Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.


Introduction
The GIE is currently used in quantum field theory, nonlinear optics, weak nonlinear dispersive water waves and in telecommunication industry. The GIE was studied by many researchers, where different techniques were explored. A new direct approach namely exp(− ( ))-expansion method was employed to seek exact solutions of GIE which include variety of models (Kadkhoda and Jafari 2017). The modulation instability and higherorder rogue waves for the GIE were investigated in Lou et al. (2021). The multi-component nonlocal reverse-time GIE was derived through nonlocal group reduction of the multicomponent GIE (Zhang and Dong 2022). The traveling wave solutions of the generalized GIE were investigated (Kudryashov 2020). An explicit N-fold Darboux transformation (DT) with multi-parameters of coupled derivative nonlinear Schrödinger equation was constructed together with a gauge transformation of a spectral problem . In Xu and He (2012), the GIE was defined, via DT, by a quadratic polynomial spectral problem with 2 × 2 matrix coefficients. The standard DT for Gerdjikov-Ivanov (GI) equation was presented (Yilma 2015). In Ji and Zhai (2020), based on the symmetric relations of the Lax pair, 2N-fold DT for the GIE was constructed. Higher order rogue wave solutions for the GIE were depicted explicitly in terms of a determinant expression (Guo et al. 2014). In , bilinear forms of the coupled GIE were derived and N-soliton solutions to the equation were obtained by Hirota's method. The solitary wave solutions of the complex perturbed GIE were investigated by employing two methods (Khater 2021). Exact solutions of the space-time conformable fractional perturbed GIE were derived in Yaşar et al. (2018). The GIE was decomposed into two systems of solvable ordinary differential equations and the solutions were derived in terms of the Riemann theta functions (Dai and Fan 2004). The GI-type derivative nonlinear Schrödinger equation, with fifth degree nonlinearity was studied for initial value problem (Guo and Liu 2019). The DT was used to derive a variety of soliton solutions of the nonlocal nonlocal nonlinear GIE . The double and triple pole soliton solutions for the GIE with zero boundary conditions and nonzero boundary conditions were studied via the Riemann-Hilbert method (Peng and Chen 2022). The perturbed GGIE consisting of group velocity dispersion and quintic nonlinearity coefficients was studied in Baskonus et al. (2020). The optical solitons to perturbed GIE with space-time dispersion were derived in Al-Kalbani et al. (2021). The GIE was investigated by the Riemann-Hilbert approach and the technique of regularization (Nie et al. 2018). In , the Riemann-Hilbert method for initial problem of the vector GIE, and obtain the formula for its N-soliton solution was considered. Soliton molecules solutions were obtained by velocity resonance for the GIE. Further, and n-order smooth position solutions were generated by means of the general determinant expression of n-soliton solution (Yang et al. 2020). In Ding and Liu-Q (2019), two types of the breathers on the periodic background, and the k-th order rogue waves were constructed for the GIE, based on the existing N-th order analytic solutions.Under suitable hypothesis for the current velocity, the GI envelope solitons were derived and discussed in Lü et al. (2015). The coupled GIE was investigated by using Lax pair, Darboux transformation (Dong et al. 2021). In Hassan et al. (2021), the collective variable technique was used to explore the GIE.The perturbed optical solitons to the time-space fractional GIE with conformable derivatives was investigated (Younis et al. 2021). New optical solutions of the conformable fractional perturbed GIE were explored in Zulfiqar and Ahmad (2021). By using the Darboux transformation and some limit technique of higher-order algebraic soliton solutions of the GIE, the determinant representation was derived in  .Using a truncated M-fractional derivative the perturbed GIE was studied inZafar et al. (2022). A new coupled GIE was proposed and its integrability aspects were studied via the inverse scattering transform (Wu 2019). In Zhu et al. (2017), the Fokas unified method was used to analyze the initial-boundary value problem of two-component GIE on the half-line (Zhu et al. 2017). In , a spectral problem and the associated GI hierarchy of nonlinear evolution equations was presented. The perturbed GGIE with full nonlinearity was considered and periodic, quasi-periodic, bifurcation and chaotic patterns were shown (Rafiq et al. 2023). Exact single traveling wave solutions to the nonlinear fractional perturbed GIE were captured by the complete discrimination system for polynomial method and the trial equation method (Xiao and Yin 2021). Further, some relevant works on complex field equation were achieved (Nizovtseva et al. 2022;Wazwaz 2016Wazwaz , 2017Lu et al. 2015).
In the present work, we propose a modified GIE (MGIE) and the UM is utilized to study the MGIE (Abdel-Gawad 2022a, b, c; Abdel-Gawad and Abdel-Gawad 2022; Abdel-Gawad 2012). Our objective is to finding approximate analytic soliton solutions to MGIE. By the UM the maximum error is controlled.
The organization of this work is as in what follows. Sect. 2 is concerned with the model equation and a brief account of the UM. Polynomial solutions are derived in Sect. 3, while in Sect. 4 rational solutions are found. Sections 5 and 6 are devoted to discussions and modulation instability respectively. conclusions are presented in Sect. 7.

The model equation
We consider the GI derivative nonlinear Schrödinger equation Nie et al. 2018).
where w = w(x, t) = is the complex field function and w * is the complex conjugate of w. The terms can be identified as the time rate, the quadratic dispersion, the quantic power nonlinearity and the last term may be considered to stand for Raman scattering. Eq. (1) generalizes to, Here, we propose a modified GIE by, where the last term in (3) describes nonlinear dispersion.
Here, we are interesting with obtaining approximate analytic solution of (3). First, we introduce a transformation with complex amplitude, which describes soliton-periodic wave collision, 298 Page 4 of 18 into (3) and we get for the real and imaginary parts, (5) and (6) reduce to,

Brief account of the UM
The UM asserts that the solutions of NLEEs are written in polynomial and rational form in an auxiliary function with an adequate auxiliary equation.

Polynomial solutions (PSs)
The PSs of (7) and (8)  The integrability of (7) and (8) can be tested by the Painleve' analysis. Here, integrability is tested in the sense of existence of (9). A necessary condition for the solutions in (9) to exist, is that there exist integers n, m, and k.To this issue, two conditions are examined, the balance condition (BC) and the consistency condition (CC). We consider the case when p = 1.
The BC results from balancing the nonlinear term with the nonlinear dispersion (NLD), we get n = m = k − 1. To determine k, we use the CC. We need to calculate the following; (i) The number of equations that results when inserting (9) in (7) (or (8)) and by setting the coefficients of g i (z), i = 0, 1, 2, ...etc. equal to zero (which is (5k − 4)). (ii) The number of arbitrary parameters in (9), (which is 2k + 1).The CC reads 5k − 4 − (2k + 1) ≤ m + r − 1,where m is the highest order derivative and r the degree of the NLD, that is r = 3. Finally, we get 0 ≤ k ≤ 3.

Rational solutions (RSs)
To avoid repetition, we write directly the PS in the form, The balance condition is; r = k − 1.
It is wort noticing that here, when inserting (9) (or (10)) into (7) and (8), and by setting the coefficients of g i (z), i = 0, 1, 2, ...etc. equal to zero. When the solutions of the equations that result, are non trivial, then an exact solution exists. Thus (7) and (8) are integrable. If the solutions of some equations are trivial, then (7) and (8) are not integrable. Which holds in the present case. The strategy of finding the solutions of (7) and (8) is to take into consideration of the non-trivial solution, while non-zero ( residue) terms are considered as errors. The maximum error (ME) is controlled via an adequate choice of the relevant parameters.
In the literature, it is established that a necessary condition for a complex field equation (cf. (3)) to be integrable is that the real and imaginary parts are linearly dependent. But this condition is not in general sufficient Lu et al. 2015).

When p = 1
Here, we consider two cases.
Case (i).When k = 2 , ( n = 1) We write the solution of (7) and (8) in the form, where the last condition leads to that U(z) and V(z)are linearly dependent.
By inserting (11) into (7) and (8) and by setting a part of the coefficients of g i (z), i = 0, 1, 2, ...etc.equal to zero, we get, The residue terms are lengthy and will not produced here. By taking, we find the errors are (Table 1), We find that the ME is 1.5 × 10 −4 . Finally the approximate solutions of (5) and (6) are, By using (14), Rew(x, t) is displayed in Fig. 1i and ii. Case (ii).When k = 3, (n = 2) In this case the solution is written, From (15) into (7) and (8) and by the same way as in Sect. 3.1, gives rise to, u(x, t) = − 75b 1 c 2 1 p 16c 2 k (1+ tanh( b 2 1 t(−6400c 2 1 k 4 p 2 +101250c 4 1 k 2 p 4 +1265625c 6 1 p 6 −512k 6 ) 1536c 2 2 k 2 p 2 In the residue terms, we take, and the errors are given in Table 2. We find the the ME is 1.2 × 10 −4 . The solutions of (5) and (6) are, where q is given in (16). The results in (18) are used to display Rew(x, t) in Fig. 2i and ii.

When p = 2
We consider the case when k = 2, (n = 1 ) In this case, we write, From (19) into (7) and (8) leads to, In the residue terms, by taking, the errors are given in Table 3. The ME is 5 × 10 −4 . The solutions of (5) and (6) are,  The results in (22) are used to display Rew(x, t) in Fig. 3i and ii.

RSs
We distinguish the two cases when p = 1 and p = 2.

When p = 1
We consider two cases.

Case (i-2)
When the AE is, From (10) and (35) into (7) and (8), we get, By taking, By the same way as in the above, the ME is −6.06147 × 10 −11 . The solutions of (5) and (6) are, By using (38) Rew(x, t) is displayed in Fig. 7i and ii.

Discussions
Here, it is found that the MUM, which is a new method, is of low time cost in symbolic computation. Further, in this method, the ME is controlled. Many soliton structures are revealed in the following. Figure 1i shows M-shaped soliton on x > qt∕p and breathers on x < qt∕p . Figure 1ii shows self-phase modulation. Figure 2i and ii show soliton with double kink-periodic wave, with no insignificant change against . Figure 3i and ii show exotic M-shaped soliton on x > qt∕p and periodic wave on x < qt∕p.Significant change for higher occurs. Figure 4i shows two-level complex M-shaped soliton forming cable-shape soliton, while Fig. 4ii M-shaped soliton. Figure 5i and ii show M-shaped soliton with trapping. Figure 6 i and ii show complex chirped solitons. Figure 7i and ii sow solitons with alternative amplitude.

MI
The modulation stability is concerned of studying the stability of normal mode. This holds for a complex field system.
(i) To this issue, we nominate a parameter that characterizes a relevant phenomena.
(ii) We search for the (critical) value of the parameter, so that, above this value modulation instability (MI) triggers. (iii) The MI, in turn, enforce the dominance of this phenomena.
In the present case, we take the parameter which is the coefficient of the nonlinear dispersion.
(3) has the solution, when Ω = −A 4 + A 2 K 2 + K 2 . Now, we use the perturbation expansion, The solution of (40) is detM = 0, which leads to the eigenvalue equation, We solve the eigenvalue problem in (40) subjected to the boundary conditions (BCs) U(±∞) = 0 and V(±∞) = 0 . Thus, we can take, From (42) into (41), we find that, In (43), we remark that is the dominant coefficient. When < − A 2 , < 0,then modulation stability holds. Now we consider the case > − A 2 and the sign of Δ is crucial. When Δ > 0 , MI occurs and the bifurcation is Hopf bifurcation for = − A 2 . When Δ < 0, we determine 0 where 0 > > − A 2 . The value of 0 for MI to trigger is determined by displaying Δ against for different values of the relevant parameters in Fig. 8i-iii. Figure 8i-iii show that Δ < 0 when 0 > 0.6, and in this case MI triggers. (42)

Conclusions
A novel modified Gerdjikov-Ivanov equation is presented. It is found that the aforementioned equation is not integrable. So, only, approximate analytic solutions are amenable. The solutions of this equation are derived by using the modified-unified method, where they are found in polynomial and rational forms in an auxiliary function. For utilizing this method it is required to consider some non-zero coefficients of the auxiliary function (residue terms) as errors. The maximum error can be controlled by an adequate choice of the relevant parameters. Multiple solitons structures, breathers solitons, chirped soliton and M-shape soliton with trapping, complex chirped solitons, among others, are observed. The modulation instability is analyzed and it is shown that it triggers when the coefficient of the nonlinear dispersion exceeds a critical value.