Exact Soliton Solutions for Nonlinear Perturbed Schrödinger Equations with Nonlinear Optical Media
Round 1
Reviewer 1 Report
The author constructs exact traveling wave type solutions to a generalized NLS equation with higher-order nonlinearity. The topic is of current scientific interest and the results would supplement the existing literature well. There are a few suggestions for revision. The system (8) can be presented by a single ODE for g by using the second equation (8), and the whole system presents an ODE constraint on a second-order derivate of g, which is $ B g_{\xi\xi}= g ( (B^2 \delta +A^2 (\delta +2) ) B g^2 -\delta ( A^2 +2 B^2 ) g +\delta B ) $. Therefore, all the solutions of (9)-(12) need a condition A^2=B^2. The adopted expansions throughout the manuscript could be viewed as applications of the transformed rational function method or more generally the multiple exp-function method. Such expansion ideas have been applied to seeking exact solutions, including soliton and rogue wave solutions, for the standard NLS equation (see, e.g., Appl Math Comput, 215(2009), 2835-2842). Another kind of exact solutions called lumps has been studied extensively recently (see, e.g., Pramana J Phys, 94(2020), 43), which provide best approximations to real physical nonlinear waves, and could exist in even linear wave models (see, e.g., Mod Phys Lett B, 33(2019), 1950457). Those are all very interesting in the studies of nonlinear dispersive waves. It is expected that the author could revise and amend the manuscript to improve the statements so that the interested audience could benefit more from reading such an article, and a revised manuscript would be recommended for publication in the journal.
Author Response
Respond to Referee 1.
At the First, I thank the first referee for his constructive remarks and for his efforts in reviewing this paper.
The system (8) can be presented by a single ODE for g by using the second equation (8), and the whole system presents an ODE constraint on a second-order derivate of g, which is $ B g_{\xi\xi}= g ( (B^2 \delta +A^2 (\delta +2) ) B g^2 -\delta ( A^2 +2 B^2 ) g +\delta B ) $. Therefore, all the solutions of (9)-(12) need a condition A^2=B^2.
Although I have verified that the solution (9) satisfies the system (8) when Also I have verified that the solution (11) satisfies the (8) when for any A and B. But, when, we reduced the system (8) to the one trial equation must be added the condition . We have mentioned this part in remark 1 in section2. I have thanked the referee for that outstanding remark.
The adopted expansions throughout the manuscript could be viewed as applications of the transformed rational function method or more generally the multiple exp-function method. Such expansion ideas have been applied to seeking exact solutions, including soliton and rogue wave solutions, for the standard NLS equation (see, e.g., Appl Math Comput, 215(2009), 2835-2842). Another kind of exact solutions called lumps has been studied extensively recently (see, e.g., Pramana J Phys, 94(2020), 43), which provide best approximations to real physical nonlinear waves, and could exist in even linear wave models (see, e.g., Mod Phys Lett B, 33(2019), 1950457).
I have mentioned this in the section 1 and section2. I have added the references [23]-[26].
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The manuscript written by Khaled Gepreel describes the solution of the nonlinear perturbed Schrödinger equation with applied q-deformed hyperbolic function and q-deformed trigonometric functions methods. The work is the discussion with the article published by Arshed (Optik : Int. J. Light and Electron Optics 220 (2020) 165123).
In the article there was presented the solution of nonlinear Schrödinger equation divided into families, which successfully present e.g. Kerr law, power law, parabolic law, dual-power law, saturating law and exponential law. In my opinion the presented topic is worth of publishing, as indicated with the increasing interest in solition studies. An extensive work was done in order to acquire the solution to the nonlinear Schrodinger equation. At the same time, there is no broader reference to the conducted research especially for optical solitons, with lack of some classical works (Stegeman, George I., and Mordechai Segev. "Optical spatial solitons and their interactions: universality and diversity." Science 286.5444 (1999): 1518-1523.; Segev, Mordechai, et al. "Spatial solitons in photorefractive media." Physical Review Letters 68.7 (1992): 923.; Chowdury, Amdad, Wieslaw Krolikowski, and N. Akhmediev. "Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits." Physical Review E 96.4 (2017): 042209.), but also in application of q-deformation of hyperbolic and trigonometric functions for nonlinear Schrodinger equation (Alina Dobrogowska, "The q-deformation of Hyperbolic and Trigonometric Potentials", International Journal of Difference EquationsISSN 0973-6069, Volume 9, Number 1, pp. 45–51 (2014) http://campus.mst.edu/ijde; Eğrifes, Harun, Doğan Demirhan, and Fevzi Büyükkılıç. "Exact solutions of the Schrödinger equation for the deformed hyperbolic potential well and the deformed four-parameter exponential type potential." Physics Letters A 275.4 (2000): 229-237.)
While the equations presented in the articles are based on the previous work of Arshed it would be necessary to present the differences in equation (4) while some of parameters become negative either the wave function in comparison to the cited work. I understand there was takes such a convention, which should be indicated, however it does not affect the soliton velocity.
The quality of the English language in the article should be increased. Followed by the correction of multiple typos. Below I present some discovered errors that should be corrected. While describing Kerr law, please use capital letter for "Kerr" unlike in line 56. Line 196 inappropriate word "researcher" describing few articles published by many people. Line 251 please remove one bracket. In the description of soliton solutions behavior please double check the referred figures and solutions numbers, which are mixed and figures have the equations numbers. Reference number 10 should be detailed.
I am considering publishing the article after the indicated changes.
Author Response
Respond to Referee 2.
At the First, I thank the second referee for his constructive remarks and for his efforts in reviewing this paper.
- At the same time, there is no broader reference to the conducted research especially for optical solitons, with lack of some classical works (Stegeman, George I., and Mordechai Segev. "Optical spatial solitons and their interactions: universality and diversity." Science 286.5444 (1999): 1518-1523.; Segev, Mordechai, et al. "Spatial solitons in photorefractive media." Physical Review Letters 68.7 (1992): 923.; Chowdury, Amdad, Wieslaw Krolikowski, and N. Akhmediev. "Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits." Physical Review E 96.4 (2017): 042209.), but also in application of q-deformation of hyperbolic and trigonometric functions for nonlinear Schrodinger equation (Alina Dobrogowska, "The q-deformation of Hyperbolic and Trigonometric Potentials", International Journal of Difference EquationsISSN 0973-6069, Volume 9, Number 1, pp. 45–51 (2014) http://campus.mst.edu/ijde; Eğrifes, Harun, Doğan Demirhan, and Fevzi Büyükkılıç. "Exact solutions of the Schrödinger equation for the deformed hyperbolic potential well and the deformed four-parameter exponential type potential." Physics Letters A 275.4 (2000): 229-237.)
We have added some of related references, especially for optical solitons and the q-deformation of Hyperbolic - Trigonometric Potentials please see references [18-22].
- While the equations presented in the articles are based on the previous work of Arshed it would be necessary to present the differences in equation (4) while some of parameters become negative either the wave function in comparison to the cited work. I understand there was takes such a convention, which should be indicated, however it does not affect the soliton velocity
I thank the referee very much for this remark. I have improved the Typing error in Eq. (4) Please see the end of Section1. 3. The quality of the English language in the article should be increased. Followed by the correction of multiple typos. Below I present some discovered errors that should be corrected. While describing Kerr law, please use capital letter for "Kerr" unlike in line 56. Line 196 inappropriate word "researcher" describing few articles published by many people. Line 251 please remove one bracket. We have improved the English and the arrangement of equations in all my paper. In the description of soliton solutions behavior please double check the referred figures and solutions numbers, which are mixed and figures have the equations numbers. Reference number 10 should be detailed. I have improved the description of soliton solutions and checked referred figures and solutions numbers. I have improved reference 10.
I hope this new version of the article responds to all the comments by referees . If there is anything else, please do not hesitate to contact me.
Please Accept my science agreeing .
Dr. Khaled A. Gepreel
Author Response File: Author Response.pdf
