Exploring the impact of Brownian motion on novel closed-form solutions of the extended Kairat-II equation

Khaled Aldwoah; Alaa Mustafa; Tariq Aljaaidi; Khidir Mohamed; Amer Alsulami; Mohammed Hassan

doi:10.1371/journal.pone.0314849

Peer Review History

Original SubmissionSeptember 22, 2024
30 Oct 2024 Decision Letter - Rab Nawaz, Editor PONE-D-24-42240Exploring the Impact of Brownian Motion on Novel Closed-Form Solutions of the Extended {Kairat-II} EquationPLOS ONE Dear Dr. Aljaaidi, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Dec 14 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. 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Please ensure that you refer to Table 1 in your text; if accepted, production will need this reference to link the reader to the Table. 5. 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 If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references.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 If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.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. 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: Yes Reviewer #2: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: I Don't Know Reviewer #2: 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: Yes Reviewer #2: 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 ****** 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: The research article titled "Exploring the Impact of Brownian Motion on Novel Closed-Form Solutions of the Extended Kairat-II Equation" presents a valuable contribution to the study of stochastic systems by incorporating Brownian motion into the deterministic Kairat-II equation. The authors employ two mathematical approaches—the modified Tanh method and the Sardar Sub-Equation Method (SSEM)—to derive a variety of closed-form solutions. These solutions, expressed through trigonometric and hyperbolic functions, are visualized in 2D and 3D to demonstrate distinct wave behaviors, such as bright, dark, singular, and periodic solitons. The article provides interesting insights into how random perturbations affect wave dynamics, showing that high-energy regions are more vulnerable to noise than low-energy regions, which remain relatively robust. The incorporation of Brownian motion into the Kairat-II equation through Ito calculus represents a novel approach to studying the dynamics of stochastic systems. This framework is promising for practical applications in plasma physics, optical communications, and marine environments, where random perturbations play a significant role. Additionally, the comparison of singular solutions generated by the modified Tanh method with the non-singular solutions obtained through the SSEM method provides a rich mathematical insight into the nature of the solutions. While the article offers several strengths, some improvements in the literature review could enhance the study’s theoretical depth and relevance. In particular, the authors could strengthen their work by citing recent studies on nonlinear and stochastic differential equations. Despite the article’s strengths, a few limitations are noted. The model assumes Brownian motion as the primary source of noise, but real-world systems often involve more complex noise types, such as colored noise or Lévy processes. Discussing the limitations of using Brownian motion and suggesting potential extensions to more complex stochastic environments would provide a more comprehensive outlook for future research. In addition, a comparative analysis with other stochastic systems, if possible, could contextualize the findings and highlight the novel aspects of the current study more effectively. More so, while the use of advanced mathematical techniques like the modified Tanh method and SSEM is commendable, the article could benefit from providing step-by-step explanations or an appendix with detailed derivations. This would make the work more accessible to readers who may not be familiar with these specialized techniques. Reviewer #2: The article explores the impact of Brownian motion on the extended version of the Kairat-II equation, which is a non-linear differential equation used to model wave dynamics. The primary objective is to incorporate stochastic behavior (random perturbations) into the deterministic equation and derive closed-form analytical solutions using two distinct methods—the modified Tanh method (linked with the Riccati equation) and the Sardar sub-equation method (SSEM). These solutions are analyzed to understand the influence of noise intensity on wave behavior, which has implications for modeling real-world wave dynamics, such as in plasma physics, optical communication, and marine environments. Overall, the article is well written and justifies its impact for possible publication in PLoS ONE provided the following observations are responded positively: 1. Although several types of solutions (e.g., bright, dark, and singular solitons) are presented, the article lacks details related to physical interpretations of these solutions in real-world systems, such as plasma or marine environments. A more explicit connection between the solutions and their potential physical implications would enhance the work's impact. 2. The study observes that high-energy regions are more affected by noise, but it lack the reasons behind this happening. 3. The literature review part can be improved, particularly, by discussing exact solutions, soliton dynamics, and algebraic properties related to non-linear and fractional differential equations, which align closely with the objectives of current study. To enhance the theoretical foundation of the research and to demonstrate the broader relevance of the findings in the field of non-linear wave analysis and stochastic integrable systems, I would like to suggest the inclusion of the following recent studies: Kai, Y., & Yin, Z. (2021). On the Gaussian traveling wave solution to a special kind of Schrödinger equation with logarithmic nonlinearity. Modern Physics Letters B, 36(02), 2150543. https://doi.org/10.1142/S0217984921505436 Kai, Y., Chen, S., Zhang, K., & Yin, Z. (2022). Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation. Waves in Random and Complex Media. https://doi.org/10.1080/17455030.2022.2044541 Zhu, C., Al-Dossari, M., Rezapour, S., & Gunay, B. (2024). On the exact soliton solutions and different wave structures to the (2+1) dimensional Chaffee–Infante equation. Results in Physics, 57, 107431. https://doi.org/10.1016/j.rinp.2024.107431 Guo, S., & Wang, S. (2024). Twisted relative Rota-Baxter operators on Leibniz conformal algebras. Communications in Algebra, 52(9), 3946-3959. https://doi.org/10.1080/00927872.2024.2337276. ****** 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 ******** [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. Attachments Attachment Submitted filename: Review rep..docx https://doi.org/10.1371/journal.pone.0314849.r001
Revision 1
6 Nov 2024 Author Response Reviewer #1: The research article titled "Exploring the Impact of Brownian Motion on Novel Closed-Form Solutions of the Extended Kairat-II Equation" presents a valuable contribution to the study of stochastic systems by incorporating Brownian motion into the deterministic Kairat-II equation. The authors employ two mathematical approaches—the modified Tanh method and the Sardar Sub-Equation Method (SSEM)—to derive a variety of closed-form solutions. These solutions, expressed through trigonometric and hyperbolic functions, are visualized in 2D and 3D to demonstrate distinct wave behaviors, such as bright, dark, singular, and periodic solitons. The article provides interesting insights into how random perturbations affect wave dynamics, showing that high-energy regions are more vulnerable to noise than low-energy regions, which remain relatively robust. The incorporation of Brownian motion into the Kairat-II equation through Ito calculus represents a novel approach to studying the dynamics of stochastic systems. This framework is promising for practical applications in plasma physics, optical communications, and marine environments, where random perturbations play a significant role. Additionally, the comparison of singular solutions generated by the modified Tanh method with the non-singular solutions obtained through the SSEM method provides a rich mathematical insight into the nature of the solutions. While the article offers several strengths, some improvements in the literature review could enhance the study’s theoretical depth and relevance. In particular, the authors could strengthen their work by citing recent studies on nonlinear and stochastic differential equations. Despite the article’s strengths, a few limitations are noted. The model assumes Brownian motion as the primary source of noise, but real-world systems often involve more complex noise types, such as colored noise or Lévy processes. Discussing the limitations of using Brownian motion and suggesting potential extensions to more complex stochastic environments would provide a more comprehensive outlook for future research. In addition, a comparative analysis with other stochastic systems, if possible, could contextualize the findings and highlight the novel aspects of the current study more effectively. More so, while the use of advanced mathematical techniques like the modified Tanh method and SSEM is commendable, the article could benefit from providing step-by-step explanations or an appendix with detailed derivations. This would make the work more accessible to readers who may not be familiar with these specialized techniques. Response: We are really thankful for the positive comments and appreciation of our work we have tried our best to present clear and easily readable work. We have taken the above comments into consideration and added some recent works in this area for comparison. Besides this we have added that in future this work can be re-visited using a more complex noise and more importantly can use neural network to study some more novel behaviors in the proposed model. The method is straightforward presented that how we applied so due to the length of the article we omitted to present some more details. Reviewer #2: The article explores the impact of Brownian motion on the extended version of the Kairat-II equation, which is a non-linear differential equation used to model wave dynamics. The primary objective is to incorporate stochastic behavior (random perturbations) into the deterministic equation and derive closed-form analytical solutions using two distinct methods—the modified Tanh method (linked with the Riccati equation) and the Sardar sub-equation method (SSEM). These solutions are analyzed to understand the influence of noise intensity on wave behavior, which has implications for modeling real-world wave dynamics, such as in plasma physics, optical communication, and marine environments. Overall, the article is well written and justifies its impact for possible publication in PLoS ONE provided the following observations are responded positively: 1. Although several types of solutions (e.g., bright, dark, and singular solitons) are presented, the article lacks details related to physical interpretations of these solutions in real-world systems, such as plasma or marine environments. A more explicit connection between the solutions and their potential physical implications would enhance the work's impact. Response: We have now elaborated on the potential physical implications of the derived solutions, particularly their relevance in plasma and marine environments. In plasma systems, the stability and structure of soliton solutions can represent energy transfer and wave propagation under perturbative influences. In marine environments, solitons reflect tidal and surface wave dynamics, which could be subject to noise, as seen in our stochastic model. 2. The study observes that high-energy regions are more affected by noise, but it lack the reasons behind this happening. Response: We have expanded the discussion to include potential reasons why high-energy regions are more affected by noise, drawing from principles of wave dynamics and energy distribution in stochastic systems. 3. The literature review part can be improved, particularly, by discussing exact solutions, soliton dynamics, and algebraic properties related to non-linear and fractional differential equations, which align closely with the objectives of current study. To enhance the theoretical foundation of the research and to demonstrate the broader relevance of the findings in the field of non-linear wave analysis and stochastic integrable systems, I would like to suggest the inclusion of the following recent studies: Kai, Y., & Yin, Z. (2021). On the Gaussian traveling wave solution to a special kind of Schrödinger equation with logarithmic nonlinearity. Modern Physics Letters B, 36(02), 2150543. https://doi.org/10.1142/S0217984921505436 Kai, Y., Chen, S., Zhang, K., & Yin, Z. (2022). Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation. Waves in Random and Complex Media. https://doi.org/10.1080/17455030.2022.2044541 Zhu, C., Al-Dossari, M., Rezapour, S., & Gunay, B. (2024). On the exact soliton solutions and different wave structures to the (2+1) dimensional Chaffee–Infante equation. Results in Physics, 57, 107431. https://doi.org/10.1016/j.rinp.2024.107431 Guo, S., & Wang, S. (2024). Twisted relative Rota-Baxter operators on Leibniz conformal algebras. Communications in Algebra, 52(9), 3946-3959. https://doi.org/10.1080/00927872.2024.2337276. Response: The literature review is carefully revised as suggested. We are very thankful to the reviewers for their positive remarks on our proposed work, which enhanced the quality of the present work. Attachments Attachment Submitted filename: Kairat-II model Response letter.docx https://doi.org/10.1371/journal.pone.0314849.r002
19 Nov 2024 Decision Letter - Rab Nawaz, Editor Exploring the Impact of Brownian Motion on Novel Closed-Form Solutions of the Extended {Kairat-II} Equation PONE-D-24-42240R1 Dear Dr. Tariq Aljaaidi, 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. 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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 #1: All comments have been addressed Reviewer #2: 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 #1: Yes Reviewer #2: Yes ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: N/A Reviewer #2: 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 #1: Yes Reviewer #2: 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 #1: Yes Reviewer #2: 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 #1: The authors have addressed all comments constructively and thoroughly, making the manuscript suitable for acceptance in its current form. Reviewer #2: I have reviewed the revised manuscript which is prepared in accordance with the observations made during the initial review. Thus, I recommend the publication of the revised manuscript in the current form. ****** 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 #1: No Reviewer #2: No ******** https://doi.org/10.1371/journal.pone.0314849.r003
Formally Accepted
4 Dec 2024 Acceptance Letter - Rab Nawaz, Editor PONE-D-24-42240R1 PLOS ONE Dear Dr. Aljaaidi, 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 Dr. Rab Nawaz Academic Editor PLOS ONE https://doi.org/10.1371/journal.pone.0314849.r004

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