Breather and Rogue Wave Solutions of a New Three-Component System of Exactly Solvable NLEEs
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper generalizes the Zaharov-Shabat system, based on a Lax pair with an L operator linear in the spectral parameter λ, by considering a higher order polynomial dependence. Then they derive a system of nonlinear evolution equation related to a Lax pair in the third and fifth order in λ. The system is fully analyzed in the paper and soliton-like solutions are presented. The paper is written in a technically rigorous manner and some further explanation, may be concerning the origin of Lax pairs idea, would be appreciated by the reader. Overall, I think it's a good work, offering interesting insights into the topic. For this reason, it deserves publication.
Author Response
Thank you for your kind remarks. Indeed, expanding upon the history and origin of Lax pairs would be extremely valuable to the reader. To this end, we have significantly expanded the introduction, explaining the difference between scalar Lax pairs and the ZCR, see lines 50-61 in the revised manuscript. We have also added numerous references ([38-43]).
Please find attached our authors' response file. It includes our response to the editor and other reviewers, as well as a full changelog, listing all the changes made to the manuscript.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsSee attached
Comments for author File: Comments.pdf
Author Response
We are extremely grateful for the time you took to read and evaluate our work. We agree with all your points and have done the following to rectify the situation:
- This is indeed true, and it is an error on our part to have omitted those classical examples. We have revised the Introduction to include the following sentences (along with the relevant references):
Lines 22-25: Added “Classical examples include the derivative NLS equations - the Kaup-Newell [26], Chen-Lee-Liu [27] and Gerdjikov-Ivanov models. Note that they are all examples of one-component NLEE related to higher-order energy dependent Lax operators [28,29].” The references read as follows:
26. D. J. Kaup and A.C. Newell, An exact solution for a derivative nonlinear Schrödinger equation. Journal of Mathematical Physics 1978, 19, 798–801. https://doi.org/10.1063/1.523737
27. H. H. Chen, Y C Lee and C S Liu. Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method. Physica Scripta 1979, 20, 490. https://doi.org/10.1088/0031-8949/20/3-4/026
28. M. Antonowicz and A.P. Fordy, Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems. Communications in Mathematical Physics 1989, 124, 465–486. https://doi.org/10.1007/BF01219659
29. M. Antonowicz, A. P. Fordy and Q. P. Liu. Energy-dependent third-order Lax operators. Nonlinearity 1991, 4, 669. https://doi.org/10.1088/0951-7715/4/3/003 - Also true. The text was revised as follows:
Lines 14-16: Added "Athorne and Fordy also showed that a large class of multi-component KdV and mKdV equations can also be related to symmetric spaces [11]." along with the relevant citation:
11. C. Athorne and A.P. Fordy. Generalised KdV and MKdV equations associated with symmetric spaces. Journal of Physics A: Mathematical and General 1987, 20, 1377. https://doi.org/10.1088/0305-4470/20/6/021 - We have corrected the mentioned typos and errors. In fact, we found a bit more which were corrected as well. As a result, the quality of the used language and the text in general have been improved significantly.
Please find attached our authors' response file. It includes our response to the editor and other reviewers, as well as a full changelog, listing all the changes made to the manuscript.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsExactly solvable problems play a special role because precise statements and conclusions can be formulated about them. That is why they will always be at the forefront of research regardless of the field of science examined. The research carried out in this article does all this in the field of scattering problems and soliton solutions. The article is well structured, the calculations are neatly carried out and correct. The results achieved are worthy of publication.
In a concise introduction quoting the most important literature, the authors point out the mathematical tools used in the article, namely the application of Lax operators, and the set goal, to find soliton solutions with the method. After a brief overview of some basic knowledge, the necessary information about Lax operators is presented (Sec. 3). Despite the many mathematical connections, the explanation is easy to understand. The analytical discussion of scattering problems is presented within a formal framework (Sec. 4). This may later be suitable for discussing specific cases. In any case, the conditions necessary for the solution are presented. The authors also show a computational example through the soliton solution (Sec. 5). Thus, the actual application can also be seen through the numerical procedure. This is a valuable part of the work. The completed figures are illustrative. The above explanation of a large group of nonlinear evolutionary equations can contribute to the mapping of the dynamics of physical, chemical, and biological systems.
Please, check Eq. (77).
Author Response
Thank you for your review and valuable insights. We agree that the correctness on Eq. (77) (now (81)) might not be immediately visible, which is an omission on our part. To see that this is true, we need to consider the properties of C under Hermitian conjugation. To this end, the preceding sentence to the equation in question was corrected as follows:
Line 224 (was 203): Added "(and using the fact that $C^\dagger = C^{-1} =-C$)".
Please find attached our authors' response file. It includes our response to the editor and other reviewers, as well as a full changelog, listing all the changes made to the manuscript.
Author Response File: Author Response.pdf
