On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation
Round 1
Reviewer 1 Report
This is an interesting manuscript which describes a construction of obtaining solutions to P_2 via the solutions of coupled set of non-linear Schrödinger equations, which reduce to a homogeneous set for a particular choice of coefficients, obtained by appealing to their Schlesinger symmetry. These are shown to be equivalent to the Bäcklund trafos for P_2. The interesting feature is that the homogeneous Schrödinger system implies a homogeneous fourth-order equation for its first component function u whose symmetries P0,+-1 satisfy a certain algebra. This equation can be reduced to second order (first integral=Wronskian, Cole-Hopf transformation) which then takes the form of P_2 with alpha not=0. Indeed, the symmetries P0,+-1 then turn out to be the Bäcklund trafos.
This method of constructing solutions to P_2 via the symmetries of a fourth-order equation, emerging from a coupled set of particular non-linear Schrödinger equations, is applied to a matrix P_2 and to the vectorial P_2, and ideas on how to generalize this method to construct solutions to P_4 are presented.
As far as I can see, this is viable work. However, prior to publication the authors are strongly advised to improve on their style of presentation (avoid use of colloquial statements, add important references, scan text for repetitions, define quantities). Also, the ms would benefit from a discussion of a few physical applications of P_2 (top, diffusion, etc.)
Apart from that, I have noticed the following problems:
In general: check spelling (focused -> focussed, etc) and capitalizations. Sometimes P_2, sometimes II Painleve, sometimes Second Pailneve -> use same name for the same thing throughout
L. 19: also -> also is
L. 24 : Eq.(10) -> Eq. (1)
L. 25: rational solutions if alpha in Z requires key references: H. Airault, Stud. Appl. Math. 61, 31 (1979), N. A. Lukashevich, Differ. Equations 7, 853 (1971).
L. 29 : pls spell out KdV and KP here (and not much later)
L. 47: Schlesinger Trafos require reference
L. 49: from for -> to
Eq. (3): and not slanted
Remark 1: sloppy formulation, proof is not given, therefore reference needs to included
E.q (6): unexpected surprise ?! (tautological)
Eqs. (7) and (8): no definition of D_x
Eq. (20): Property 4 is never used to construct new solutions
L 145,146 : Unnecessary statement!
As it turns out, this approach is capable of generalization
Author Response
The article has been proofread, fixing the typos and altering the notation in accordance with the referee’s suggestions (in particular, applying uniform notation when referring to the Painlevé II). The introduction has been significantly reworked to include a much more detailed discussion of the history and applications of the Painlevé equations (including but not limited to Painlevé II); accordingly, the list of references has been expanded from 31 to 56 articles. In addition, the conclusion has been rewritten in order to better emphasize the results of the article and to discuss their possible ramifications.
Reviewer 2 Report
In this paper, the authors demonstrated the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS). The proposed algorithm allows for a construction of Jordan algebras-based completely integrable multiple-fields generalizations of the Second equation while also producing the corresponding Bäcklund transformations. The idea seems good, however, its importance in the field of Mathematical Physics is still not clear. It would be better if the authors declare the benefits of their obtained results, especially, when solving PDEs in Mathematical Physics.
Author Response
The article has been proofread, fixing the typos and altering the notation in accordance with the referee’s suggestions. The introduction has been significantly reworked to include a much more detailed discussion of the history and both mathematical and physical applications of the Painlevé equations (including but not limited to Painlevé II) to better ground the proposed approach; accordingly, the list of references has been expanded from 31 to 56 articles. In addition, the conclusion has been rewritten in order to better emphasize the results of the article and to discuss their possible ramifications.
Round 2
Reviewer 2 Report
The revised version of this paper is ready for publication.
