Prolongation Structure of a Development Equation and Its Darboux Transformation Solution
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
In the paper under consideration, authors discuss the Ohta-Hirota coupled KdV equation, where the Lax pair representation is derived by using prolongation method. Next Backlund transformation and Darboux transformation are discussed. Finally, the recursion operator is given. It seems that the results here are important. But there are many some questions which are not clear, given as follows.
1. Please compare the result with the reference below
Yang G Y and Liu Q P, A Darboux transformation for the coupled Kadomtsev-Petviashvili, Chin Phys Lett 25 (2008) 1
2. Please show how to obtain Ohta--Hirota equation from Lax operator $L=\partial^2+u\partial^{-1}v\partial$ in Section 4, so that we can judge the recurion operator is the one for Ohta--Hirota equation.
3. The English is really a big problem. For example in the title,
3.1 "Extended structure" should be prolongation structure
3.2 "transform " should be "transformation"
3.3 in the second line of Section 2, "continuation" should be "prolongation".
3.4 Capitalization issues (The third line below (59), ohta should be Ohta), typesetting （e.g. the place around (41)）, and other English expressions should be carefully checked.
Author Response
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Response to Reviewer 1 Comments
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1. Summary |
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We would like to express our sincere appreciation for your valuable time in reviewing this scientific manuscript. Please see the detailed response below and note that relevant revisions and corrections have been highlighted or tracked as changes in the resubmitted document. |
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2. Questions for General Evaluation |
Reviewer’s Evaluation |
Response and Revisions |
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Does the introduction provide sufficient background and include all relevant references? |
Yes |
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Is the research design appropriate? |
Yes |
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Are the methods adequately described? |
Yes |
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Are the results clearly presented? |
Can be improved |
This study will review the existing results. It will also aim to provide more detailed and accurate information.
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Are the conclusions supported by the results? |
Yes |
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3. Point-by-point response to Comments and Suggestions for Authors |
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Comments 1: Please compare the result with the reference below Yang G Y and Liu Q P, A Darboux transformation for the coupled Kadomtsev-Petviashvili, Chin Phys Lett 25 (2008) 1
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Response 1: Dear reviewers. Greetings! Thank you very much for your meticulous review and valuable comments on our manuscript.We have corrected them one by one in the revised manuscript. In comparing our work with that of Professor Liu Qingping's team, although we are working in the same field, we have adopted a novel continuation structure method to solve for analytical solutions. This approach significantly differs from the methods and tools used by Professor Liu’s team. We acknowledge that each method has its specific application scenarios and limitations, and we believe our approach offers unique contributions to this field. We appreciate your suggestions and will continue to refine our research efforts. |
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Comments 2: Please show how to obtain Ohta--Hirota equation from Lax operator $L=\partial^2+u\partial^{-1}v\partial$ in Section 4, so that we can judge the recurion operator is the one for Ohta--Hirota equation.
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Response 2: Dear reviewers. Greetings! We sincerely thank you for your valuable comments on the inadequacy of the exposition in this paper, which are of vital importance for improving our scientific paper. We have attached great importance to your suggestions and deleted the inadequate parts of the exposition, as well as fine-tuned and revised the relevant expressions. Based on the matrix form previously obtained, the recursion relation between $u_{tm} $ and $u_{tm + 1} $ of the coupled $\mathrm{KdV}$ equation under the n-reduction condition is given. This is to find an operator $R_{n}$ such that the equation $u_{tm + 1}=R_{n}u_{tm} $ holds. |
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4. Response to Comments on the Quality of English Language |
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Point 1:The English is really a big problem. For example in the title, 3.1 "Extended structure" should be ‘’prolongation structure’’ 3.2 "transform " should be "transformation" 3.3 in the second line of Section 2, "continuation" should be "prolongation". 3.4 Capitalization issues (The third line below (59), ohta should be Ohta), typesetting （e.g. the place around (41)）, and other English expressions should be carefully checked. |
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Response 1: Dear reviewers. Greetings! First of all, thank you for your careful review of our manuscript and your valuable comments. In response to the issues you have raised, we have undertaken a detailed review and correction process. The specific revisions are as follows. 1: 1. Concerning the terminology in the title, we have noticed the translation problems of 'extended structure' and 'transformation' and have unified them into accurate expressions. The consistency and accuracy of the terminology has been ensured. 2. Correction to the second line of section 2: replacing 'continuation' with 'extension' to more accurately convey the original meaning. 3. English presentation and formatting issues: All capitalisation issues have been checked and corrected, e.g. 'ohta' has been changed to the correct 'Ohta'. Typographical errors have been corrected, in particular to ensure that the text in and around formula (41) is correctly positioned. We have also undertaken a thorough review of the English language throughout the document, with the aim of making the language fluent and natural, while maintaining the rigour expected of scientific literature. We believe that these revisions not only address the issues you raised, but also improve the overall quality of the paper. Thank you again for your guidance and suggestions, which are essential to improving our work.
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5. Additional clarifications |
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Thank you very much for your letter and the comments from the referees about our manuscript submitted . We have revised the paper according to the referee’s comments and your advice. The attached files are our revised manuscripts (Latex and PDF). If you have any question about this paper, please don’t hesitate to contact us. |
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Reviewer 2 Report
Comments and Suggestions for Authors
The paper is devoted to a popular topic in the theory of integrable systems. This important subject attracts a lot of researchers both in pure and applied mathematics and also a big population of theoretical physicists. Such great popularity of the subject opens up possibilities both for the appearance of really important, ‘breakthrough’ results and methods and, unfortunately, for a huge number of weak, secondary articles proposing applications of standard methods and well-known ideas to some specific equations, models, and systems. Such papers are good for introducing young students to research, and to their training during master programmes.
Alas, the article under consideration suffers from all the disadvantages of a poorly prepared student's manuscript: a superficial introduction replete with strange errors, poor English, complete lack of motivation for certain methods of computation, a lack of definitions and formulations of both technical and principled statements necessary in a mathematical paper (lemmas, propositions, theorems).
I will give only two or three examples of such deficiencies. It is the author's scientific supervisor's business to proofread and correct all of them.....
1. In the introduction, the authors, for some reason, decided to retell the story of the ‘discovery of the soliton’ for the hundred thousandth time. In doing so, they tell it in an extremely sloppy manner, omitting the name of the sole discoverer (John Scott Russell) and twisting the name of the Edinburgh-Glasgow Union Canal.
2. Apparently, the authors used a standard Google translator because I was not immediately able to understand what is behind the terminology ‘extended structures’ (most likely ‘Wahlquist-Estabroek extensions’) and ‘cloned transformations’ (similarity transformations?).
3. I couldn't figure out what "detailed examination and expansion based on the Lax of the tutor’s thesis, Professor Jiayangjie" meant.
Comments on the Quality of English Language
See my previous comments.
Author Response
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Response to Reviewer 2 Comments
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1. Summary |
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We would like to express our sincere appreciation for your valuable time in reviewing this scientific manuscript. Please see the detailed response below and note that relevant revisions and corrections have been highlighted or tracked as changes in the resubmitted document. |
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2. Questions for General Evaluation |
Reviewer’s Evaluation |
Response and Revisions |
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Does the introduction provide sufficient background and include all relevant references? |
Must be improved |
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Is the research design appropriate? |
Must be improved |
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Are the methods adequately described? |
Can be improved |
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Are the results clearly presented? |
Must be improved |
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Are the conclusions supported by the results? |
Must be improved |
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3. Point-by-point response to Comments and Suggestions for Authors |
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Comments 1: In the introduction, the authors, for some reason, decided to retell the story of the ‘discovery of the soliton’ for the hundred thousandth time. In doing so, they tell it in an extremely sloppy manner, omitting the name of the sole discoverer (John Scott Russell) and twisting the name of the Edinburgh-Glasgow Union Canal. |
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Response 1: Dear reviewers. Greetings! First of all, thank you very much for your careful reading of our manuscript and your valuable comments. Regarding the problems you pointed out in the treatment of the story of ‘Discovering the Orphan Son’ in the introduction, we fully agree with you and have taken appropriate measures to correct them. 1. Repetition of the narrative: We recognise that it is clearly inappropriate to retell the story of the discovery of the orphan 100,000 times, which not only adds to the redundancy of the text, but can also be boring for the reader. Therefore, we have streamlined the section, retaining only one succinct and precise description, with the aim of providing the reader with the necessary background information without detracting from the reading experience. 2. Omission of names of key characters: we apologise for failing to mention the name of John Scott Russell, the sole discoverer of the soliton phenomenon. We have explicitly included recognition of his contribution in the revised version to ensure that this important historical detail is accurately presented. 3 Accuracy of geographical names: we have also noted an incorrect description of the name of the Edinburgh-Glasgow Union Canal. In order to rectify this error, we have double-checked the information to ensure that all geographical names are written in the correct manner and reflect the true historical context. We believe that with these corrections, our paper will be more rigorous, accurate, and reflect a greater respect for scientific historical facts. Thank you again for your guidance and suggestions, they are of great significance to us in improving our work. |
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Comments 2: I couldn't figure out what "detailed examination and expansion based on the Lax of the tutor’s thesis, Professor Jiayangjie" meant. |
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Response 2: Dear reviewers. Greetings! Thank you for your question about our manuscript. Regarding the sentence "Detailed examination and extension of Lax based on the thesis of our supervisor, Prof Kayang Jie", we understand that this expression may not be clear enough, which has led to confusion. We would like to clarify and improve this description. The original intention was to mean that our study was based on Prof Kayangjie's previous work, in particular his research paper on Lax pairs. We have analysed it in depth and tried to extend his findings to explore new theories and methods. To make this clearer, we have changed the sentence to read: 'This study builds on, analyses in depth, and extends Professor Kayangjie's previous research on Lax pairs. |
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4. Response to Comments on the Quality of English Language |
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Point 1:Apparently, the authors used a standard Google translator because I was not immediately able to understand what is behind the terminology ‘extended structures’ (most likely ‘Wahlquist-Estabroek extensions’) and ‘cloned transformations’ (similarity transformations?). |
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Response 1: Dear reviewers. Greetings! First of all, thank you very much for your careful review of our manuscript and your valuable comments. Regarding the terms ‘extension structure’ (possibly ‘Wahlquist-Estabrook extension’) and ‘clone transformation’ (possibly ‘similar transformation’), we apologise for the use of standard translation tools. ‘We apologise for the inaccuracies caused by the use of standard translation tools. Below are our specific responses and modifications: 1. Correction on ‘extension structure’: The term ‘extension structure’ used in the original text does refer to ‘Wahlquist-Estabrook extension’. 2. Clarification on ‘clone transformation’: ‘clone transformation’ in the original text should actually refer to ‘similarity transformation’. This is a mathematical operation that simplifies the form of an equation or system by changing the proportions of the variables. To eliminate confusion, we have carefully revised all occurrences to ensure terminological accuracy. We recognise that the direct use of translation tools may have led to misunderstandings and mistranslations of terminology and apologise for any confusion this may have caused. In the revised version, we will pay special attention to the selection and translation of terminology to ensure that it conforms to academic norms and is easy for readers to understand. Thank you again for your patience and guidance. |
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5. Additional clarifications |
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Thank you very much for your letter and the comments from the referees about our manuscript submitted . We have revised the paper according to the referee’s comments and your advice. The attached files are our revised manuscripts (Latex and PDF). If you have any question about this paper, please don’t hesitate to contact us. |
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Reviewer 3 Report
Comments and Suggestions for Authors
Authors have studied the nonlinear coupled KdV equation by using the extension structure method. Based on Lax pairs, the Darboux transform and the Blacklund transform were derived.
Authors are requested to process the following issues before any further processing:
1. English needs to be polished. Many typos and grammatical errors were found. See the highlighted in the attached file.
2. Support the mentioned information in the Introduction part with related resources. See and cite:
2.1. https://doi.org/10.1007/s11071-024-09992-z
2.2. Peakon and solitary wave solutions for the modified Fornberg-Whitham equation using simplest equation method. Int. J. Math. Comput. Sci, 14(3) (2019), 635-645.
3. The applications of listed models in Section 2?
4. “By eliminating from equation (6),” explain such assumption. As well, “the following must be considered:” in Eq.(19)!?
5. “ … calculations indicate that sl(3, C) is the optimal selection.” Show your work.
6. Check the concrete expressions of F and G have been derived in Eq.(22).
7. Is T a constant matrix as I isn’t?
8. Authors claimed that the characteristics and dynamics of the considered model have been analyzed. Highlight these findings in the text.
9. What are the benefits of the used method over the existing method? Future recommendations? Novelty? and study limitations?
Comments for author File: 
Comments.pdf
Comments on the Quality of English Language
See "Comments and Suggestions for Authors"
Author Response
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Response to Reviewer 3 Comments
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1. Summary |
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We would like to express our sincere appreciation for your valuable time in reviewing this scientific manuscript. Please see the detailed response below and note that relevant revisions and corrections have been highlighted or tracked as changes in the resubmitted document. |
||
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2. Questions for General Evaluation |
Reviewer’s Evaluation |
Response and Revisions |
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Does the introduction provide sufficient background and include all relevant references? |
Must be improved |
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Is the research design appropriate? |
Must be improved |
|
|
Are the methods adequately described? |
Must be improved |
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Are the results clearly presented? |
Must be improved |
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Are the conclusions supported by the results? |
Must be improved |
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3. Point-by-point response to Comments and Suggestions for Authors |
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Comments 1: Support the mentioned information in the Introduction part with related resources. See and cite: 2.1. https://doi.org/10.1007/s11071-024-09992-z 2.2. Peakon and solitary wave solutions for the modified Fornberg-Whitham equation using simplest equation method. Int. J. Math. Comput. Sci, 14(3) (2019), 635-645. |
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Response 1: Dear reviewers Greetings! Thank you for your valuable comments on our manuscript. Based on your suggestions, we have revised the introduction section by adding relevant citations to support the information mentioned in the text. Specifically: We have carefully read the two texts and found that it provides important background information and supporting data that help to enhance our discussion of the research topic. In the revised introduction, we have cited this resource and linked it to our research to provide a more solid theoretical foundation. This not only enhances the explanatory power of our methodology, but also provides the reader with pathways for further exploration. We believe that by bringing in this key literature as support, we can present a fuller picture of the context of our research and its significance, whilst enhancing the academic value of the paper. Thank you again for your guidance and suggestions, which are important for improving our work. The changes in this section are in the first reference on page 17 of the revised manuscript. |
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Comments 2:The applications of listed models in Section 2? |
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Response 2: Dear reviewers. Greetings! Thank you for your valuable comments on our manuscript. In response to your reference to the application of the coupled equation models listed in Section 2, we have made detailed additions and explanations to more fully demonstrate the practical use and importance of these models. In the revised version, we will specifically introduce the application areas of these coupled equation models, including but not limited to the following: 1. Nonlinear wave dynamics: These coupled equation models are widely used to describe nonlinear wave behaviour in a variety of physical phenomena, such as water waves, plasma waves, and optical pulse propagation in optical fibres. For example, by studying these models, we can better understand and predict the evolution of wave forms and their stability. 2. engineering and technical applications: these models also have important engineering applications, such as optimising signal transmission in communication systems, studying structural stability in materials science, and simulating complex flow patterns in fluid mechanics. 3. Theoretical Physics: From the perspective of theoretical physics, these models provide a framework for studying complex dynamical systems, helping scientists to explore phenomena such as phase transitions, bifurcations, and sensitivity analyses. We believe that by adding these application scenarios, we can more fully demonstrate the value of these coupled-equation models and enhance the reader's understanding and interest in them. Thank you again for your guidance and suggestions, which are important for improving our work. |
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Comments 3:“By eliminating from equation (6),” explain such assumption. As well, “the following must be considered:” in Eq.(19)? |
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Response 3: Dear reviewers. Greetings! We sincerely thank you for your valuable comments on the inadequacy of the exposition in this paper, which are of vital importance for improving our scientific paper. We have attached great importance to your suggestions and deleted the inadequate parts of the exposition, as well as fine-tuned and revised the relevant expressions. In the process of revision, we have re-examined the logical structure and adopted a more precise and clearer way of presentation in order to make the content of this part more reasonable and coherent. These changes are reflected on pages 4 and 6 of the revised draft. |
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Comments 4:Check the concrete expressions of F and G have been derived in Eq.(22). |
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Response 4: Dear reviewers. Greetings! We are grateful for your interest in the F and G expressions in this paper. After rigorous compatibility verification, we confirm that these expressions are derived in equation (22) in the text. In addition, in order to further enhance the clarity and transparency of this section, we plan to provide a more detailed elucidation in the subsequent revision process. |
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Comments 5:Is T a constant matrix as I isn’t? |
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Response 5: Dear reviewers. Greetings! I would like to thank the reviewers for their careful review. The property of the T-matrix pointed out by the reviewer is extremely important, and it is hereby clarified that T essentially belongs to the category of canonical transformation matrices rather than function matrices. In subsequent revisions of the paper, we will place special emphasis on this property of T matrices and make it explicit in the relevant sections. These changes are reflected on pages 7 of the revised draft. |
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Comments 6:What are the benefits of the used method over the existing method? Future recommendations? Novelty? and study limitations? |
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Response 6: Dear reviewers. Greetings! We begin with the sought-after Lax pair and perform a canonical transformation on the spectral problem. By employing the compatibility conditions of the canonical transformation, we complete and analyze the Bäcklund transformation and the Darboux transformation from a multitude of perspectives, thereby ensuring the utmost rigor and precision in our research outcomes.This section is devoted to an investigation of the fundamental characteristics of the Ohta − Hirota equation and its recurrence operator. By employing the Lax operator derived from the given Lax equation, a comprehensive analysis is conducted to ascertain the recurrence operator of the Ohta − Hirota equation. |
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4. Response to Comments on the Quality of English Language |
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Point 1: English needs to be polished. Many typos and grammatical errors were found. See the highlighted in the attached file. |
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Response 1: Dear reviewers. Greetings! I sincerely thank you for taking time out of your busy schedule to point out the grammatical and vocabulary errors in this paper, I am deeply impressed and have benefited from your meticulous review. After receiving your feedback, I went through it sentence by sentence and made fine corrections to the grammatical errors. I have carefully examined and replaced inappropriate phrases with more precise ones, and optimised the sentence structure to make the presentation of this paper clearer and more fluent, thus enhancing its readability and professionalism. Once again, I would like to express my sincere gratitude to you, and in the future, I will pay more attention to the detailed treatment of language during the writing process, and strive to improve the quality of my writing in order to prevent the recurrence of similar problems. Relevant changes have been reflected in the revised draft. |
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5. Additional clarifications |
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Thank you very much for your letter and the comments from the referees about our manuscript submitted . We have revised the paper according to the referee’s comments and your advice. The attached files are our revised manuscripts (Latex and PDF). If you have any question about this paper, please don’t hesitate to contact us. |
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Round 2
Reviewer 1 Report
Comments and Suggestions for Authors
I believe that authors have revised their manuscript according to reviewers' reports.
Author Response
We would like to emphasize that extensive and meticulous revisions have been made to the English language and grammar in our manuscript. We have taken great care to ensure clarity and precision throughout the text. Should there be any aspects that remain unclear or require further refinement, we sincerely invite the editors and reviewers to provide their critical feedback and corrections. Thank you very much for your invaluable guidance and support.
Reviewer 2 Report
Comments and Suggestions for Authors
The revised version looks much better but still need to be seriously re-elaborated.
One of my main suggestions was to reframe the paper in a format more suitable for publication in a mathematical journal. The journal Mathematics, as its name implies, emphasizes this publishing sector. I proposed rewriting the main body of the text by structuring the computational flow into definitions, lemmas, propositions, and, ultimately, theorems to highlight the main and significant results. This approach, in my opinion, would improve the clarity and rigor of the initial presentation.
Unfortunately, the authors did not accept this suggestion.
I would also like to emphasize another recommendation I made, which, although terminological in nature, I believe is equally important. In the MDPI online form, my suggestion appeared in a slightly modified form and was ultimately adopted by the authors in their revised version. Specifically, I proposed using "Estabrook-Whalquist prolongation," which is the only correct terminology for this crucial technique aligned with the authors' objectives, rather than "Estabrook-Whalquist extension."
I propose that, while rewriting the paper, the authors should discuss the questions of locality and non-locality concerning the transformations and operators they used, particularly before introducing the recursion operator in Chapter 4.
Comments on the Quality of English Language
The language and presentation flaws still need to be addressed.
I recommend avoiding jargon or argot expressions such as "a closed ideal on the manifold." The term "ideal" is strictly an algebraic concept and exists only within the context of a ring or an algebra.
The sentence following equation (3) is grammatically unacceptable. Additionally, the next page contains similarly poor expressions, such as:
"The variable yi is a continuation variables, and it is necessary to form a new closed ideal with αi, to form a new ideal closed, requiring ωk to meet the conditions."
The first part of this sentence fails to align the plural "variables" with the singular "the variable yi." It would be better to split this into two sentences, beginning the second with "To form...".
I would also like to point out the poor stylistic choice of using "Upon..." four times consecutively on the same page. This repetition leaves a negative impression and should be revised for variety and readability.
Once again, I would like to stress that it is NOT the Referee's job to find ALL language, sense and presentation flaws. I have stopped on the page 4.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for Authors
Authors have worked seriously on modifying their contribution. I recommend this version for publication in Mathematics.
Author Response
We would like to emphasize that extensive and meticulous revisions have been made to the English language and grammar in our manuscript. We have taken great care to ensure clarity and precision throughout the text. Should there be any aspects that remain unclear or require further refinement, we sincerely invite the editors and reviewers to provide their critical feedback and corrections. Thank you very much for your invaluable guidance and support.
