1. About Fig. 1: - I think it is a bit silly to use a numerical method to simulate the collision of two KdV solitons, since exact formulae are available for doing this. If the point was to contrast a soliton collision in an integrable system with a solitary wave collision in a non-integrable system, that would make more sense. - The figure would benefit from a second panel showing the (x,t) plane, so that the phase shift can be clearly observed.
2. The part
"Although Toda didn't initially examine recurrence phenomena, his lattice has subsequently been used by many to discuss near and exact recurrence. We compare the Toda and linear-plus-quadratic ("\alpha") FPU lattices side-by-side in Fig. 2. The initial condition in this figure consists of a sine wave with large displacement, as one can see based on the irregular appearance of the left panel as compared with the right one. The time evolution in the left panel eventually leads to "equipartition" of energy. Remarkably, the FPU and KdV simulations exhibited the property of near-recurrence to the initial condition, indicating that the asymptotic description was valid for very long times. Such results vindicated the prescient belief of John von Neumann and Stan Ulam that digital computers could be used solve scientific problems and led to the rapid development of nonlinear physics (Zabusky, 2005)(Porter, 2009b)".
is informative, but is not relevant for an article about solitons. It gives interesting facts about recurrence etc, but I think it should be omitted here.
3. Some care should be used when discussing solitons of multi-dimensional equations. Are line-solitons allowed? These are not really two-dimensional solitons. And true two-dimensional solitons are much much rarer: see the lump solutions of KP1. These have algebraic decay, and are often not in one's favorite function space. And many 2-D (and beyond) equations don't have them at all. For instance KP2 has no fully localized solutions. Some of the statements about the presence of solitons in multi-dimensional systems could be made more careful.
The paper is quite useful, and, generally, is a well-written piece. However, it shows some obvious omissions, which should be filled. First, as concerns historical aspects, it is necessary to mention that the integrability of the sine-Gordon equation in terms of the Baecklund transform was known back in the 19th century, in the context of differential geometry. Next, as concerns two-dimensional integrable equations, it is necessary to indicate the distinction between the KP-I and KP-II equations, and display the simplest "lump" solution to the former one (in particular, it is worthy to stress that these are weakly localized solitons and, on the other hand, collisions between them yield zero phase shifts). As an important example of a nonintegrable system where exact solutions for isolated solitons are available, it is worthy to mention the coupled-mode equations for fiber Bragg gratings in optics (and display the exact form of the soliton solution). As concerns applications, it is definitely necessary to give adequate references to the two papers which reported the creation of bright solitons in the BEC of Li-7 atoms (Rice University and Paris, 2002), and the work by the group of M. Oberthaler which reported the creation of the gap soliton in the condensate of Rb-87 (2005). Also quite relevant would be to mention the concept of skyrmions, and the related idea that real nucleons may actually be realized as solitons of the Skyrme's model. Finally, it may be quite relevant to explain that the greatest challenge to the current experimental work with solitons in nonlinear optics and BEC is the creation of stable 2D and 3D solitons in media with the cubic self-focusing nonlinearities (in fact, the challenge is to stabilize such solitons against the collapse, following diverse theoretical predictions).
Recent edit looks worse in my browser
Regarding the revision:
11:31, 26 October 2011, Leo Trottier (Fixed equation formatting)
The equations seemed correct in my browser before the edit. After the edit the equations are no longer aligned properly. Particularly, Eq 5 is indented three inches more than equation four.
Does anyone else have this problem?
Some notes on 30 Oct 2011
Mason: There seems to be a nonuniformity among browsers in the equations with the new scholarpedia formatting. They look weird but readable in my browser (I have the indentation problem too). Norm also didn't like how the equations now look. One possibility is just to remove the numbering (and references to the numbering elsewhere in the text) and circumvent the problem that way. However, the new indentation system does seem buggy, so I would prefer a better solution. For now, I am not adjusting the equation indentation (pending what the decided solution is).
There is another issue as well: I am one of the curators and my edits didn't show up even though I am a curator. (Norm asked me to add a couple of references and adjust text related to another reference.) He did not see my edits, and I think in practice he doesn't go into the wiki parts to do direct editing himself. So I think we need to resolve this on the permissions end so that my edits can show up without Norm's intervention. (I need to adjust the edits further per his e-mail comments, but the broader issue of the edits showing up without his intervention remains.)
Eugene: The new revisions were approved and they will appear in a few days. You can still see them in the history window. The equations look nice on my Safari and iPad. They overlapped with the figure, though because MathJax tries to center them. Leo, maybe, we should left-justify the equations by default?
Mason: I am also using Safari. I agree the current version looks better. I am going to do number the unnumbered equation as well because its left-justification does not look very good.