Noether Symmetries of the Triple Degenerate DNLS Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see attached. 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Minor English checking required

Author Response

Comments: The paper focuses on Lie and Noether symmetries for the triple degenerate derivative nonlinear Schr\"{o}dinger equations. Conservation laws are also found, which are distinct from [2] given that a gauge term exists. Moreover, new Lie symmetries are found for the equations for some non-vanishing integration functions. The paper is interesting and will be of interest to readers. The work adds to existing knowledge on Noether symmetries. There are some minor language errors in text that can be fixed by another read through. I recommend the paper is accepted after this.

Response:  Thank you for pointing out the language errors in text. I have fixed the errors in the manuscript as far as I found them in the revised manuscript. 

 

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript explores Noether symmetries to a generalized system of derivative nonlinear Schrödinger equations by using the least action principle. Various examples of s Lie symmetries and Noether symmetries are explicitly investigated and presented, together with their conservation laws. The authors should indicate what kind of white noise has been used in the plots. A more general setting than Noether’s theorem is to use a pair of symmetry and adjoint symmetry to establish a conservation law for non-Lagrangian equations in any dimensions (see, Discrete Continu Dyn Syst Ser S, 11(2018), no. 4, 707-721). The topic of the manuscript is of current interest, and the results contribute to the existing literature on symmetries and conservation laws.

 

On the other hand, bi-Hamiltonian systems of multi-component generalized integrable equations have been formulated via compatibility conditions of matrix spectral problems (see, e.g., Rom J Phys, 69(2024), nos.1-2, 101 and a paper at Mod Phys Lett B, https://doi.org/10.1142/S0217984924503196), which present concrete examples of four-component integrable nonlinear Schrödinger equations. Hamiltonian structures naturally exhibit a connection between symmetries and conserved quantities. It would capture the interest of research scientists in the international nonlinear mathematical physics community, if the authors could comment on those existing theories on connections between symmetries and conservation laws for multicomponent nonlinear wave equations in whatever dimensions.

In summary, the manuscript presents a diverse range of conserved densities and fluxes corresponding to the Noether symmetries and various Lie symmetries for the considered nonlinear model. it is therefore recommended that an enhanced and revised version of the manuscript be accepted for publication in the journal.

Author Response

I provide my response to the reviewer’s comments  in the attached PDF file. Please see the attachment.

Author Response File: Author Response.pdf