Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors use improved F- 12 expansion technique to explore novel soliton solutions to the stated couple of wave equations. The physical behavior of these solitons is demonstrated in detail through three-dimensional, two-dimensional, and contour representations. The impact of the fractional-order derivative on the wave profile is notable and isillustrated through two-dimensional graphs. The research content of this paper is novel, the article logic is clear, and it is worth publishing. But there are still the following small problems to improve.

1. In formula (1.1), some parameters are not given specific explanations.

2. In the last paragraph of the introduction, the case of 'section' is not uniform,such as ‘section’ and ‘Section’.

3. Is M in formula (2.3) the same or different from L in formula (2.1)? If different, please explain the meaning of M.

4. Please check the correctness of the second formula in (3.1).

5. From line 221 onwards, ƹ continues to appear, but the meaning of ƹ is not explained.

Author Response

Reviewer 1:

In this paper, the authors use an improved F-expansion technique to explore novel soliton solutions to the stated couple of wave equations. The physical behavior of these solitons is demonstrated in detail through three-dimensional, two-dimensional, and contour representations. The impact of the fractional-order derivative on the wave profile is notable and is illustrated through two-dimensional graphs. The research content of this paper is novel, the article logic is clear, and it is worth publishing. But there are still the following small problems to improve.

Comments 1: In formula (1.1), some parameters are not given specific explanations.

Response 1: Thank you very much. We appreciate your suggestion regarding the explanations of some parameters in formula (1.1). In response to your comment, we have added detailed explanations for the parameters in equation (1.1) and also provided explanations for the parameters in equation (1.2). These changes are highlighted in red in the revised manuscript. Thank you for your constructive feedback, which has helped to improve the manuscript.

Comments 2: In the last paragraph of the introduction, the case of 'section' is not uniform, such as ‘section’ and ‘Section’.

Response 2: Thank you very much for pointing out the inconsistency in the use of the word "section" in the last paragraph of the introduction. We have already corrected this typographical mistake and marked it in red color in the revised manuscript.

Comments 3: Is M in formula (2.3) the same or different from L in formula (2.1)? If different, please explain the meaning of M.

Response 3: Thank you for your query regarding the notation.  in formula (2.3) is indeed different from  in formula (2.1).  is a polynomial in  and its ordinary derivatives, which arises from the application of transformation (2.2). In the revised manuscript, we have addressed this notation and included this explanation in red for clarity. We appreciate your attention to this detail.

Comments 4: Please check the correctness of the second formula in (3.1).

Response 4: We are very sorry for the typographical mistake. We appreciate your careful review of the manuscript. We have reviewed the second formula in (3.1) and identified a typographical mistake. The necessary corrections have been made and highlighted in red in the revised manuscript to ensure its accuracy.

Comments 5: From line 221 onwards, ƹ continues to appear, but the meaning of ƹ is not explained.

Response 5: Thank you for your observation. The meaning of the parameter  has been provided immediately after the governing equation (1.3). This explanation has been highlighted in red in the revised manuscript for added clarity. Please check the revised manuscript.

We have resubmitted the revised article for your kind consideration. We look forward to your positive response.

Reviewer 2 Report

Comments and Suggestions for Authors

An analytical soliton solutions are proposed for the fractional regularized long wave equation and the fractional nonlinear shallow water  wave equation by utilizing F-expansion technique. Authors applied known techniques, but got some known and some new soliton solutions. 

Remarks

1.  General remark: differential equations should be considered together with initial value or boundary value conditions, especially in the case of particular equations.  These equations remains uncomplete. Thorough paper just in few places is referred that zero boundary conditions are considered.

2. Formula 2.2, the … terms are not well explained, please write in more explicit form, please check +- signs.

3.  In eq. 2.3 seems also justified add after 3 dots some  final value (here is not infinity).

4. Line 264, one of variables should be time (not both x).

5. Since paper include analytical solutions here are not mesh problems. However, nonuniform mesh can be used for depicting.

6. For rapidly changing soliton solutions is used often nonuniform mesh (DOI: https://doi.org/10.3846/mma.2021.12920). It can be explained in results and discussion section that due to analytical form of the soliton solutions the nonuniform and/or moving mesh is not here needed, but can be used in figures to describe more precisely rapid change of solution solutions.

 

Author Response

Reviewer 2:

An analytical soliton solution is proposed for the fractional regularized long wave equation and the fractional nonlinear shallow water wave equation by utilizing the F-expansion technique. Authors applied known techniques, but got some known and some new soliton solutions.

Remarks

Comments 1 (General remark): Differential equations should be considered together with initial value or boundary value conditions, especially in the case of particular equations.  These equations remain incomplete. Thorough paper just in few places is referred that zero boundary conditions are considered.

Response 1: Thank you for highlighting the importance of boundary conditions. We have addressed this by including the boundary conditions after Eq. (3.2) and Eq. (3.6) in the revised manuscript to enhance the quality of the article. The conditions are marked in red. Please check the revised manuscript.

Comments 2: Formula 2.2, the … terms are not well explained, please write in more explicit form, please check +- signs.

Response 2: Thank you for your detailed observation. In the revised manuscript, we have added one term to the formula and highlighted it in red to clarify the summation. This is a traveling wave transformation, where the coefficient of  (time) can be either positive or negative. However, to ensure the wave propagates in the positive direction, we use the negative sign. Please review the updated manuscript to confirm that the changes address your concerns.

Comments 3: In Eq. 2.3 seems also justified add after 3 dots some final value (here is not infinity).

Response 3: Thank you for your comment. In the revised manuscript, we have clarified the notation in equation (2.3) by explicitly including the highest-order derivative term, . This change makes it clear that the ellipsis ("...") represents the continuation of terms up to a specific final derivative, not an infinite series. The revision has been highlighted in red for your review.

Comments 4: Line 264, one of variables should be time (not both x).

Response 4: Thank you for pointing out the typographical mistake in line 264 (in revised manuscript, 272). There was indeed an error where both variables were labeled as  when one will be  (time). We have corrected this in the revised manuscript and highlighted the change in red for your review. Please check the revised manuscript for accuracy.

Comments 5: Since the paper include analytical solutions here are not mesh problems. However, non-uniform mesh can be used for depicting.

Response 5: In this work, we focus on obtaining analytical solutions (solitary waves) for two fractional nonlinear evolution equations, which successfully satisfy the corresponding governing equations. The use of non-uniform meshes was not included in this study, as the analytical approach does not involve numerical methods. We appreciate your understanding of the analytical nature of our study.

Comments 6: For rapidly changing soliton solutions is used often non-uniform mesh (DOI: https://doi.org/10.3846/mma.2021.12920). It can be explained in results and discussion section that due to analytical form of the soliton solutions the non-uniform and/or moving mesh is not here needed, but can be used in figures to describe more precisely rapid change of solution solutions.

Response 6: Thank you for your insightful suggestion and for highlighting the referenced article. Since we have derived exact solutions using an analytical method, the use of non-uniform meshes is not applicable in this approach, as it does not involve numerical techniques. Instead, we have employed 3D, 2D, and contour plots to illustrate the behavior of the solutions. The referenced article is relevant, as it deals with both numerical and analytical methods. Therefore, we have studied this article attentively and improved our article, and thus we have cited the article whose citation number is [23]. We plan to work on these interesting ideas into our future work, particularly for rapidly changing systems.

We have modified the article as per the reviewers’ comments and suggestions. Therefore, the present version has improved considerably and will be able to make a significant contribution to the existing literature.

We have re-submitted the revised manuscript for your consideration. We look forward to your positive response.

Thanks again for your insightful comments and constructive feedback!