not change. Figure 9 shows the collision of the first-order RW and the second-order semi-rational soliton. When g 1 = 0, the second-order semi-rational soliton almost fuses together under the given free parameters and spectral parameters, and the sharp wave peak appears after the collision with the first-order RW and then continues to propagate in the original direction; when g 1 is given a certain value, the second-order semi-rational soliton starts to separate slowly and collides with the RW as well, but as the separation distance becomes larger, the initial cut RW becomes wrapped RW and the fusion phenomenon gradually disappears. Figure 10 shows the collision process of the second-order RW with the first-order semi-rational soliton, and we give the two cases of η. When η = 0, the two waveforms collide with each other after three small wave peaks, similar to the previous one, and the phenomenon of fusion and fission appears, while when η = 0, there is just a fusion phenomenon with a wave peak under the given parameters. The different values of g 1 can also be used to observe more clearly the process of two waveforms colliding and then separating afterward.
It should be noted, in particular, that a localized wave interaction is given in this paper for n = 3, as shown in Fig. 11. Figure 11 illustrates η = 0 and η = 0 two cases of the three types of wave collision phenomenon, in the appearance of the RW, one-breather and the firstorder semi-rational soliton amplitude both occurred a sharp increase, and after the disappearance of the RW and restore the original shape to continue to follow the established trajectory, in line with the law of the development of periodic waves and RWs after the collision. All these different phenomena will have a certain application value for nonlinear optics and long-distance transmission.

Conclusion
The higher-order RW solutions and collision phenomenon to the modified GI equation with self-steepening terms via the GDT are investigated. Starting from the modulational instability analysis of the solutions of the model, the specific steps of the nfold DT are first given based on the given Lax pair and extended to the (n, N − n)-fold DT and finally expanded from the higher-order RW solutions of the model to the collision phenomenon of multiple localized waves. The results obtained have potential applications in plasma physical and fiber communications, where they are able to propagate over long-distances without changing shape. In this paper, we mainly analyze two cases that n = 2, N = 2 and n = 2, N = 3, but of course there are more research possibilities in the subsequent work of this model, such as continuing the design procedure for deep learning for the n = 3 and N = 4 cases and also analyzing other forms of solutions of this model, including but not limited to multiple soliton solutions, numerical solutions, etc. It is worth mentioning that subsequently we intend to continue the study of spectral matrices with the same form with the help of 3 · 3 basis matrices and expect that the subsequent study will be of practical application value.