PONE-D-24-22626Bifurcation, chaos, and stability analysis  to the second fractional WBBM modelPLOS ONE

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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

Reviewer #3: Yes

Reviewer #4: Yes

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: N/A

Reviewer #2: N/A

Reviewer #3: Yes

Reviewer #4: Yes

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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

Reviewer #3: Yes

Reviewer #4: Yes

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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

Reviewer #4: No

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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: DESCRIPTION OF THIS ARTICLE:

In this article the authors study a fractional generalization of the WBBM (Wazwaz-Benjamin-Bona-Mahony) model. The fractional derivatives used are conformable derivatives. By proposing an adequate similarity reduction [Eq. (8)], the authors managed to transform the partial differential equation Eq. (7) into the ordinary differential equation (ODE) (9), which can be integrated once, to yield the second-order Eq. (10). This transformation is, by itself, an excellent result. Then Eq. (10) is transformed into a system of two first-order ODEs [Eqs. (12)] which depend on two parameters: “l” (afterwards called “alfa”] and “m”. From system (12) the phase portraits shown in Fig. 1 are constructed, showing three or one equilibrium points (depending on the signs of the parameters “alfa” and “m”). Then the authors show that system (12) has quasi-periodic, periodic, and chaotic solutions, which is an interesting result. The apparition of chaotic solutions is then studied by introducing a time-dependent oscillatory perturbation [shown in Eqs. (15)]. Moreover, the sensitivity of the solutions to small changes in the initial conditions [Fig. 6] is also studied. Finally, exact analytical solutions are obtained. Periodic solution and soliton-like solutions are obtained.

OPINION OF THE ARTICLE:

This is an interesting article. It is clear and well-written. I consider that it deserves to be published in PLOS ONE in its present form. Only one small detail has to be corrected. In the system (12) two coefficients appear: “l” and “m”. However, in the following, the parameter “l” is called “alpha”. Therefore, I believe that Eqs. (12) and (13), and the paragraph below Eq. (13), should be corrected, writing “alpha” instead of “l”.

RECOMMENDATION:

I consider that this article is suitable for publication in PLOS ONE (once the small error mentioned above is corrected).

Reviewer #2: See attachment.

Manuscript Number: PONE-D-24-22626

Full Title: Bifurcation, chaos, and stability analysis to the second fractional WBBM model

By Mohammad Safi Ullaha, M. Zulfikar Ali, Harun-Or Roshid

In this manuscript, the authors presented the bifurcation, chaos, and stability analysis for a significant the second 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) model in the research of shallow water waves. In addition to improving our understanding of shallow water nonlinear dynamics, including waveform features, bifurcation analysis, sensitivity, and stability, this study reveals insights into dynamic properties and wave patterns. The obtained results are interesting and novel. However, the following issues must be modified before it is considered for publication.

1. In Eq. (11), it must be emphasized that . Otherwise, the bifurcation analysis mentioned below cannot occur.

2. For the literature on the Nizhnik Novikov Veselov model and various others, please introduce the latest achievements: Chaos, Solitons and Fractals 176 (2023) 114075 and Chinese Physics Letters 41, 044201 (2024).

3. In Eq. (8), to prevent singularities in future analysis, it is necessary to propose limiting conditions , , , .

4. On the next line of Eq. (12), it is mentioned that , but Eq. (12) does not have this symbol at all. Please carefully check.

5. If Eq. (12) is , the label in Fig. 1 must be modified, from change to .

6. After Eq. (12), all symbols must be carefully checked to prevent mixing of and .

In conclusion, if the author can make revisions according to the above suggestions, I recommend publishing this article in PLOS ONE.

Reviewer #3: The paper investigated theoretically and numerically the bifurcation, chaos, and stability analysis to the second 3D fractional WBBM model, stressing the findings of some solitary waves as bright solitons, dark solitons, kink waves, and anti-kink waves. The novelty of the results was justified. This work can be recommended to accept in PLOS one, while before that a critical issue should be well taken into consideration.

Such issue is the stability of the diverse soliton structures predicted in the manuscript, although the stability assessment was done in the Section 9, it is instructive to include at least one stable example and an unstable one in the propagation course.

Reviewer #4: In this manuscript “Bifurcation, chaos, and stability analysis to the second fractional WBBM model”, authors had investigated the Bifurcation, chaos, and solitons in the fractional 3D WBBM equation.

The issues studied in this paper have been analyzed and presented in other papers, such as:

1.“Periodic and solitary wave solutions to a family of new 3D fractional WBBM equations using the two-variable method” published in “Partial Differential Equations in Applied Mathematics 3 (2021) 100033”.

2.“Bifurcation analysis and new waveforms to the first fractional WBBM equation”, published in “Scientifc Reports (2024) 14:11907”.

The theoretical results presented in this paper have been does not introduce new concepts or significant advancements in the field. The issues and model discussed in this paper are similar to the previous work of “Scientifc Reports (2024) 14:11907”, however, the authors did not provide a detailed comparison with research.

After a thorough review of the manuscript, I regret to inform you that I cannot recommend its publication.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: Yes: Jianhua Zeng

Reviewer #4: No

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Attachments

Attachment

Submitted filename: comment-PONE-D-24-22626.docx

Bifurcation, chaos, and stability analysis  to the second fractional WBBM model

PONE-D-24-22626R1

Dear Dr. Ullah,

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.

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Kind regards,

Boris Malomed

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

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 #1: All comments have been addressed

Reviewer #2: All comments have been addressed

Reviewer #4: All comments have been addressed

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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

Reviewer #4: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: N/A

Reviewer #2: Yes

Reviewer #4: 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: No

Reviewer #4: Yes

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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

Reviewer #4: 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: This is an interesting article. It is clear and well-written. Moreover, the suggestion that I mentioned in my previous review has already been taken into account in the revised version of the articloe. Therefore, I consider that it deserves to be published in PLOS ONE in its present form.

Reviewer #2: Dear editors,

Due to the author's revisions following my suggestions,

I recommend publishing this article.

Reviewer #4: The major comments about the revised manuscript have been clarified. My conclusion is that the revised manuscript is scientifically sound, with the main results easy to catch. The results and the subject of the paper should be of interest for researchers. I recommend the publication in Plos One.

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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

Reviewer #4: Yes: Pengfei Li

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