Talk:Dym equation

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[edit] Alternative formulation?

The Dym equation

u_t = u³u_xxx

appears to be written in many places as

u_t = (u ^{− 1 / 2})_xxx

and it is claimed for example in

• Olver, Peter J. (1993). Applications of Lie groups to differential equations, 2nd ed.. New York, NY; London: Springer-Verlag. ISBN 0387940073. 

that the two forms are equivalent. Can an expert confirm that? Oleg Alexandrov (talk) 04:10, 26 January 2007 (UTC)

Well, Peter Olver is an überexpert so his word is law, but for your edification: Substitute u = v ^{− 1 / 2} in u_t = v³u_xxx to get (v ^{− 1 / 2})_t = v ^{− 3 / 2}(v ^{− 1 / 2})_xxx and do the differentiation with respect to t to get − (1 / 2)v_t = (v ^{− 1 / 2})_xxx. Finally, do some rescaling to get rid of the factor − 1 / 2; for instance, set v(t,x) = w( − (1 / 2)t,x). Any mistakes in my computations are left as an exercise for the reader. -- Jitse Niesen (talk) 08:40, 26 January 2007 (UTC)

Thanks! Frankly I was not expecting an answer to the above, it was more of a note to myself. (but big brother Jitse is watching. :)

I added the alternative formulation in the article (that one, u_t = (u ^{− 1 / 2})_xxx is actually used much more often from what I saw in my google books search). If you have more to add to the article, you are more than welcome to. Oleg Alexandrov (talk) 16:47, 26 January 2007 (UTC)