Exploration of Soliton Solutions to the Special Korteweg–De Vries Equation with a Stability Analysis and Modulation Instability

Comment 1:

  abstract: line 15, instead of verify it should be verified.

Response:

Dear, we done.

Comment 2:

  line 57, it is better to write (where) instead of (here)

Response:

Dear, we done.

Comment 3:

Spacing between Eq.(2) and similar other should be corrected, Like for example Eq. (2) and similarly i have found similar mistakes in all over the paper.

Response:

Dear, we corrected all mistakes in all over the paper.

Comment 4:

   Line 90-91, If the equation numbers are already given to equation, then why we need to mention the roman numbers also?

Response:

Dear, we corrected.

Comment 5:

  In line 99, I think the authors statement is confusing e.g. Supposing the solutions for Eq. (14) are [29–32]: Here the references are redundant and should be either removed or can be adjusted in another suitable place.

Response:

Dear, we done.

Comment 6:

Similar to line 99-106 and others the same in paper. Authors started sentences with a small alphabet after full stop and sometime after comma started with a capital word.

Response:

Dear, we corrected.

Comment 7:

Authors have used ODE, but they must first define it when they have used it for the first time like ordinary differential equations (ODE) and then they can use it like ODE everywhere.

Response:

Dear, we corrected.

Comment 8:

Line 131, the involvement of multiplication of (-iota) in tan and (iota) in cot is doubtful. Please can you show me step by step that how this iota is multiplied within it.

Response:

e^(ix)=cos(x)+isin(x)…(1),   e^(- ix)=cos(x)-isin(x)…(2)

By solving (1) and (2), we get

Cos(x)= [e^(ix)+ e^(-ix)]/2,    sin(x)= [e^(ix)- e^(-ix)]/2i

Tan(x)=sin(x)/cos(x)=    [e^(ix)- e^(-ix)]/ i[e^(ix)+ e^(-ix)]= -i[e^(ix)- e^(-ix)]/ [e^(ix)+ e^(-ix)]

Or

Tan(x)=[i e^(-ix)-i e^(ix)]/ [e^(ix)+ e^(-ix)]

Similarly, we can explain the multiplication of (iota) in cot.

Comment 1:

Since, many higher dimensional KdV equation has already investigated by several

researchers, authors need to discuss the features of (1+1) KdV equation.

Response:

Dear, we discussed the features of (1+1) KdV equation.

Comment 2:

Authors have mentioned that breather-wave, 1-Soliton, 2-Soliton, three-wave, new

periodic-wave results are obtained through this work. But they failed to present the 3D figures for these solutions.

Response:

Dear, we modified the figures.

Comment 3:

It is essential to illustrate the solutions via drawing figures for better understanding of solutions.

Response:

Dear, we done.

Comment 4:

Authors arrived 1 soliton, 2 solitons and 3 solitons. However, they portrayed figures corresponds to only the one soliton solution.

Response:

Dear, we modified the figures.

Comment 5:

Use any one nonlinear or non-linear in the manuscript.

Response:

Dear, we done.

Comment 6:

More explanations needed for all figures.

Response:

Dear, we added.

Comment 7:

No interpretations observed for figure 8.

Response:

Dear, we added the interpretations for figure 8.

Comment 8:

Intensity of wave is |g(?, ?)|^2. Use this notation in 3D graph.

Response:

Dear, we done.

Comment 9:

For MI analysis, 3D representation needed.

Response:

Dear, we added the 3D representation for MI analysis.

Comment 10:

Authors may provide a table for comparison of methods used in this paper with type of

soliton solutions for readers to understand the work easily.

Response:

Dear, we added the types of solutions corresponding to each method in the motivation paragraph in introduction.

Comment 11:

In reference section, authors cited many references which are irrelevant. Authors

mainly focused KdV equation in the present work. But they cite many papers related

to NLS equation. Authors may eliminate irrelevant references.

Response:

Dear, we done.

Comment 12:

Authors may provide recent references related to KdV equations.

Response:

Dear, we done.

Comments:

I have reviewed the revised paper and feel that authors have made changes in the revised manuscript and tried their best to enhance quality of their paper.

Response:

We appreciate the referees for spending time and taking care of our manuscript.

Thank you for your useful comments and suggestions on the structure of our manuscript.

The revised version of our manuscript has been submitted to your journal.

We look forward to your positive response.

Comments:

Thanks for incorporating my comments. However, Authors should eliminate irrelevant references. Since, authors investigated KdV equation, they should eliminate unwanted references.

Response:

We eliminated irrelevant and unwanted references.

We appreciate the referees for spending time and taking care of our manuscript.

Thank you for your useful comments and suggestions on the structure of our manuscript.

The revised version of our manuscript has been submitted to your journal.

We look forward to your positive response.