Comment on ″Optical soliton to multi-core (coupling with all the neighbors) directional couplers and modulation instability″ [Eur.Phys.J.Plus(2021)136:325]

In a recent paper, Abbagari et al. (EPJP 136:325, 2021) investigated the nonlinear multi-core coupler (coupling with all neighbors) with the parabolic law nonlinearity via the so called new sub-ODE method. In this comment, we show that all solutions presented by Abbagari et al. (EPJP 136:325, 2021) do not satisfy the considered equation and therefore are incorrect. Some important attentions to the authors (EPJP 136:325, 2021) have been presented.


Introduction
Recently, Abbagari et al. [1] investigated the nonlinear multi-core coupler (coupling with all neighbors) with the case of parabolic law nonlinearity in the form where 1 ≤ j ≤ M and κ j m are the coupling coefficients with all neighbors, while the remaining parameters are provided in the case of twin-core couplers [2]. The authors [1] introduced the traveling wave transformation ( j) (x, t) φ j (ξ )e i θ (x, t) (2) and ξ K (x − v t). They take θ (x, t) in the form which represents phase component, while λ, ω and θ 0 represent the frequency, wave number and phase constant, respectively. The authors [1] have used Eqs. (2) and (3) (which are Eqs.(67) and (68) in [1]) and transformed Eq. (1) to an ordinary differential equation in the form with the constraint conditions and v −2a j λ (6) where v is the velocity of the soliton. The authors [1] have adopted the new sub-ODE method [3] and assume the formal analytic solution where p is an arbitrary constant and F(ξ ) satisifies the generalized first order ODE

Comment 1
Comparing our results (11) with the results (9) of the authors [1], we find that they got the correct values of the constants A, B, C, D, E, μ, a j , but they made mistakes when they found the values of ω, ξ j , ς j . We note here that "η j " appear in Eq. (9)of the results of the authors [1],while η j are not originally existed in Eq. (4) to be solved (which is Eq. (72) in [1]). In addition, the constraint condition (12) is missed by the authors [1].

Comment 2
The authors [1] provided the solutions (76)-(88) of Eq. (1) which are all incorrect. This can be easily checked by substituting these solutions directly into the Eq. (4), as follows: Let us take for example, the following solutions of Eq. (4) (which is Eq. (72) in [1]) where A B D 0: for ε 1 and the other solutions of [1] are omitted here for simplicity. In case of Eq. (13), the frequency of the obtained optical bright multi-core couplers directional ω becomes.

Comment 3
It should be mentioned here that we can derive (amend the incorrect solutions (1) via the same used method which are not found by the authors [1], but this is not the purpose of this comment paper and so this is left to the interested readers.

Important attention to the authors [1]
The incorrect solutions (76)-(88) are found in the paper of the authors [1] and these are not actual solutions of Eq. (4) (which is Eq. (72) in [1]). This indicates that serious mistakes have been occurred in the commented paper [1]. By this comment paper, these mistakes have been cancelled for the search of extracting exact solutions of nonlinear physical systems found in [1].
We notify the authors who work in the field of nonlinear optical birefringent fibers and optical metamaterials not to deal with the results of the commented paper [1] because they are incorrect as well as not general as claimed by the authors [1] than those contained in [2,4,5]. We draw the attention of the authors [1] that they did not pay attention to the sixth error described in [6] by Kudryashov which states that "some authors do not check solutions of differential equations. " We hope our comment paper be a good contribution to the authors [1] when they adopting the new sub-ODE method to nonlinear equations in physical problems in future work.

Conclusion
We have shown some errors occurred in [1] and have indicated that all the presented solutions in [1] are not actual solutions of the considered equation. In addition, the authors [1] do not use any analytics, but simply assume the used architecture of the solution to the ODE Eq. (4) (which is Eq. (72) in [1]) and presented incorrect solutions of the considered model equation.