Talk:Korteweg–De Vries equation/Archive 1

Archive 1

Soliton Solution

Purely an ease thing, but for the soliton solution, the easiest way to solve it is to integrate again using f' as an integration factor 128.232.237.41 (talk) 16:17, 14 February 2009 (UTC)

This corresponds with the kinetic and (cubic) potential energy interpretation indicated in the article: multiply the equation with f' and integrate again. -- Crowsnest (talk) 19:19, 14 February 2009 (UTC)

Homogeneity of a PDE and classification of KdV

The following statement does not make sense to me: "The equation is homogeneous of degree 5 if x has degree 1, φ has degree 2, and t has degree 3."

I agree that the KdV equation is homogene, but for another reason, namely: there is no expression on the right hand side of the pde. So my suggestion is to change the above mentioned line into: The equation is homogeneous - in the usual sense of partial differential equation - which means roughly that there is no (nontrivial) expression on the right hand side which does not involve the solution φ itself.

By contrast the inhomogeneous version of the KdV equation looks like

$\partial _{t}\phi +\partial _{x}^{3}\phi +6\,\phi \,\partial _{x}\phi =g(x,t),\,$

with a known function g(x,t) depending only on x and t on the right hand side.

Some further words about classification: KdV is a third order, semilinear equation. By the way: Is the equation hyperbolic? —Preceding unsigned comment added by Aklaiber (talk • contribs) 20:37, 12 August 2009 (UTC)

Standard form

The standard form of the Korteweg-de Vries equation is ${\frac {\partial u}{\partial t}}-6u{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial t^{3}}}=0$, and not the one given. —Preceding unsigned comment added by 64.198.242.52 (talk • contribs) 21:37, 5 November 2004

Except, that the dispersive term should be differentiated with respect to x.

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

—Preceding unsigned comment added by 137.205.162.8 (talk • contribs) 17:16, 6 September 2005

That is surely a matter of taste. At least in his textbook Partial Differential Equations (2002) Evans uses the version with $+$ sign as written in the article.

Aklaiber (talk) 20:42, 12 August 2009 (UTC)

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Does anyone interested in the article written by Korteweg & de Vries ?

I am reading the article and have some questions(see picture)

File:PBNX((3B)567) `FXBS )J9.png

I can't find the two formula in fulid mechanics .I need help,I need the answer ,Thank you— Preceding unsigned comment added by Davi689 (talk • contribs) 06:25, 24 April 2018 (UTC)

Connection to waves

A nice addition would be motivation of the KdV equation from physical principles (a 'derivation' of the wave equation). Does this exist? Chris2crawford (talk) 23:17, 8 November 2019 (UTC)

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)