Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors use the approach of scaling homogeneity and related software to symbolically compute conserved densities, fluxes, higher-order symmetries, Lax pairs, bilinear forms, and Miura-type transformations for polynomial systems of evolution equations. They consider the Gardner equation as a prototype example, provide a detailed review of the integrability analysis and solutions of this equation, and demonstrate the applicability of the scaling symmetry approach for investigating the complete integrability of polynomial nonlinear partial differential equations. This paper is a well-written and an accessible review to non-experts readers. Therefore,

I recommend that this article be published in the journal Mathematical and Computational Applications in its current form.

 

Author Response

Dear referee:

Thank you very much for your review. We are glad to learn that you find our paper valuable and recommend publication. We have made some changes requested by the other reviewer. 

Sincerely yours, 

Willy Hereman and Unal Goktas 

Reviewer 2 Report

Comments and Suggestions for Authors

Please see pdf document attached.

Comments for author File: Comments.pdf

Author Response

Dear referee, 

We very much appreciate your thoughtful suggestions and comments. We have taken into account your suggestions in the revision and also addresses your technical questions. We also corrected the typos you had listed (and a few other ones).  

A detailed response is provided in the file we have uploaded which also has a version of the paper in which all the changes are colored in blue. 

Thanks again for your careful review of our paper. 

Sincerely yours, 

Willy Hereman and Unal Goktas 

Author Response File: Author Response.pdf