New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra ${\mathit{A}}_{\mathbf{2}}^{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}$

Round 1

Reviewer 1 Report

In this paper, the authors studied a new system of integrable derivative non-linear Schrodinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac-Moody algebra. The content of the paper is interesting and the results obtained are novel, but there are many details in the paper, which need to be modified and improved before the paper is published.

There are also the following details that need to be corrected.

1) The Introduction, line 8: “effort” should be written as “efforts”.

2) The Introduction, line 15: “of ” should be deleted.

3) Page 2, line 6: “are” should be written as “were”.

4) Page 2, line 10: “equation” should be written as “equations”.

5) Page 2, line 12: “dividied” should be written as “divided”.

6) Page 3, line 9, “in the form” should be written as “in the form of”.

7) Page 4, line 24: “.” should be written as “,”.

8) Page 5, line 1: “solution” should be written as “solutions”.

9) Page 6, line 25: “idea” should be written as “ideas”.

10) Page 8, line 14: “,” should be written as “.”.

11) Page 9, line 3: The second “that” should be deleted.

12) Page 9, “,” after equation (37) should be written as “.”.

13) Page 10, line 1: “determining” should be written as “determined”

14) Page 10, line 19: “we” should be written as “We”

15) Page 11, line 12: “took” should be written as “take”

16) Page 12, line 15: “ There are however some details” should be written as “There are, however, some details”.

The English expression needs moderate modification and improvement.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper is not written well. It needs major revision. Now, I'll give some comments:

1.        The novelty of the paper is not well defined. In Introduction some known results are discussed. But nowhere in the paper, the obtained results in this paper are compared with existing ones. It is written on l. 31 that the results are in some sense generalizations, so, definitely, these results have to be discussed and compared.

2. Eq. (1) and elsewhere in the paper- what does [X,Y] mean?

3.       The sentence on lines 51-52 is useless.

4.       Figure 1- is it author’s or it is taken from another paper? If it not author’s one, then the source has to be cited. Also, why did you take the number to be 11?

5.       Section 3- what is the author’s result what is taken from [4]. It has to be defined. Now, the readers have to read [4] to answer of this question. The written has to be compared with the ones in [4].

6.       Line 107- if it is not obvious, it has to be proved.

7.       To be written a section with just one system of equations is not  serious (see Section 4).

8.       English is terrible. For example, l. 76, l 213, l. 217,etc.

9.       Eq, (59)- what does I in the subscript mean?

10.    Section 7- no conclusions, no discussions are given. Only some future work is given.

11.   The application of the obtained results has to be given. The importance of the obtained results has to be proved.

12.   As an overall- the used notations are strange and not one and the same on different lines. For example, the subscripts of S on line 211 differ the ones on line212.

English-speaking person has to correct the paper. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors did not point out the changes in the revised version. Do they assume that i have to compare line by line the first and the second version of the paper?

It was so difficult for me to compare line by line both versions, but I had no other choice. So, I've realized, that the authors answered to my questions in the review but they did not change appropriately the paper. For example, I wrote about English, I gave them some examples, and they corrected only them. Also, I pointed out that the notations have to be one and the same, that all used notations have to be defined , but the authors did not correct. For example, in (62) what are k and j? Also, look at T- it has two variables. But on line 274 it has 3 variables. It is absolutely unacceptable in Mathematics.  

English is fine. 

Author Response

Point-by-point response to Comments and Suggestions for Authors
"It was so difficult for me to compare line by line both versions, but I had no other choice."

This indeed is an error on my side. The reason that there wasn't an exact record of all the changes is that the text has been significantly reworked and expanded, especially the introduction (around 1000 words were added). I apologize for any inconvenience this has caused you, and I am grateful for the work you have done in reviewing the article.

"So, I've realized, that the authors answered to my questions in the review but they did not change appropriately the paper. For example, I wrote about English, I gave them some examples, and they corrected only them."

As mentioned, a significant part of the text has been reworked with the purpose of improving the language and quality of the text. The structure of the paper has been kept the same - section numbering, equation numbers and so on should be the same. I have tried to eliminate all the grammatical and punctuation errors, but it is possible that some still remain (if this is the case, those will be corrected in the final proofreading).

I would also like to address here the first question from the questions for general evaluation: Does the introduction provide sufficient background and include all relevant references? As mentioned the introduction has been significantly expanded, which includes adding additional references. Adding more information is always possible, but this probably will move the focus away from the goal of the paper. As to the references, I have tried to cite all the relevant articles, but it is possible that something was missed. If this is the case, please provide some suggestions and I will consider them.

"Also, I pointed out that the notations have to be one and the same, that all used notations have to be defined , but the authors did not correct. For example, in (62) what are k and j? Also, look at T- it has two variables. But on line 274 it has 3 variables. It is absolutely unacceptable in Mathematics."

I tried to correct all the mismatched notations in the revised version, but apparently missed the one in question. The T- does not depend on x, which has been corrected in the second revision. As for k and j, the equations in question are written for the components of the corresponding matrices. 
In the second revision, this has been written explicitly for (62) and also for (32) where component-wise notation is also used. The preliminaries section also specifies the used notation for the unit matrix and the trace operation.
Additionally some changes were made to further improve the quality of the text. This is highlighted in the text and the full changelog is:

Changes in text:

• line 46 - replaced
  "...where most of the low dimensional cases were studied..."
  with
  "...where integrable models, related to low rank Kac-Moody algebras were studied...".
  This is more mathematically precise, since Kac-Moody algebras are infinite dimensional. 
• line 54 - replaced
  "Generalizing the mentioned results to the polynomial case presents some difficulties."
  with
  "Studying exactly solvable models related to the polynomial case presents some difficulties.".
  This improves the flow of the introduction. 
• line 94 - added "where tr denotes the trace."
• line 124 - 126 - merged the two sentences into one. This is to show that the used definition somewhat differs from the usual one - the reason being that the central extension has no relation on the derived exactly solvable model.
• line 130 - removed ",i.e" as it was redundant.
• line 140 - added the notation for the unity matrix. Note that this adds another line to the revised text.
• line 195 (now 197) - explicitly specified that eq (32) is written component-wise. This adds an additional line.
• line 274 (now 276) - removed the x dependence of T and an erroneous partial derivative sign. Those were typos that were missed during the first revision.
• line 277 (now 279) - replaced "in summary we get" with "results in".
• line 279 (now 281) - explicitly specified that eq. (62) is written component-wise.

Changes in formulas:

• In formula (17), specified that k is taken mod 3 (where relevant).
• The indices that denote the matrix components in eq. (32) (were ij) and eq. (62) (were kj) were rewritten as jk. Since i is used for the imaginary unit, using it as an index was not appropriate.

Round 3

Reviewer 2 Report

The authors revised the paper according to my suggestions.

Author Response

Thank you again for the time you took to review the paper.