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)

"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)

"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,

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

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)