Analytic Investigation of a Generalized Variable-Coefficient KdV Equation with External-Force Term

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

While the abstract effectively outlines the study's theoretical scope, it would benefit from mentioning quantitative or parametric results. For instance, specifying how the external-force term alters soliton amplitude or velocity would strengthen its impact.

Although the mathematical derivations are comprehensive, the connection to real-world physical systems is only briefly mentioned. A more detailed discussion of applications (e.g., shallow water waves, plasmas) and how the results influence our understanding of those systems would add depth.

The choice of seed solution u=−W(t)−M(t)x−Q(t) in Section 3.4 is critical for soliton construction. However, the paper does not justify why this particular form is appropriate or optimal, nor how it aligns with physical boundary conditions.

Figures 1 and 2 provide useful visualizations of soliton propagation and interaction. However, axes are unlabeled, and no numerical scale is given for amplitude or time evolution. Including these would greatly improve interpretability.

The paper relies entirely on symbolic and analytical methods. While these are elegant, it would significantly strengthen the work to include a numerical simulation to validate the soliton-like solutions and interactions derived through Bäcklund transformations.

While the authors build logically on prior works (e.g., equations from [11], [15]), some derivations (especially Lax pairs) seem to be extensions rather than fundamentally new contributions. The novelty should be clarified or emphasized more directly.

The derivation of integrability conditions in Section 2 is mathematically thorough but lacks a schematic summary or flowchart to help readers understand the steps from leading-order analysis to compatibility conditions and final constraints.

In Section 5, the effects of coefficient variation on soliton properties are only qualitatively discussed. A comparative table showing how different forms of α0(t),α1(t),h(t) affect soliton amplitude, width, and speed would make the analysis clearer and more practical.

Author Response

Dear reviewer,

We firstly would like to thank you for your timely and valuable comments and suggestions. We have more carefully checked our manuscript, and have corrected some minor grammatical errors and misprints, and modifications have been made in accordance to your comments as the response PDF attached.

All revisions made in the manuscript in response to your comments are highlighted in red text for easy identification.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript is well structured, follows the framework of classical integrable systems, and presents a complete and correct derivation. The authors systematically apply classical methods including Painleve analysis, Lax pair construction based on the AKNS system, Backlund transformations, and the derivation of soliton-like solutions and conservation laws.  This paper focuses on the integrability of a generalized variable-coefficient KdV (gvcKdV) equation with an external force term. The approach represents a direct extension of known results for homogeneous or variable-coefficient KdV equations. If the following concerns are addressed, the originality of the manuscript will be further highlighted:

 

1. Introduction (lines 15–41)

The introduction outlines the importance of the variable-coefficient KdV (vcKdV) equation but does not clarify the physical or mathematical significance of introducing the external forcing term h(t). The manuscript should clearly explain the relevance of the inhomogeneous term and relate it to real-world phenomena wherever possible. For example, that h(t) can simulate background forcing or external driving in physical systems, such as controlled solitary wave propagation in shallow water channels.

 

2.Painleve analysis (lines 72–121)

Although the derivation is mathematically correct and detailed, the role of the external term h(t) in the integrability condition is only briefly mentioned. The manuscript notes that h(t) is not a necessary condition for the Painleve property, but this point is not emphasized or discussed. Since h(t) does not affect the integrability structure and leads to the same form as the unforced case, it would be valuable to consider whether h(t) can be interpreted as a perturbation term.

 

3.When h(t=?) = 0, does the Lax pair reduce to the same structure as that of a known unforced integrable case? A brief clarification would be helpful.

 

4.The manuscript repeatedly refers to the solutions as "soliton-like" but does not explain why they are not strict solitons. Do the time-dependent coefficients and external force affect the conditions for strict integrability, and if so, how?

 

5.It is recommended to add a short table near lines 227–231, or at least provide some representative numerical values, to illustrate how the amplitude determines the wave speed. In addition, for the two-soliton case shown in Figure 2, the authors should briefly discuss whether deformation or phase shift occurs after the interaction.

 

6.Since the equation passes the Painleve test and admits infinitely many conservation laws, it is very likely to be integrable. To enhance the manuscript's originality, the authors could consider extending the classical integrability framework, for example by addressing perturbative integrability or quasi-integrable behavior.

 

7.In line 252, the AKNS system is defined as the Ablowitz–Kaup–Newell–Segur system. This explanation should be introduced where the term first appears, preferably in the abstract.

Author Response

Dear reviewer,

We firstly would like to thank you for your timely and valuable comments and suggestions. We have more carefully checked our manuscript, and have corrected some minor grammatical errors and misprints, and modifications have been made in accordance to your comments as the response PDF attached.

All revisions made in the manuscript in response to your comments are highlighted in blue text for easy identification.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have improved the manuscript and responded to my comments. I suggest accepting it as is.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors provide a comprehensive and well-organized response to the review comments. The analytical framework, including the Painleve test, the Lax pair construction, and the conservation laws derivation, is clearly stated and mathematically justified. The authors appreciate and carefully handle the introduction of the external force term and its impact on the integrability conditions.

Even so, I encourage the authors to emphasize their work's novelty further. The combination of time-dependent coefficients with external force terms in a fully integrable framework could be a good extension that goes beyond many existing studies. Explicitly stating how this generalization differs from previous variable coefficients or unforced KdV-type models helps highlight their contribution more effectively.

In addition, the author's suggestion to study perturbative or quasi-integrable extensions in future work is also a good start. I support the authors' intention to pursue these directions and suggest that they more clearly position the present results as a foundation for future developments, especially regarding physical applications, such as wave propagation in controlled conditions or soliton systems under external forces.

Overall, I am satisfied with the revised manuscript and the responses received.