Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearity

Md. Mamunur Roshid; Mohammad Safi Ullah; M. M. Rahman; Harun-Or- Roshid

doi:10.1371/journal.pone.0314681

Peer Review History

Original SubmissionJuly 10, 2024
6 Aug 2024 Decision Letter - Boris Malomed, Editor PONE-D-24-28379Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearityPLOS ONE Dear Dr. Roshid, Thank you for submitting your manuscript to PLOS ONE. Reviews provided by three referees imply that the manuscript is definitely inappropriate for the publication in the present form. In particular, one review, presented by a top expert in the field, is outright negative. If you believe that you can thoroughly revise the paper and properly address critical comments, you have an option to resubmit the paper. Please submit your revised manuscript by Sep 20 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. 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Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Partly Reviewer #2: Partly Reviewer #3: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: No Reviewer #2: N/A Reviewer #3: Yes ****** 3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: No Reviewer #2: Yes Reviewer #3: Yes ****** 4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes Reviewer #3: Yes ****** 5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: I have read this paper carefully. I see that the authors work very hard but unfortunately they have not found new results. My remarks on this paper are the following. 1) I believe that the authors have to give references on Eq. (1) where this equation was first derived. 2) After equation (1) the authors wrote that the expression in the y direction but I cannot see any function which depends on y direction. I see the only function R(x,z,t). 3) The Section (2.2) is devoted to consideration Eq. (2), but Eq. (2) is nonlinear ordinary differential equation. 4) Solution of Eq. (2) is well-known but the authors present a few of exact solutions. 5) The looked for of Equation (2) using formula (3) but they did not say what q_1, q_2 and p_1.The authors present different formulas for solution of Eq. (2) but these solutions are the same. More interesting Section 2 of paper. They present the consideration the chaotic nature. 6) However the approach by authors is well known as well. See , for example, the book “Chaotic Transitions in deterministic and Stochastic Dynamical Systems” by Emil Simiu, Prinston University Press. Taking into account the remarks above unfortinately I do not recommend this paper for publication. Reviewer #2: Review of the article: Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearity. Manuscript Number: PONE-D-24-28379 ESSENTIALS OF THIS ARTICLE: In this paper the authors find several solutions of a fractional version of the 2D NLS (two-dimensional nonlinear Schrödinger) equation (which the authors call “paraxial wave equation”). In this article “x” is the evolution variable, and “t” and “z” are the “transversal” variables. The second-order time derivative has been replaced by an M-fractional derivative, which is a generalization of the conformable derivative. Proposing a cleverly chosen similarity reduction, defined by the equations shown in (6), and the form of R(x,z,t) written above Eq. (6), the authors manage to reduce the fractional 2D NLS equation (1) to the ODE (ordinary differential equation) shown in (7). Then several particular solutions of this ODE are obtained by means of the EJEFE method. Afterwards the authors study a perturbed form of Eq. (7) [shown in (17)], and obtain quasiperiodic and chaotic solutions of this perturbed equation. OPINION OF THE ARTICLE: The fractional 2D NLS equation studied in this article is an interesting equation, the solutions obtained are interesting, and the procedure to obtain these solutions is also interesting. The reduction of the fractional 2D NLS equation (1) to the ODE (7) is an excellent result (the most interesting result of the paper, from my point of view). HOWEVER, the paper contains some errors and undefined symbols, which are probably the consequence of a hasty and careless writing. Moreover, some of the statements made in this article are, in my opinion, incorrect. ISSUES THAT SHOULD BE CORRECTED: ISSUE 1: In the fourth line below Eq. (1) it is written “in the y and t directions”. This statement is wrong. It should say “in the t and z directions”. ISSUE 2: In pag. 3 of the paper the authors mention References [34]-[43]. But the list of references presented at the end of the article only contains 32 references!!! ISSUE 3: In the definition of the M-fractional derivative the meaning of the “t” with subscript kappa (which appears within the argument of the function “u”) is not explained. Moreover, in the right-hand-side (rhs) of the equation which defines this derivative, the parameter “M” does not appear. Please explain. ISSUE 4: Below the definition of the M-derivative, in the section “Features”, in the equations (a), (b), (c) and (f), the meaning of the letters with the subscript kappa (le letters l, m, u and v) is not explained. ISSUE 5: Two lines below Eq. (2) the authors say that: “Balancing H’’ and H3, yields M=1” but the authors have not said what is “M”. As I said in the Issue 3, the parameter M does not appear in the rhs of the equation which defines de M-derivative. So, it is incomprehensible how the equation M=1 is obtained. ISSUE 6: In the first line below Eq. (5) the authors mention the equations (6)-(8), but these equations have not been presented at this point of the article. Moreover, it is quite clear that in this line [below Eq. (5)], the authors are not referring to the Eqs. (6)-(8) which appear in Section 3. ISSUE 7: Two lines below Eq. (9) the equation M=1 appears again. As I said in Issue 5, we don´t know what is M, and therefore the origin of the equation M=1 is incomprehensible. ISSUE 8: Two lines above Eq. (17) the authors mention a “superficial component”. This term is a misleading, as it is not related to any “surface” at all. Moreover, as far as I could see, this term is never mentioned in Ref. [31]. ISSUE 9: In the line above Eq. (17) the authors mention the “system Eq. (30)”. But the article contains 17 equations, consequently the mention of Eq. (30) is obviously incorrect. ISSUE 10: The numbering of the sections of this article is an absolute disaster, revealing an extremely careless work. In page 3 the Sec. 2 begins. In page 5, Sec. 3 begins. Then, in page 6, another “Section 2” begins. Then, in page 8, the Section 8 begins!. And in page 9, the Section 5 begins! Finally, in page 14, the Section 6 closes the paper! ISSUE 11: In page 6, in the Section entitled “Chaotic nature”, the authors modify their Eq. (7), introducing an oscillatory perturbation in time [the last term in the second equation shown in (17)]. And then they show that the solutions of this new ODE [Eq. (17)] exhibit a chaotic behavior. But the chaotic behavior of the solutions of Eq. (17) does not imply that Eq. (7) [or Eq. (1)] is a chaotic system. The chaotic nature of a system (i.e., of an equation), is proved by perturbing THE INITIAL CONDITIONS USED, and NOT by perturbing the equation itself. Even though some authors may consider that an equation is “chaotic” if the solutions exhibit a huge change when the equation is perturbed, THIS IS NOT THE USUAL DEFINITION OF CHAOS. At least, it is not the definition of “chaos” used by Edward Lorenz (who was one of the founders of the theory of chaos). Consequently, I consider that the authors must explain clearly that the chaotic solutions of Eq. (17) do not imply that the system described by Eq. (7) [or Eq. (1)] is chaotic. RECOMMENDATION: I consider that if the authors correct the eleven issues mentioned above, the article will be suitable for publication in PLOS ONE. Reviewer #3: The authors investigated various types of wave solutions in the M-fractional paraxial wave equation with Kerr law nonlinearity, including periodic waves, lumpperiodic waves, periodic breather waves, kink-bell waves, kinky-periodic waves, anti-kinky-periodic waves, double-periodic waves. In addition, the chaotic phenomena are also studied in this work. The wave equations with M-fractional derivative attracted many attentions in recent years and the results presented in this work are interesting. However, some issues should be fixed, and my comments are as follows: 1. The physical scene (settings) for M-fractional derivative may be a good supplement to this work. 2. I am wondering if such M-fractional derivative can be expressed by other easier forms that can be solved numerically. 3. Where is the Eq. (30) the authors pointed out before Eq. (17) ? 4. I assume the references of [34] to [42] in the Introduction should be wrong. The authors should check it. 5. Some review should be added for the better understanding of fractional derivative. [1] B. A. Malomed. Optical solitons and vortices in fractional media: a mini-review of recent results. Photonics 8(9), 353 (2021). [2] B. A. Malomed. Basic fractional nonlinear-wave models and solitons. Chaos 34, 022102 (2024). [3] D. Mihalache. Localized structures in optical media and Bose-Einstein condensates: An overview of recent theoretical and experimental results. Rom. Rep. Phys. 76, 402 (2024). ****** 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No Reviewer #2: No Reviewer #3: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. https://doi.org/10.1371/journal.pone.0314681.r001
Revision 1
21 Aug 2024 Author Response Reviewer Response Reviewer #1: I have read this paper carefully. I see that the authors work very hard but unfortunately they have not found new results. My remarks on this paper are the following. 1) I believe that the authors have to give references on Eq. (1) where this equation was first derived. Response: The necessary correction is done in section 1 and we added the reference in [19]. 2) After equation (1) the authors wrote that the expression in the y direction but I cannot see any function which depends on y direction. I see the only function R(x,z,t). Response: Thanks. The necessary correction is done. 3) The Section (2.2) is devoted to consideration Eq. (2), but Eq. (2) is nonlinear ordinary differential equation. Response: The necessary correction is done. 4) Solution of Eq. (2) is well-known but the authors present a few of exact solutions. Response: We appreciate the reviewer's comment. While it is true that the general solution to equation (2) is well-known, our contribution lies in presenting a few exact solutions in specific forms that may not be widely recognized or easily derived. These solutions provide additional insight into the behavior of the governing model under conditions, and we believe they add value to the existing body of knowledge. 5) The looked for of Equation (2) using formula (3) but they did not say what q_1, q_2, and p_1.The authors present different formulas for solution of Eq. (2) but these solutions are the same. More interesting Section 2 of paper. They present the consideration the chaotic nature. Response: The necessary correction is done in section 2.2 and 3. 6) However the approach by authors is well known as well. See , for example, the book “Chaotic Transitions in deterministic and Stochastic Dynamical Systems” by Emil Simiu, Prinston University Press. Taking into account the remarks above unfortinately I do not recommend this paper for publication. Response: The reviewer’s mentioned paper described Melnikov's theory for chaotic dynamics. However, our paper uses a planar dynamical system to analyze the chaotic behavior of the governing model. Reviewer #2: Review of the article: Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearity. Manuscript Number: PONE-D-24-28379 ESSENTIALS OF THIS ARTICLE: In this paper the authors find several solutions of a fractional version of the 2D NLS (two-dimensional nonlinear Schrödinger) equation (which the authors call “paraxial wave equation”). In this article “x” is the evolution variable, and “t” and “z” are the “transversal” variables. The second-order time derivative has been replaced by an M-fractional derivative, which is a generalization of the conformable derivative. Proposing a cleverly chosen similarity reduction, defined by the equations shown in (6), and the form of R(x,z,t) written above Eq. (6), the authors manage to reduce the fractional 2D NLS equation (1) to the ODE (ordinary differential equation) shown in (7). Then several particular solutions of this ODE are obtained by means of the EJEFE method. Afterwards the authors study a perturbed form of Eq. (7) [shown in (17)], and obtain quasiperiodic and chaotic solutions of this perturbed equation. OPINION OF THE ARTICLE: The fractional 2D NLS equation studied in this article is an interesting equation, the solutions obtained are interesting, and the procedure to obtain these solutions is also interesting. The reduction of the fractional 2D NLS equation (1) to the ODE (7) is an excellent result (the most interesting result of the paper, from my point of view). HOWEVER, the paper contains some errors and undefined symbols, which are probably the consequence of a hasty and careless writing. Moreover, some of the statements made in this article are, in my opinion, incorrect. ISSUES THAT SHOULD BE CORRECTED: ISSUE 1: In the fourth line below Eq. (1) it is written “in the y and t directions”. This statement is wrong. It should say “in the t and z directions”. Response: The necessary correction is done. ISSUE 2: In pag. 3 of the paper the authors mention References [34]-[43]. But the list of references presented at the end of the article only contains 32 references!!! Response: The necessary correction is done. ISSUE 3: In the definition of the M-fractional derivative the meaning of the “t” with subscript kappa (which appears within the argument of the function “u”) is not explained. Moreover, in the right-hand-side (rhs) of the equation which defines this derivative, the parameter “M” does not appear. Please explain. Response: The necessary explanation is done in section 2.1 ISSUE 4: Below the definition of the M-derivative, in the section “Features”, in the equations (a), (b), (c) and (f), the meaning of the letters with the subscript kappa (le letters l, m, u and v) is not explained. Response: The necessary correction is done. ISSUE 5: Two lines below Eq. (2) the authors say that: “Balancing H’’ and H3, yields M=1” but the authors have not said what is “M”. As I said in the Issue 3, the parameter M does not appear in the rhs of the equation which defines de M-derivative. So, it is incomprehensible how the equation M=1 is obtained. Response: We explain it in section 2.2 and section 3. ISSUE 6: In the first line below Eq. (5) the authors mention the equations (6)-(8), but these equations have not been presented at this point of the article. Moreover, it is quite clear that in this line [below Eq. (5)], the authors are not referring to the Eqs. (6)-(8) which appear in Section 3. Response: Thanks. The necessary correction is done. ISSUE 7: Two lines below Eq. (9) the equation M=1 appears again. As I said in Issue 5, we don´t know what is M, and therefore the origin of the equation M=1 is incomprehensible. Response: we explain it in section 2.2. ISSUE 8: Two lines above Eq. (17) the authors mention a “superficial component”. This term is a misleading, as it is not related to any “surface” at all. Moreover, as far as I could see, this term is never mentioned in Ref. [31]. Response: We modified it. ISSUE 9: In the line above Eq. (17) the authors mention the “system Eq. (30)”. But the article contains 17 equations, consequently the mention of Eq. (30) is obviously incorrect. Response: We modified it. ISSUE 10: The numbering of the sections of this article is an absolute disaster, revealing an extremely careless work. In page 3 the Sec. 2 begins. In page 5, Sec. 3 begins. Then, in page 6, another “Section 2” begins. Then, in page 8, the Section 8 begins!. And in page 9, the Section 5 begins! Finally, in page 14, the Section 6 closes the paper! Response: We modified it. ISSUE 11: In page 6, in the Section entitled “Chaotic nature”, the authors modify their Eq. (7), introducing an oscillatory perturbation in time [the last term in the second equation shown in (17)]. And then they show that the solutions of this new ODE [Eq. (17)] exhibit a chaotic behavior. But the chaotic behavior of the solutions of Eq. (17) does not imply that Eq. (7) [or Eq. (1)] is a chaotic system. The chaotic nature of a system (i.e., of an equation), is proved by perturbing THE INITIAL CONDITIONS USED, and NOT by perturbing the equation itself. Even though some authors may consider that an equation is “chaotic” if the solutions exhibit a huge change when the equation is perturbed, THIS IS NOT THE USUAL DEFINITION OF CHAOS. At least, it is not the definition of “chaos” used by Edward Lorenz (who was one of the founders of the theory of chaos). Consequently, I consider that the authors must explain clearly that the chaotic solutions of Eq. (17) do not imply that the system described by Eq. (7) [or Eq. (1)] is chaotic. Response: We include the following paragraph in the “Chaotic nature” section: It is important to note that the chaotic behavior observed in the solutions to Eq. (17) arises due to the introduction of an oscillatory perturbation to the system. This does not necessarily imply that the original system described by Eq. (7) [or Eq. (1)] is chaotic in the traditional sense of chaos theory, where chaos is typically characterized by sensitivity to initial conditions rather than sensitivity to perturbations in the system itself. This distinction aligns with the definition of chaos introduced by Edward N. Lorenz and is crucial for accurately characterizing the system's dynamics [37]. [37] Lorenz EN. The Essence of Chaos. University of Washington Press. 1993; 181–206. This clarification should help address the reviewer's concern while maintaining the integrity of your analysis. Reviewer #3: The authors investigated various types of wave solutions in the M-fractional paraxial wave equation with Kerr law nonlinearity, including periodic waves, lumpperiodic waves, periodic breather waves, kink-bell waves, kinky-periodic waves, anti-kinky-periodic waves, double-periodic waves. In addition, the chaotic phenomena are also studied in this work. The wave equations with M-fractional derivative attracted many attentions in recent years and the results presented in this work are interesting. However, some issues should be fixed, and my comments are as follows: 1. The physical scene (settings) for M-fractional derivative may be a good supplement to this work. Response: Thanks for your response. 2. I am wondering if such M-fractional derivative can be expressed by other easier forms that can be solved numerically. Response: Thanks 3. Where is the Eq. (30) the authors pointed out before Eq. (17) ? Response: we modified it. 4. I assume the references of [34] to [42] in the Introduction should be wrong. The authors should check it. Response: we modified it. 5. Some review should be added for the better understanding of fractional derivative. Response: We cited some related article in [16, 17, 18]. ________________________________________ 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Attachments Attachment Submitted filename: Reviewer Response.docx https://doi.org/10.1371/journal.pone.0314681.r002
4 Sep 2024 Decision Letter - Boris Malomed, Editor PONE-D-24-28379R1Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearityPLOS ONE Dear Dr. Roshid, Thank you for resubmitting your manuscript to PLOS ONE. A new review suggests that an additional revision is necessary. Please submit your revised manuscript by Oct 19 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. 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Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols. We look forward to receiving your revised manuscript. Kind regards, Boris Malomed Academic Editor PLOS ONE Journal Requirements: Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #2: (No Response) Reviewer #3: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #2: Yes Reviewer #3: Yes ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #2: N/A Reviewer #3: Yes ****** 4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #2: Yes Reviewer #3: Yes ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #2: Yes Reviewer #3: Yes ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #2: Review of the revised version of the article: Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearity. Manuscript Number: PONE-D-24-28379R1 In the revised version of this article the authors have taken into account the observations mentioned in my previous review. However, two of the issues mentioned in that review have not been answered satisfactorily: Issues 3 and 4. In page 4 the authors added a new paragraph entitled “Features”, aimed to clarify the Issues 3 and 4. However, the “explanations” contained in this paragraph are completely unsatisfactory. The statement: “M indicates the truncation point or order” is absolutely ambiguous, and it does noy clarify how the parameter M enters in the calculation of the limit which appears in the rhs (right hand side) of the equation which defines the M-derivative. Additionally, the statement: “kappa represents a scaling factor or a characteristic constant associated with the operator” is also imprecise and ambiguous, as the letter “kappa” enters in the rhs of the definition of the M-derivative as the name of a subindex, and it does not occupy the position of a numerical factor or a constant. Therefore, it is necessary to explain with absolute clarity how the symbol t_kappa depends on kappa. And, in a similar way, the authors do not explain how the parameters l_kappa, m_kappa, u_kappa and v_kappa, which appear in the equations (a), (b) and (c) presented in page 4, depend on kappa. Therefore, it is necessary to explain with absolute clarity how these four symbols depend on kappa. Recommendation: As the essential novelty of the equation studied in this article is the use of the M-derivative, the definition and the properties of this operator must be absolutely clear in the article. Therefore, in my opinion, this article should not be accepted for publication in PLOS ONE until these issues are presented with absolute clarity. Reviewer #3: The authors have replied all my comments. This manuscript looks much better now. I have no further comments on this work. ****** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #2: No Reviewer #3: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. https://doi.org/10.1371/journal.pone.0314681.r003
Revision 2
8 Nov 2024 Author Response Reviewer #2: Review of the revised version of the article: Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearity. Manuscript Number: PONE-D-24-28379R1 In the revised version of this article the authors have taken into account the observations mentioned in my previous review. However, two of the issues mentioned in that review have not been answered satisfactorily: Issues 3 and 4. In page 4 the authors added a new paragraph entitled “Features”, aimed to clarify the Issues 3 and 4. However, the “explanations” contained in this paragraph are completely unsatisfactory. The statement: “M indicates the truncation point or order” is absolutely ambiguous, and it does noy clarify how the parameter M enters in the calculation of the limit which appears in the rhs (right hand side) of the equation which defines the M-derivative. Additionally, the statement: “kappa represents a scaling factor or a characteristic constant associated with the operator” is also imprecise and ambiguous, as the letter “kappa” enters in the rhs of the definition of the M-derivative as the name of a subindex, and it does not occupy the position of a numerical factor or a constant. Therefore, it is necessary to explain with absolute clarity how the symbol t_kappa depends on kappa. And, in a similar way, the authors do not explain how the parameters l_kappa, m_kappa, u_kappa and v_kappa, which appear in the equations (a), (b) and (c) presented in page 4, depend on kappa. Therefore, it is necessary to explain with absolute clarity how these four symbols depend on kappa. Ans.: In this operator(_κ^ )D_(M,t)^(σ,ϕ) , κ is real constant, known as a scaling factor or a characteristic constant associated with the operator. It is related to the operator through the truncated Mittag-Leffler function with one parameter, as discussed in [40] (_κ^ )E_ϕ (t)=∑_(n=0)^κ▒t^n/Γ(ϕn + 1) . In the feature, u=u(t),v=v(t) are the function of time t. We justify the properties of truncated M-fractional operator. The parameter 𝑀 is used to denote that the function to be derived involves a Mittag-Leffler function with one parameter. Further discussion can be found in [41]. 〖(_κ^ )D_(M,t)^(σ,ϕ) 〗^ u(t)=lim┬(ϵ→0)⁡〖(u(t_K E_ϕ (ϵt^(-σ) ))-u(t))/ϵ〗,t>0,ϕ>0. Here, (_κ^ )E_ϕ (.) is a truncated Mittag-Leffler function of one parameter. In left side M covert into (_κ^ )E_ϕ (.) In right side. (See [41] (definition 2.1, 2.2)) Recommendation: As the essential novelty of the equation studied in this article is the use of the M-derivative, the definition and the properties of this operator must be absolutely clear in the article. Therefore, in my opinion, this article should not be accepted for publication in PLOS ONE until these issues are presented with absolute clarity. Reviewer #3: The authors have replied all my comments. This manuscript looks much better now. I have no further comments on this work. Thanks for your recommends. Attachments Attachment Submitted filename: Reviewer Response.docx https://doi.org/10.1371/journal.pone.0314681.r004
15 Nov 2024 Decision Letter - Boris Malomed, Editor Chaotic behavior, sensitivity analysis and Jacobian elliptic function solution of M-fractional paraxial wave with Kerr law nonlinearity PONE-D-24-28379R2 Dear Dr. Roshid, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. If you have any questions relating to publication charges, please contact our Author Billing department directly at authorbilling@plos.org. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Boris Malomed Academic Editor PLOS ONE Additional Editor Comments (optional): Comments from PLOS Editorial Office: We note that one or more reviewers has recommended that you cite specific previously published works in an earlier round of revision. As always, we recommend that you please review and evaluate the requested works to determine whether they are relevant and should be cited. It is not a requirement to cite these works and you may remove them before the manuscript proceeds to publication. We appreciate your attention to this request. Reviewers' comments: https://doi.org/10.1371/journal.pone.0314681.r005
Formally Accepted
29 Nov 2024 Acceptance Letter - Boris Malomed, Editor PONE-D-24-28379R2 PLOS ONE Dear Dr. Roshid, I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now being handed over to our production team. At this stage, our production department will prepare your paper for publication. This includes ensuring the following: * All references, tables, and figures are properly cited * All relevant supporting information is included in the manuscript submission, * There are no issues that prevent the paper from being properly typeset If revisions are needed, the production department will contact you directly to resolve them. If no revisions are needed, you will receive an email when the publication date has been set. At this time, we do not offer pre-publication proofs to authors during production of the accepted work. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few weeks to review your paper and let you know the next and final steps. Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. If we can help with anything else, please email us at customercare@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Prof. Boris Malomed Academic Editor PLOS ONE https://doi.org/10.1371/journal.pone.0314681.r006

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