Optical soliton solutions of dispersive Schrödinger–Hirota equation with chromatic and inter-modal dispersion in a couple of law medium

In this paper, for the first time, we investigated the Kerr and parabolic law forms of the dispersive Schrödinger-Hirota equation in the presence of chromatic and inter-modal dispersions via the Sinh-Gordon equation expansion method. We gained dark, bright, kink and singular soliton solutions and we depicted their 3D, 2D, contour views. As an additional investigation, the effect of parameters on the model was examined and various 2D graphical simulations were portrayed.


Introduction
The nonlinear Schrödinger equation is one of the fundamental equations of many branches of engineering sciences. Many physical events in quantum, optics, ocean engineering, acoustic, biology, statistics and particle physics can be modeled with Schrödinger type equations. Such as, motion of a charged particle in the magnetic field (Chen et al. 2017), nonplanar waves in complex plasma (El-Tantawy et al. 2022), unidirectional waves in deep water (Adcock and Taylor 2009), chaos observation in plasma-driven damped oscillators (Nozaki and Bekki 1986), wave model in quantum field (Akinyemi et al. 2022), pulse propagation in optical fiber Khodadad et al. 2021;Houwe et al. 2023), superfluids (Berloff 1999).
Analytical methods are used to obtain solutions of PDE models and can produce various solutions according to the use of auxiliary functions. Therefore, dark, bright, singular, kink or periodic soliton solutions can be obtained depending on the chosen method. Some of the methods frequently used in the literature like unified method (Ali et al. 2020;Osman et al. 2019), mapping method (Seadawy and Cheemaa 2020), Hirota bilinear method , two variable G � ∕G, 1∕G -expansion method (Wang et al. 2023;Seadawy 2014), ansatz method (Biswas et al. 2012), extended algebraic method (Seadawy 2016) and so on. In this paper, we have used the ShGEEM to obtain the optical soliton solution of the models in Eqs. (4) and (5). The method was introduced by Yan in 2003 (Yan 2003). The method includes the Jacobi elliptical functions as arbitrary functions. Some studies which used ShGEEM are as follows; cubic-quartic Fokas-Lenells equation (Onder et al. 2022), Kundu-Eckhaus model (Kumar et al. 2018) and Konopelchenko-Dubrovsky equation (Yang and Tang 2008). The reasons why this method is frequently used by many researchers are that it is easy to use, gives reliable results and it is possible to acquire various soliton forms through the Jacobi elliptic auxiliary functions it uses.
The nonlinear evolution equation which models the pulse propagation in optical fibers with the third-order dispersion is given as follows (Biswas 2004); where the first term is the linear temporal evolution, the second is the group velocity dispersion (GVD), the third comes from Kerr-law and the last one is the third-order dispersion (3OD) with real constant . Assuming the following Lie symmetry (Bhrawy et al. 2014;Biswas 2004); Eq. (1) takes the following form: where is real coefficient of nonlinear dispersion term. The model in Eq. (3) is called as Schrödinger-Hirota (SH) equation which is one of the basic models used in nonlinear optics. In the literature, there are various studies regarding the SH models. Such as complexiton-typed soliton of SH model (Biswas et al. 2012), optical solitons of nonautonomous SH equation (Osman et al. 2019), optical solitons of conformable fractional SH model (Rezazadeh et al. 2018), variable coefficient SH model (Kaur and Wazwaz 2019), bifurcation analysis of the SH equation (Tang 2022), optical solitons of perturbed SH with spatio-temporal dispersion .
The dispersive Schrödinger-Hirota equation with Kerr law in the presence of group velocity dispersion and inter-modal dispersion is given as (Ekici et al. 2017): u = u(x, t) is soliton profile, a, c, are the coefficients of GVD, 3OD, inter-modal dispersion terms, respectively. Besides, b, d are the coefficients of Kerr law nonlinearity nonlinear dispersion terms.
The Schrödinger-Hirota equation with parabolic law nonlinearity in the presence of group velocity dispersion and inter-modal dispersion is presented as: where b 1 , d 1 are the coefficients of cubic, b 2 , d 2 are the coefficients of quintic nonlinearities terms from self-phase modulation, respectively. Equations (4) and (5) represent the propagation of optical dispersive pulse propagation in fiber. As far as we know, the models presented with the Eqs. (4) and (5) have not been studied.
Let we take a quick look at the recent studies related to Eqs. (4) and (5). In Bhrawy et al. (2014), derived the dark, bright and singular optical soliton solutions in the presence of perturbation with full nonlinearity. In Biswas et al. (2012), studied the power law form by tanh method and gained the bright and dark 1-soliton solution. Osman et al. (2019) investigated the nonautonomous SH model with power-law power law nonlinearity via unified method. In Rezazadeh et al. (2018), nonlinear conformable fractional SH model was presented and studied via (G�∕G)-expansion technique. Kaur and Wazwaz (2019) also studied the variable coefficient SH equation using different ansatz methods. In Tang  When a literature review, which is not limited to the studies listed above, is made, it is seen that the Eqs. (4) and (5) forms of the SH models have not yet been presented and studied. In addition to the investigation of the optical soliton solutions of these models, the examination of the effect of model parameters on the soliton behavior is another point that adds value to the study.
As an added ring to the aforementioned studies, this study is dedicated to not only obtaining optical soliton solutions of the dispersive Schrödinger-Hirota with Kerr and parabolic law form, presented with Eqs. (4) and (5), but also examining the effect of model parameters.
The organization of the paper is as follows; Sect. 2 includes some mathematical analysis and the extraction of the nonlinear ordinary differential equation (NODE) form of the presented model, description of the proposed method. Also, working steps of the ShGEEM are included in Sect. 2.3. Implementation of the method to the Eqs. (4) and (5) are included in Sects. 3.1 and 3.2 respectively. In addition, graphs and discussion on the obtained results are placed in Sect. 4. Lastly, Sect. 5 is devoted for Conclusion.

Extracting NODE form of Eq. (4)
Let we give the following wave transformation: where v is velocity, , are wave number and frequency. Besides, x, t, 0 are the spatial, temporal coordinates and phase constant, respectively. By injecting Eq. (6) into Eq. (4) allows the following equations which are generated from the real and imaginary parts: where U = U( ) , superscripts indicate the ordinary derivative of U w.r.t . If we integrate the Eq. (8) with respect to once and assume the integration constant as zero, then we derive: Using the homogeneous balance between the Eqs. (7) and (9), we get: and the following constraints: Substituting Eq. (11) into Eq. (7), we reach the NODE form of Eq. (4) as: Utilizing the homogeneous balancing principle, with the terms U ′′ and U 3 , the balancing constant N is computed as N = 1.

Extracting NODE form of Eq. (5)
Assume Eq. (6) and substitute into Eq. (5), then separate the real and imaginary parts as follows: where U = U( ) . Integrating the Eq. (14) with respect to once and supposing the integration constant as zero, we retrieve: Balancing the Eqs. (13) and (15), we write: Equation (16) serves the following restrictions: where Δ 1 = 2cd 2 + 5b 2 . Lastly, we obtain the following NODE form of Eq. (5) by substituting Eq. (17) into Eq. (13): Balancing the terms U 5 and U ′′ , the balance constant is computed as N = 1 2 . Thus we need to define the following equality: Then substitute Eq. (19) into Eq. (18), we obtain the following NODE form of Eq. (5): In Eq. (20), considering the homogeneous balancing principle between the terms V 4 and V ′′ V , the balance constant is computed as N = 1.

Introduction of the ShGEEM
This section presents the basic features and application of the ShGEEM (Yan 2003). The ShGEEM is easy to apply, effective, reliable and widely used. ShGEEM has the following steps: • Step I: Suppose the following general partial differential equation (PDE).

Application of ShGEEM to the model-1 in Eq. (4)
Remembering that the balance constant N = 1 in Eq. (12), turns into following form: where A 1 and B 1 can not be zero simultaneously. Substitute Eq. (29) into Eq. (12) by considering Eq. (26), collect the terms cosh(W) i sinh(W) j and equate all coefficients to zero, get the following algebraic system:  Equations (33) and (34) represent bright and singular soliton solutions and they are depicted by Figs. 1 and 2, respectively. On the other hand, the following solutions are obtained via using SET 2 , Eqs. (27) and (28), respectively:

Results and discussion
(39) Figure 3 shows 3D, contour and 2D plots of |u 1,3 (x, t)| 2 in Eq. (35). The plots were obtained with = 2, v = 1, c = 1, b = 2, d = −1 and 0 = 0 values. 3D (Fig. 3a) and contour (Fig. 3b) sign the dark soliton. On the other hand, blue, red and green plots represent the traveling waves for t = 1, 2, 3 in Fig. 3. According to the Fig. 3c, the traveling direction of soliton is the positive-x axis. In Fig. 3d, we investigated the effect of the parameter d which is the nonlinear dispersion coefficient on the u 1,3 (x, t) in Eq. (35). Figure 3d has three different waves and these waves were plotted with fixed = 2, c = v = 1, b = 2, 0 = 0, t = 1 and d = −1, −2, −3 respectively. According to the Fig. 3d, decreasing parameter d affects the wave behavior in terms of increasing wave amplitude.
The modulus of u 2,1 (x, t) is shown by 3D, contour, and 2D graphs in Fig. 4. The plots in Fig. 4a-d, which correspond to the 2D, contour, and 3D depictions of |u 2,1 (x, t)| , were sketched for = 2, c = 2, b 2 = 2, v = 2, d 2 = −1 and 0 = 0 values. The u 2,1 (x, t) solution is a kink soliton solution, as shown by the graphs. There are three distinct waves with colorings blue, red, and green plotted for t = 1, 2, 3 , respectively in Fig. 4c. In this way, Fig. 4c illustrates that the direction of movement is in the positivex axis. Contrarily, Fig. 4d illustrates the impact of the parameter d 2 on the u 2,1 (x, t) solution in Eq. (41). In Fig. 4d, there are three waves that were produced using fixed  Fig. 4d. Figure 5 shows the sketches of modulus of u 2,3 (x, t) in Eq. (43). 3D, contour and 2D plots were obtained with = 2, c = 2, b 2 = 2, v = 2, d 2 = 1 and 0 = 0 values in Fig. 5a-c. In addition, Fig. 5c has blue, red and green waves which were plotted for t = 1, 2, 3 respectively. According to the Fig. 5c, the soliton travels through the positive-x axis. Lastly, in Fig. 5d, we investigated the d 2 parameter effect on u 2,3 (x, t) solution in Eq. (43). Figure 5d has three waves and the waves are plotted with fixed = c = b 2 = v = 0 = 2, t = 1 and d 2 = 1, 2, 3 values. According to the Fig. 5d, two different effects appear. The first effect is the amplitude of bright soliton and the second effect is shifting. An increase in d 2 affects the wave by both increasing the amplitude of the wave and shifting the wave in the positive-z axis.
As the final emphasis of this section, we would like to point out that all solution functions obtained in the article are checked for providing Eqs. (4) and (5) by backsubstituting with Maple and Mathematica symbolic computation programs.

Conclusion
In this study, we successfully investigated Kerr and parabolic laws of self-phase modulation forms of the dispersive Schrödinger-Hirota equation in the presence of group velocity dispersion and intermodal dispersion terms using the ShGEEM method. The fact that the Schrödinger-Hirota model investigated in the article have not been studied before, shows that the results and analysis will contribute to the literature. As a result of the application, bright, singular and dark soliton types were obtained for the first model. In contrast, kink, bright and singular soliton solutions were obtained for the second model. In addition to all these, the effect of the change in the parameters of the model and the change in the wave behavior were also examined. Fractional and stochastic forms of the investigated model that are the subject of the article are open topics that may be the target of both us and other researchers working in this field in the future.
Author contributions All parts contained in the research carried out by the authors through hard work and a review of the various references and contributions in the field of mathematics and Applied physics.
Funding No funding for this article.
Data availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Declarations
Conflict of interest This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. The authors did not have any competing interests in this research.

Ethical approval
The Corresponding Author, declare that this manuscript is original, has not been published before, and is not currently being considered for publication elsewhere. The Corresponding Author confirm that the manuscript has been read and approved by all the named authors and there are no other persons who satisfied the criteria for authorship but are not listed. I further confirm that the order of authors listed in the manuscript has been approved by all of us. we understand that the Corresponding Author is the sole contact for the Editorial process and is responsible for communicating with the other authors about progress, submissions of revisions, and final approval of proofs.