An analog of the lattice KdV equation of Nijhoff et al. is constructed on a hexagonal lattice. The resulting system of difference equations exhibits soliton solutions with interesting local structure: there is a nontrivial phase shift on moving between adjacent lattice sites, with the magnitude of the shift tending to zero in the continuum limit.
Publication information: A typical story of what happens when a journal does not get an article reviewed in a timely fashion.....
The following analysis of the resulting system is, in my view, not completely satisfying. The author separates an (uninteresting) set of solutions (10) and claims that "the other component" of the set consists of (11),(12). Then, it turns out that in this "othger component" there is another uninteresting set (13) which is completely analogous to (10). This leaves the reader wondering whether the rest does not, by chance, contain also further (or, may be, consist only of) uninteresting solutions. There is a valid mathematical task to describe a well-posed Cauchy problem for the author's equation, both on a single plaquette and on larger portions of the hexagonal lattice. This not being done, the model remains mathematically not well formulated.
Even more important, there seem to be serious reasons to abandon this model altogether. It is more or less clear that "natural" lattices for KdV-like systems to live are those with quadrilateral plaquettes, or quad-graphs, as described in: A.Bobenko, Yu.Suris. Integrable systems on quad-graphs. IMRN,2002, 11, p.573-611. The lengthy and ugly equations governing HexaKdV seem to confirm this. Actually, it is easy, by adding vertices and edges, to extend the hexagonal lattice to a quad-graph which is a natural ambient place for discrete KdV. To this end, one should add to all hexagons as on Fig.2 their centers and the edges coonnecting these centers with the points (n1,n2+1,n3-1), (n1+1,n2,n3-1), (n1,n2,n3) (in the notations of Fig.2). The resulting lattice will be dual to the kagome lattice, ad the KdV on this new lattice is much more aesthetically satisfying and is much easier to analyze. In particular, all equations on all plaquettes are identical with the quad-KdV (1) (but with three various parameters $q$, $h$, $k$), and it is absolutely clear how a correct Cauchy problem should be posed. At the same time, any solution of the KdV on the dual kagome lattice projects (just by forgetting the added vertices) to the solution of HexaKdV, and it is very plausible that any solution of HexaKdV can be extended to the dual kagome lattice (although I did not check this).
For the reasons above, I recommend to reject the paper.
Dear Professor Fordy, Thank you for correspondence of 24th April concerning my paper "HexaKdV". I must say, though, that I am more than a little disappointed at receiving a rejection, after 7 months of waiting, on the basis of a single negative report to which I have no difficulty responding. I have already submitted the paper to a different journal. But since there is a real possibility that it might be sent again to the same referee, I would ask you to please forward the comments below to the referee for his consideration. Reply to points of the referee ------------------------------ Before replying to the 2 criticisms of the referee, I would like to point out that he/she essentially relates just to the 1st half of the paper. Later in the paper I give explicit soliton solutions of the new "hexaKdV" system, which exhibit a new physical phenomenon - a nontrivial phase shift between different lattice sites. This is significant result that I feel has been ignored. The first criticism of the referee is essentially "how are we to know that the system studied is well-formulated and non-trivial?" Non-trivialness follows from the existence of the soliton solutions (and the host of other solutions that can be obtained by dressing techniques). As far as well-formulatedness is concerned, I willingly admit that the only evidence I can give for this is that the counting of the degrees of freedom is right, and a full proof would be very hard indeed. But such a proof should certainly not be a requirement of publication in PLA! After all, there are systems of substantial interest for which well-posedness remains a big question. The second criticism of the referee is that it is maybe redundant to study integrable systems on anything else but quad graphs, and he/she points out that the hexagonal lattice can be embedded in such a quad graph. I have to agree that studying the relationship of hexaKdV with KdV on the quad graph the referee suggests is evidently interesting. A little experimentation shows, however, that the solution set of hexaKdV is actually richer, with the difference being precisely in the "small" sets of solutions the referee labels "uninteresting". This is quite remarkable, as hexaKdV has full hexagonal symmetry, which the system on the quad graph does not; this symmetry is apparently restored by the "small" sets! There is much material for further study here. But, even if, say, all solutions of hexaKdV could be obtained from solutions of various quad graph equations, would that be a reason never to document it in the literature? Surely on physical grounds (hexagonal lattices do occur) and on symmetry grounds, hexaKdV is the more natural object to start with! I find it particularly bizarre that the referee cites a paper of Bobenko and Suris as the rationale for only looking at quad graphs - the same Bobenko and Suris (ref [1] in my paper and elsewhere) who study systems on other graphs and encourage the pursuit! It seems to me that underlying the referee's criticisms has a nonverbal message that he/she found the paper raised several serious issues that merit further study. I would certainly feel I had contributed to science if in wake of my paper the referee were to write a paper formalising the relationship of hexaKdV solutions and quad-graph solutions. The referee's advice "to abandon this model" sounds more like censorship than scientific assessment. Sincerely, Jeremy Schiff
From schiff@math.biu.ac.il Thu Jul 31 21:30:54 2003 +0300 Date: Thu, 31 Jul 2003 14:29:37 -0400 From: JNLS Editorial <jnlsed@math.princeton.edu> To: schiff@macs.biu.ac.il CC: abloch@umich.edu Subject: JNLS#580 Manuscript Submission REF: JNLS#580 TITLE: HexaKdV BY: Schiff EDITOR: Bloch Dear Dr. Schiff: We have now completed our review of your paper 'HexaKdV,' submitted to the Journal of Nonlinear Science. I attach copies of the referee reports along with the summary of the editor responsible for the paper. I regret to say that we do not find the paper acceptable in its present form. In my own view the most critical issues that you have not satisfactorily addressed are physical motivation, and the continuum limit of the system (two related issues, in my opinion). Both referees remark on the weak motivation and lack of physical detail, and referee 2 notes the apparant (paradoxical) linear nature of the continuum limit. Referee 1 implies that you may have essentially rediscovered a result of Bobenko and Suris; this must also be answered. Overall, the paper appears to be a work in progress rather than the kind of definitive study that we aim for at JNLS. While we are rejecting the paper in its present form, we believe that you should have an opportunity to answer the referees' and editorial comments. I am therefore inviting you to prepare a significantly revised version. If you decide to do this, you should bear in mind the 'Aims and Scope' of JNLS (see the Springer Link website). For example, it is NOT sufficient that Nijhoff and Bobenko have suggested studying lattice systems; you must convince a multi-disciplinary audience that the problem is more generally relevant. This may require significantly more work than you have done thus far. If you do decide to rewrite and resubmit, you should send either three hard copies of the revised paper, or a single .pdf or .ps electronic file, on which all major changes must be highlighted, along with a letter detailing your revisions and, should the following apply, explaining your reasons for not taking action on any of the specific points made by the referees (especailly the numbered points of referee 2). Please let me know what you intend to do. Sincerely yours, Philip Holmes Managing Editor _____________________________________________ [Editorial Summary:] In my opinion this paper has some interesting results but does not appear to rise to the level at which it can be accepted by JNLS in its current form. The author has made some progress on what appears to be an interesting problem but more work is needed. The referees have given several suggestions which should be helpful to the author. There are a number of directions in which it can be improved. Firstly, can it be formulated as a well defined evolution equations as suggested by one referee? The model does not seem well motivated physically -- more needs to be said about this to justify publication. The nature of the coupling is important in this regard. The continuum limit seems puzzling. There is very little analysis of this limit -- more should be given as well as explanation of its nature and relationship to KdV. I think it would also be helpful to have more explanation of the Nijhoff model and its physical (or otherwise) justification. More justification (method) for how the author arrives at this solutions would be good too -- it all seems ad hoc on the face of it. _____________________________________________ [Referee # 1 Report:] Report on Schiff "HexaKdV" A integrable equation on the regular hexagonal lattice is suggested. I am sceptical that this system alone can be considered as an evolution equation (the corresponding Cauchy problem is not formulated and probably can hardly be formulated). On the other hand the author obtaines a one-soliton solution for the system. I think that the system under consideration is a corollary of a corresponding system considered on a special quad-graph which is dual to the kagome lattice. The corresponding systems are treated in Bobenko, Suris, Integrable systems on quad-graphs, IMRN, 2002, 11, 573-611. Scientific quality of the paper: 3 If the model is improved so that the system becomes a well defined evolution equation (as suggested above) then the paper achieves the level of a typical paper in a good journal. I suggest to reject the paper in its present form. _____________________________________________ [Referee # 2 Report:] Referee's Report on: "HexaKdV" by J. Schiff This is an interesting short paper proposing an apparently integrable discrete (actually discrete both in space and time, hence a coupled map lattice) discretization of the KdV equation. The construction is following the lines of Nijhoff's construction on a square lattice generalizing it for a hexagonal lattice. There is a number of concerns that I have with the manuscript: 1) Supporting physics/applications: I believe the motivation of the paper is weak (in my view, quite weak). The fact that Bobenko et al. suggested this program does not give the reader a sense of where this program (of finding integrable discretizations of non-square lattices) would be applicable/relevant. Moreover the two "physicsy" references that the author mentions involve a significantly different setting than the one discussed here. Here, space and "time" are treated in the same footing, whereas there it is *space* which is genuinely two-dimensional, while time has a different dependence. Physically speaking, it is understandable to have waveguides which are placed in a fiber in a hexagonal pattern, or to have propagation of waves in a crystallic structure which is genuinely two-dimensional. On the other hand, it is very difficult to understand what a hexagonal structure of space-time would entail or why it would be relevant (?) In that sense, I feel it is not simply relevant but essential for the author to justify the physical provenance of the model at hand and its potential applications. 2) Again something related to physics, but more mathematical: it is not very straightforward to understand why it would be relevant to have a coupling (even if both dimensions were space, but also if one was space and one was time) with all the other vertices of a hexagonal plaquette. Let's take a simple example of waveguide arrays in a fiber. They would be coupled with their nearest neighbors -- they would not prefer instead of talking to a near neighbor to talk to one which is further away, neglecting the near ones. In view of that, the potential physical relevance of the model is even more questionable in my mind. 3) Another issue that puzzles me considerably (while I find that the phase shift around plaquettes of the soliton solutions and its disappearance in the continuum limit) is the hyperbolic tangent nature of the resulting solitary wave. How does that tie in with the square hyperbolic secant of the KdV that we the authors will be well-familiar with ? Further explanation of this and of mention of other potentially interesting solutions would be relevant. 4) Another item that is rather "worrisome" is the linearity of the continuum limit of the problem. I do not understand: why is the continuum version of the equation a *transport* problem rather than the KdV equation ?? This further seems suggestive that not just tanh but other solutions of the travelling wave frame could potentially be relevant in other functional forms -- is that true ? In conclusion: while I find the construction cute and interesting, at the moment I have significant concerns/doubts about the physical motivation of the problem and do not fully understand the continuum limit of the problem, the connection of the model at that limit with the KdV and the connection of the solitons of the model with the KdV solitons. I would be very interested in seeing a comprehensive revision of the manuscript addressing these items in detail. ____________________________ Journal of Nonlinear Science Princeton University Fine Hall, Washington Road Princeton, NJ 08544-1000 USA
Date: Fri, 1 Aug 2003 07:51:18 +0300 (IDT) From: Jeremy Schiff <schiff@macs.biu.ac.il> To: JNLS Editorial <jnlsed@Math.Princeton.EDU> Cc: abloch@umich.edu Subject: Re: JNLS#580 Manuscript Submission Dear Professors Bloch and Holmes, Thank you both for your time and effort reviewing my paper. Your comments and those of the referees (particularly referee 2) all seem quite reasonable. Some I can answer right away, some I know I can answer with a bit of work, and some I need to think about carefully. I would like to attempt a revision, but as I am away right now, and because of the issues raised, this is going to take some time. I hope to get back to you within a few months with a revised version, I hope this time frame is OK. Thanks again, Jeremy Schiff