Korteweg-de Vries equation: Difference between revisions

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The '''Korteweg-de Vries (KdV) equation''' is
The '''Korteweg-de Vries (KdV) equation''' is


<center><math>u_t + u_x _x + 6uu_x = 0.</math></center>
<center><math>u_t + u_x_x + 6uu_x = 0.</math></center>


The factor of 6 is convenient for reasons of [[complete integrability]], but can easily be scaled out if desired.
The factor of 6 is convenient for reasons of [[complete integrability]], but can easily be scaled out if desired.

Revision as of 19:00, 28 July 2006

The Korteweg-de Vries (KdV) equation is

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle u_{t}+u_{x}_{x}+6uu_{x}=0.}

The factor of 6 is convenient for reasons of complete integrability, but can easily be scaled out if desired.

The equation is completely integrable, and has infinitely many conserved quantities. Indeed, for each non-negative integer k, there is a conserved quantity which is roughly equivalent to the H^k norm of u.

The KdV equation has been studied on the line, on the circle, and on the half-line.

The KdV equation is the first non-trivial equation on the KdV hierarchy.