Review Reports - Conservation Laws, Soliton Dynamics, and Stability in a Nonlinear Schrödinger Equation with Second-Order Spatiotemporal Dispersion

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I have noticed an excessive self-citation in this manuscript. For example, Naila Nasreen has about six self-citations. Muhammad Arshad has more than 15 self-citations. I strongly recommend these authors to remove all these references and keep only one reference from each of them. Authors should be discouraged from inflating their citation count through excessive self-citations. Other comments to attend to

a) The authors should fix Eq(??) just above equation (67).

b) What makes the authors to stop at Third Conservation Law Derivation.

c) The authors should take their time and explain the physical meaning of their graphs

I can recommend this manuscript for publication once the above comments have been attended to.

Author Response

Response: We thank the reviewer for this insightful suggestion. We have reviewed the reference list and removed all non-essential self-citations, keeping only those directly relevant to the present work to avoid unnecessary citation inflation.

Other comments to attend to

a) The authors should fix Eq(??) just above equation (67).
b) What makes the authors to stop at Third Conservation Law Derivation.
c) The authors should take their time and explain the physical meaning of their graphs

I can recommend this manuscript for publication once the above comments have been attended to.

Response

We thank the reviewer for this insightful suggestion. According to honorable referee comments, the referenced equation number has now been corrected to ensure consistency and clarity in the manuscript.
We thank the reviewer for this insightful suggestion. The first three conservation laws related to Equation (6) are presented in this work. The conservation of mass (or power), momentum, and energy (Hamiltonian) are the three quantities that correlate to the most physically significant invariants of the model, which is why the third conservation law was chosen. Stability analysis and physical interpretation of the solutions rely heavily on these invariants, which are generally considered the fundamental conserved quantities in nonlinear wave propagation and soliton dynamics. Although there may be other higher-order conservation laws in theory, they typically involve more complex differential terms and lack a direct physical interpretation within the current model. In order to capture the essential physical behavior and to support the discussions on the energy characteristics and stability of the soliton solutions reported in the manuscript, we restrict our derivation to the first three conservation laws.
We thank the reviewer for this insightful suggestion. We have revised the manuscript to expand the physical interpretation of all plotted results. For each figure we added a brief explanation in the main text and clarified the captions, describing the observed waveforms (e.g., dark/bright, multi-peak, breather), their origin (balance of dispersion and nonlinearity), and their significance for energy localization and stability.

I read the whole paper carefully and remove some typing and grammar errors. Thanks, the reviewers for their valuable comments to improve our manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors, my comments are provided in the attached review. Please find my detailed remarks below

Comments for author File: Comments.pdf

Author Response

This paper explores a broad class of exact soliton solutions of the generalized nonlinear Schrödinger equation with second-order spatiotemporal dispersion using a modified extended functional transformation. The authors obtained both known and new types of bright, dark, breather, and multipeak solitons, including trigonometric and hyperbolic functions. An analysis of stability and balance conditions between nonlinearity and dispersion is conducted. The results, including conservation of mass, momentum, and energy, are valuable for studying nonlinear dynamics in optical media. This paper is scientifically novel and of interest to specialists; however, in my opinion, it requires further revision before publication. Here are my comments:

Some of the sources cited in the introduction require clarification of their relevance to the problem stated in the paper. In particular, the content of sources [1] and [2] only partially corresponds to the topic of the study, since they do not consider issues related to the population dynamics of individuals. It is recommended to review all sources again and more clearly justify the choice of these references or replace them with more relevant literary sources.

Response: We appreciate the reviewer's insightful comment. Reevaluating references [1] and [2], we concur that they have nothing to do with the population dynamics context that was first discussed. We have eliminated the population-dynamics statements and substituted references [1] and [2] with more pertinent literature pertinent to nonlinear evolution equations and optical wave propagation to remain consistent with the focus on NLSE models and spatiotemporal wave dynamics. These changes improve the introduction's coherence and clarity.

Furthermore, it would be appropriate for the authors to provide specific references to works that substantiate the stated claims. For example, the introduction states, "This method successfully generates a rich spectrum of exact analytical solutions, including periodic waves, bright and dark solitons, breather-type structures, and more generalized soliton profiles." Such claims require confirmation by citing relevant publications demonstrating the application of the method and the resulting classes of solutions.

Response: We appreciate the reviewer's insightful comment. In response, we have included references to back up the statements about the applied method's ability to produce generalized soliton profiles, periodic waves, bright and dark solitons, and breather structures. The updated introduction now includes a few pertinent publications that show how similar analytical methods can be successfully applied to obtain these classes of solutions. These additions guarantee that the assertions made are adequately supported and grounded in the body of current literature.

To increase the transparency and reproducibility of the results, and to allow the reader to navigate the presented data more quickly and accurately, the authors should indicate in the captions to the solution graphs the values of the functions involved, parameters and ranges of change of variables, as well as the types of solutions presented (bright or dark soliton, breather, multi-peak structure, etc.).

Response: We appreciate the reviewer's insightful comment. As a result, we updated all figure captions to reflect the full range of variables, functional values, and parameters that were used to create the solution graphs. Additionally, we have clearly defined each soliton structure's type (e.g., bright soliton, dark soliton, breather, multi-peak structure). These additions enhance the results' reproducibility and transparency and facilitate readers' more effective interpretation of the figures.

I read the whole paper carefully and remove some typing and grammar errors. Thanks, the reviewers for their valuable comments to improve our manuscript.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The comments are attached.

Comments for author File: Comments.pdf

Author Response

The authors study a generalized nonlinear Schrödinger equation (NLSE) incorporating second-order spatiotemporal dispersion and cubic nonlinearity. They apply the modified exponential rational function method (mERFM) to construct a variety of exact travelling-wave solutions, including bright and dark solitons, trigonometric, hyperbolic, rational, exponential and singular profiles. The manuscript also derives three conservation laws (mass, momentum-like invariant, and energy/Hamiltonian) and presents a linear modulation-instability (MI) analysis to discuss solution stability. Several illustrative plots are provided for representative parameter choices demonstrating soliton, multi-peak and breather-type structures.

The study has potential scientific value, but significant improvements in presentation, notational clarity, language, and justification of key analytical steps are essential. A major revision is therefore recommended before the paper can be reconsidered for publication.

The dispersion relation used in the MI analysis is not shown. Please derive its full analytical form and specify how the instability regions are obtained.

Response: We appreciate the reviewer's insightful comment. According to the honorable referee’s comment, we have revised Section 6 accordingly.

In deriving Eq. (6), integration constants seem neglected. Would including them generate additional travelling-wave families?

Response: We appreciate the reviewer's insightful comment. The integration constants arising when reducing Eq.(6) to the ODE were set to zero to enforce the usual localized /decaying travelling-wave boundary conditions and to remove redundant translation/phase offsets; nonzero constants merely produce shifts (in or overall phase/amplitude) or non-decaying families, but do not introduce qualitatively new localized solution families under the decay conditions used.

The choice of the auxiliary function φ(ξ) in the mERFM lacks justification. Why was this form preferred over other standard transformations (e.g., tanh–coth, sine– cosine)?

Response: We appreciate the reviewer's insightful comment. The auxiliary function used in the mERFM was selected because it converts the reduced ODE into a polynomial form, allowing a closed-form balance between the highest-order nonlinear and derivative terms. For the present equation this choice yields a minimal-degree polynomial structure, whereas tanh–coth or sine–cosine transformations lead to higher-degree balances and no additional solution families.

The admissible parameter ranges ensuring real and bounded soliton profiles are unclear. What explicit constraints guarantee physically meaningful solutions?

Response: We appreciate the reviewer's insightful comment. The reality and boundedness of the obtained soliton profiles follow from the sign conditions imposed on the coefficients of the reduced ODE (Eqs. (9)–(12)), which guarantee that the polynomial under the square-root remains non-negative and that the travelling-wave amplitude does not diverge. These conditions translate into explicit constraints on the dispersion–nonlinearity balance. Only parameter sets satisfying these inequalities were used in the manuscript, ensuring all reported solitons are real and bounded.

The conservation laws are stated without proof. Please verify analytically that the proposed mass, momentum, and energy integrals remain invariant under the governing NLSE.

Response: We appreciate the reviewer's insightful comment. we have checked the three conserved integrals analytically. Differentiating each conserved quantity and using the governing equation (Eq.(6)) and its complex conjugate reduces to a total spatial derivative , which vanishes under the decay/boundary conditions used (fields and derivatives → 0 as ). Thus the mass-like, momentum-like and energy/Hamiltonian integrals (Eqs.(61), (64), (67)) are indeed invariants of Eq.(6).

The authors are advised to enhance the literature review by addressing analytical approaches for deriving conservation laws and studying the structure of solutions and solitons in related nonlinear evolution equations. It is also recommended to revise the references and include recent works such as:

doi.org/10.22060/ajmc.2023.22573.1174 and doi.org/10.1142/S0219887825500677

Incorporating and discussing these studies would strengthen the theoretical foundation and contextual relevance of the present work.

Response: We appreciate the reviewer's insightful comment. In our revised manuscript, the references are added as mentioned above by reviewer and presentation of the paper is improved.

The manuscript includes a number of spelling and grammatical mistakes that require correction. Some representative examples are provided below.

Page 1 ”Conversation Laws.” → ”Conservation Laws.”
Page 3 ”wave-length” → ”wavelength”
Page 3 ”wave lengths” → ”wavelengths”
Page 3 ”denotes the usual derivatives in term of variable fi.” → ”denotes the usual derivatives in terms of variable fi.”
Page 2 ”two variables (Gfi/G, 1/G)-expansion” → ”two-variable (Gfi/G, 1/G)- expansion”
Reference 17 ”shirpay” → ”Shirpay”

Response: We appreciate the reviewer's insightful comment. We proofread thoroughly the whole paper carefully and remove the typing, grammatical errors mentioned by honorable referee and all other errors.

Thanks, the reviewers for their valuable comments to recommended our manuscript.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript is devoted to the study of a generalized nonlinear Schrödinger equation with second-order spatiotemporal dispersion. This is a relevant research topic, with potential applications in nonlinear optics and fluid mechanics, for example.
The authors' work has both minor and serious criticisms.
1. The first criticism concerns the research methodology. It is unclear why the authors choose p and q to obtain different families of solutions. The text of the article gives the erroneous impression that we can choose p and q arbitrarily and obtain an infinite number of solutions.
2. The ordinary differential equation (8) has a general solution, which is expressed in terms of elliptic functions. Special and degenerate cases lead to the solutions found by the authors of the manuscript. I would not consider this a criticism if the authors had presented the main types of degenerate solutions as exact solutions and demonstrated what types of solutions are possible. However, the authors find families of solutions that include very similar solutions. This suggests that either the authors have not sufficiently analyzed the obtained solutions or they have chosen a poor method for constructing exact solutions.
3. Many graphs show periodic singular solutions that cannot be called solitons. Furthermore, the graphs are small and of poor quality, making them difficult to study.
4. The method for constructing conservation laws is appropriate to the problem, but no references to papers using this method are provided.
5. A significant remark regarding the exact solutions found. Many of them differ simply in sign (such as Q_41 and Q_42), so the manuscript could have been significantly reduced in size and not overloaded with identical formulas that the reader is unlikely to ever check. Or, for example, by replacing ksi with -ksi (which is allowed by equation (8)), solution Q_21 becomes solution Q_22. Furthermore, in many solutions, the constants in the numerator and denominator can be reduced.
Therefore, the manuscript requires significant revision before it can be published.

Author Response

The author’s work has both minor and serious criticisms.

The first criticism concerns research methodology. It is unclear why the authors choose p and q to obtain different families of solutions. The text of the article gives the erroneous impression that we can choose p and q arbitrarily and obtain an infinite number of solutions.

Response: We appreciate the reviewer's insightful comment. In the mERFM, the parameters and are not chosen arbitrarily; rather, they are restricted by the polynomial balance between the highest-order nonlinear term and the auxiliary function . Only specific pairs satisfy the balance condition derived from the reduced ODE, and each admissible pair leads to a finite set of solvable polynomial structures. Thus, the method does not generate an infinite number of unrelated solutions only those solution families compatible with the degree balance of the governing equation is permitted.

The ordinary differential equation (8) has a general solution, which is expressed in terms of elliptic functions. Special and degenerate cases lead to the solutions found by the authors of the manuscript. I would not consider this a criticism if the authors had presented the main types of degenerate solutions as exact solutions and demonstrated what types of solutions are possible. However, the authors find families of solutions that include very similar solutions. This suggests that either the authors have not sufficiently analyzed the obtained solutions or they have chosen a poor method for constructing exact solutions.

Response: We appreciate the reviewer's insightful comment. The travelling-wave ODE Eq.(8) indeed admits a general Jacobi elliptic solution, with soliton, kink, and periodic waves arising as its standard degenerate limits. Our focus in this work is specifically on these physically relevant degeneracies under the imposed boundary conditions. Some solutions appear similar because different parameter choices in the mERFM lead to equivalent elliptic–degenerate forms. We have revised the manuscript to clarify the hierarchy of solutions, link each family to its corresponding elliptic or degenerate case, and emphasize that the method is used only to recover these meaningful reductions rather than generate redundant forms.

Many graphs show periodic singular solutions that cannot be called solitons. Furthermore, the graphs are small and of poor quality, making them difficult to study.

Response: We appreciate the reviewer's insightful comment. Some plotted profiles corresponded to periodic or singular cases rather than true solitons. In the revision, such cases have been removed or clearly relabeled, and the singularities were eliminated by adjusting the range. All figures have been regenerated with higher resolution and enlarged axes to ensure clarity and accurate interpretation of the localized, non-singular soliton profiles.

The method for constructing conservation laws is appropriate to the problem, but no references to papers using this method are provided.

Response: We appreciate the reviewer's insightful comment. The conservation-law construction follows standard multiplier and variational-identity methods. We agree that references were missing and have now added appropriate citations to established works applying these techniques to nonlinear evolution equations.

A significant remark regarding the exact solutions found. Many of them differ simply in sign (such as Q_41 and Q_42), so the manuscript could have been significantly reduced in size and not overloaded with identical formulas that the reader is unlikely to ever check. Or, for example, by replacing ksi with -ksi (which is allowed by equation (8)), solution Q_21 becomes solution Q_22. Furthermore, in many solutions, the constants in the numerator and denominator can be reduced.

Response: We appreciate the reviewer's insightful comment. Several solutions differ only by sign changes, shifts such as , or reducible constants, which correspond to equivalent travelling–wave profiles under the symmetry of Eq. (8). In the revised manuscript, redundant forms have been removed and equivalent pairs (e.g., – ) have been merged. However, the solutions in Case 1 ( ) and Case 2 ( ) are retained because they arise from different parameter sets and represent distinct solution structures despite appearing similar under the transformation . Constants have been simplified, and only genuinely distinct families are now presented to avoid unnecessary duplication.

Therefore, the manuscript requires significant revision before it can be published.

We acknowledge the reviewer’s assessment and appreciate the constructive feedback. A comprehensive revision has now been undertaken to address all raised concerns, including clarification of the methodology, refinement of solution families, correction of redundancies, improvement of figures, and strengthening of the theoretical justifications. We believe the revised manuscript is substantially improved and meets the standards required for publication.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have carried out most of the suggested comments. But I noticed that the author Muhammad Arshad is still cited three times in (refs 1,38 and 41), please keep only one reference from each of authors and only those references directly relevant to the present work.

Author Response

Reviewer comment

Accept in present form. The authors have carried out all the suggested comments.

Response: We thank the reviewer for the careful assessment. Regarding the citations of Muhammad Arshad (Refs. 1, 38, and 41), we respectfully clarify that all three references are directly relevant to the present work. Each of these papers contributes essential theoretical background or methodological developments that support the current study. Therefore, we have retained all three citations. We again thank the reviewers for their valuable comments, which have significantly improved our manuscript. We are grateful for the acceptance of our work in this prestigious journal.

Reviewer 3 Report

Comments and Suggestions for Authors

The questions have been answered and the necessary changes have been made to the article.

Author Response

Reviewer comment:

The questions have been answered, and the necessary changes have been made to the article.

Response: We thank the reviewers for their valuable comments, which have significantly improved our manuscript. We are grateful for the acceptance of our work in this prestigious journal.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have corrected most of the comments. In the figures, periodic singular solutions are still called soliton structures. This needs to be corrected. I believe that after some minor corrections, the manuscript can be published.

Author Response

Reviewer comment:

Response: We thank the reviewer for the constructive feedback. We have carefully revised the figures and corrected the terminology as suggested. The periodic singular solutions are no longer referred to as soliton structures in the revised captions. All related descriptions in the text have also been updated accordingly. We appreciate the reviewer’s positive assessment and agree that these minor corrections further improve the quality of the manuscript.

I read the whole paper carefully and remove some typing and grammar errors. Thanks, the reviewers for their valuable comments to improve our manuscript.

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

The authors cannot have such a big Self-citation rate: 6.8% hence I requested these authors to keep only one citation per author with no repeating names of authors in each of the citation.

If these authors do not agree to this, then I must reject this manuscript.

Author Response

Reviewer 1

The authors cannot have such a big Self-citation rate: 6.8% hence I requested these authors to keep only one citation per author with no repeating names of authors in each of the citations.

Response: We appreciate the reviewer’s observation regarding the high self-citation rate. Following your recommendation, we have reduced the self-citations by keeping only one citation per author. Accordingly, we retained a single reference, which is now listed as [30], and removed all repeated citations from the same authors. This adjustment keeps the self-citation rate within acceptable limits.

We again thank the reviewers for their valuable comments, which have significantly improved our manuscript. We are grateful for the acceptance of our work in this prestigious journal.