Dark Type Dynamical Systems: The Integrability Algorithm and Applications^†

Round 1

Reviewer 1 Report

The authors construct nonlinear dynamical systems possessing some integrability properties, related to “dark equations”  introduced by Boris Kupershmidt. The results obtained are new and could be useful to the reader interested in the theory of integrable systems.

Several small remarks:

Notations are sometimes redundant, however, not all the objects are clearly introduced.

It is not clear what is 'y' in formula (2).

Page 2, "Let us suppose that the defined above spectrum σ(L) ⊂ C is under the isospectral deformation by virtue of the dynamical system (3)" - (3) should be (1)?

Page 4, "2.3. Generative function of conservative quantities" - 'conservative' should be 'conserved'? 'Generative' is probably 'generating'?

The spelling 'Schrëdinger', end of page 19, is not standard.

 

Author Response

We so appreciated to Referee for pointing  remarks concerning some mentioned references to formulae, which made  a manuscript  suitably enhanced,   as well as  that concerning  the name  Schrödinger.

Many thanks !

 

, which has been corrected.

Reviewer 2 Report

See my attached report.

Comments for author File: Comments.docx

Author Response

We cordially thank to Referee for all corrections and suggestions which strongly helped to enhance  the manuscript presentation. We took into account main all remarks and comments, as well as added supplementin refrences which were suggested as related to the work, for which our separate thanks.

Sincerely, Authors.

Reviewer 3 Report

The authors conducted an impressive study on diffusive and dispersion nonlinear dynamical systems on smooth functional manifolds. Overall, I would like to recommend minor changes on the application side of this study as follows,

Prospects: The authors mention that the findings of this study can be useful in modern physics, mechanics, hydrodynamics and biology sciences. How can other researchers benefit from this work? Please include a brief statement near the end of the manuscript.

§  It would be beneficial if the authors provided targeted context for this study's practical application, e.g., for solving a certain scientific problems or improvement of an existing tech.  This would be useful to other researchers to implement and build upon the findings of the study in future research projects

Author Response

We so  thankful to  Rviewer for reading the  manuscript and suggested comments concerning virtual applications of the  algorithm  obtained in our work. The related  ans deemed applications are described within the mansucript, other ones   concern  practically important control problems of diffusion and convection processes in nuclear plant reactors, chemical reactions modeling, as well as in plasma physics where  there is solved a problem of confining  dispersive wave   packets  inside toral constructions etc.