Talk:Korteweg–De Vries equation

Sign error?

I think the equation should be (with a +6 rather than -6):

${\frac {\partial u}{\partial t}}+6u{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0$

This seems to be the case in the following two references:

[1] https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.15.240

[2] https://www.pnas.org/content/pnas/suppl/2019/07/15/1814058116.DCSupplemental/pnas.1814058116.sapp.pdf (Equation S4)

Can anyone explain this? Perhaps it should be included in the page as well for newcomers to this field (such as myself).— Preceding unsigned comment added by G-d (talk • contribs) 23:55, 31 July 2021 (UTC)

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"Korteweg-deVriesequation" listed at Redirects for discussion

The redirect Korteweg-deVriesequation has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 January 26 § Korteweg-deVriesequation until a consensus is reached. Colonies Chris (talk) 11:09, 26 January 2024 (UTC)

$\partial _{x}\phi f$

There is a back and forth over the meaning of the operator $\partial _{x}\phi$ in the Lax pair description

$A=4\partial _{x}^{3}-6\phi \partial _{x}-3\partial _{x}\phi$

Elsewhere in this formula, $\phi$ is the operator that multiplies scalars by the scalar field $\phi$, and $\partial _{x}$ is the operator of partial differentiation with respect to $\phi$. According to this interpretation, and standard conventions for operator products,

$\partial _{x}\phi f=\phi _{x}f+\phi f_{x}.$

However, this is not the correct Lax pair. Instead, what is intended is

$A=4\partial _{x}^{3}-6\phi \partial _{x}-3(\partial _{x}\phi )=4\partial _{x}^{3}-6\phi \partial _{x}-3[\partial _{x},\phi ]$

where the bracket denotes the commutator of the two operators. Tito Omburo (talk) 16:49, 27 June 2024 (UTC)

If it’s the commutator you are referring to with the square brackets, then this is not in line with your latest “explanatory note” edit. If ambiguity regarding the use of the partial derivative is what’s concerning you, then I don’t understand why this ambiguity suddenly arises halfway down the article rather than at the first description of the KdV equation Roffaduft (talk) 17:12, 27 June 2024 (UTC)

Do you disagree that $[\partial _{x},\phi ]f=(\partial _{x}\phi )f$? Tito Omburo (talk) 17:15, 27 June 2024 (UTC)

That is not the issue here (although it does look like you’re missing an $\phi _{x}f$ in there somewhere). It’s about consistent use of symbols and brackets throughout the article. I personally would prefer the use of subscript x rather than the partial symbol, because I think it makes the article look less “cluttered”. But again, addressing a “bracket issue” halfway down the article using “ambiguity” as an argument doesn’t make much sense to me Roffaduft (talk) 17:25, 27 June 2024 (UTC)

Sorry, there was a typo above, now fixed. Elsewhere, this is not a problem because everything is a scalar. But A is an *operator*, and the correct term is the commutator $[\partial _{x},\phi ]$ of the two operators, not the operator $\partial _{x}\phi$, which would take a function f to $\phi _{x}f+\phi f_{x}$. Tito Omburo (talk) 17:32, 27 June 2024 (UTC)

Yes, but that is not what you wrote down in your latest edit as “explanatory note” Roffaduft (talk) 17:34, 27 June 2024 (UTC)

It's not? Tito Omburo (talk) 17:35, 27 June 2024 (UTC)

If those are commutator brackets, then shouldn’t it be $[\partial _{x},\phi ]f=(\phi _{x}-\phi \partial _{x})f=\phi _{x}f-\phi f_{x}$. Your “explanatory note” introduces commutator brackets but then explains the basics of taking derivatives. It’s all just a bit convoluted. In fact, you don’t need to use commutator brackets (and the subsequent “explanatory note”) if you just changed the “6” with a “3” in the definition of operator “A”. Roffaduft (talk) 17:45, 27 June 2024 (UTC)

I feel like changing 6 to 3 would be even more confusing because until this point $\partial _{x}\phi$ is synonymous with $\phi _{x}$. Tito Omburo (talk) 17:49, 27 June 2024 (UTC)

Not only is it not confusing, it is also completely in line with page 32 of the Dunajski (2009) reference; discussing the lax representation of the KdV equation Roffaduft (talk) 17:53, 27 June 2024 (UTC)