User talk:Hans Lundmark

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 * Thank you! Hans Lundmark (talk) 11:37, 14 May 2008 (UTC)

Camassa–Holm equation
Hello Hans Lundmark. Someone put a lot of references to papers by prof. Constantin into this article. As already remarked by you on the talk page, this is out of balance. I think it needs some cleanup (removal of some Constantin references, or addition of references to contributions by others). Since I am not very acquainted with the subject: are you willing to have a look at it? -- Crowsnest (talk) 09:23, 11 June 2008 (UTC)


 * OK, I'll see if I can find some suitable references to add to the list. Thanks for cleaning up the article, by the way! Hans Lundmark (talk) 08:01, 12 June 2008 (UTC)


 * ...eventually. I've been quite busy with other things, but it's still on my "to do" list. Hans Lundmark (talk) 15:52, 26 August 2008 (UTC)


 * That's very nice of you. It is not urgent, since the article does not appear to be wrong, as far as I can see. It is just that the references are out of balance. -- Crowsnest (talk) 10:52, 27 August 2008 (UTC)


 * Done, finally! Hans Lundmark (talk) 11:20, 19 February 2009 (UTC)


 * That is an impressive list! Can you point out a few (2-5) review or key articles, or books, for someone who wants to get him/herself acquainted with the subject? -- Crowsnest (talk) 12:25, 19 February 2009 (UTC)

I don't know of any books, and no comprehensive review articles either, but many papers have an introduction which outlines the big picture, for example Bressan & Constantin (1997a). In any case, the original Camassa & Holm (1993) paper is of course a good starting point (and it's only 4 pages). Here are some main lines of research: Well, that was more than 2–5 papers, but I hope it helps anyway!
 * Peakon solutions (which is the only one of these items that I know really well). Some of my own talks and papers might be helpful as an introduction, but the main reference for CH peakons is Beals, Sattinger & Szmigielski (2000) which is very good although it uses an unorthodox normalization for the coefficients in the equation.
 * Water wave theory. There's some controversy regarding the validity of the derivation by Camassa & Holm as a water wave model. See the papers by Johnson, for example Johnson (2003b), and Constantin & Lannes (2007).
 * Existence, uniqueness, wellposedness, stability, propagation speed, etc. There are lots and lots of papers which deal with these analytical questions (as is clear from a glance at the list of references). What's bugging the PDE people is that some initially nice wave profiles develop singularities in finite time, whereas others do not. Peakons are a nice testbed for studying this, since the explicit n-peakon solution is known; for example, with all mk positive the solution is defined globally in time, but with both peakons and antipeakons present there are collisions after finite time where the derivative ux blows up, and then there is the question of whether the solution can be continued beyond the instant of collision. It turns out that it is possible, but not in a unique way, so one needs to impose extra conditions (like conservation of energy) to pick out the "correct" continuation. With this as a basis for the intuition, one tries to extend the results to more general solutions. Another similar question is this (peakon formation): if the initial data for u are smooth, can one give conditions guaranteeing that the derivative ux develops (or doesn't develop) a jump in finite time? Constantin & Strauss (2000) prove that a solution which initially is "almost" a peakon remains so, which is important since it indicates that peakon solutions may give some insight into how more general solutions can behave. A few other papers that might be useful (but this whole subject is quite technical, and I don't know it very well, so these are somewhat randomly chosen): McKean (2004), Holden & Raynaud (2007a) (preprint here), Bressan & Constantin (1997a).
 * Travelling waves. The classification in Lenells (2005c) shows that the CH equation admits some very peculiar and interesting travelling wave solutions.
 * Integrability structure (symmetries, hierarchy of soliton equations, conservations laws) and differential-geometric formulation. Fuchssteiner (1996), McKean (2003b), Lenells (2005a), Misiołek (1998).

Hans Lundmark (talk) 09:09, 23 February 2009 (UTC)


 * Thanks Hans! I was wondering if we can not use this to bring some order in the references. I think people unacquainted with the subject like to know where to start with reading, and the above can be used to do that. If that is OK with you (please let me know, here), we continue that discussion on Talk:Camassa–Holm equation. Best regards, -- Crowsnest (talk) 13:44, 23 February 2009 (UTC)


 * OK. Hans Lundmark (talk) 14:28, 23 February 2009 (UTC)

B-class
Hans, do you agree that the article on the Camassa–Holm equation is now B-class, see Version 1.0 Editorial Team/Assessment/B-Class criteria? Best regards, Crowsnest (talk) 10:05, 26 February 2009 (UTC)


 * Well, I guess so, with a bit of good will, although I still find many statements quite cryptic (so B6 could be questioned, and maybe B2 as well). I can't really put the finger on anything, but when I read the article, the feeling I get is that it could be a lot better, but then much of it would need to be completely rewritten, and I'm not quite up to that task at the moment. But at least the introduction looks much better now, with the nice picture! :) Hans Lundmark (talk) 10:24, 26 February 2009 (UTC)


 * Which you made! I leave it C-class then, for the moment. -- Crowsnest (talk) 10:25, 26 February 2009 (UTC)

Degasperis–Procesi equation
Just saw you did the same "Further reading" trick over here. :-) Good luck, Crowsnest (talk) 12:10, 13 March 2009 (UTC)