Korteweg-de Vries equation: Difference between revisions
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<center><math>\partial_t u + \partial_x^3 u + 6u\partial_x u = 0.</math></center> | <center><math>\partial_t u + \partial_x^3 u + 6u\partial_x u = 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 [[completely integrable|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 <math>H^k</math> norm of u. | 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 <math>H^k</math> norm of u. | ||
Revision as of 03:47, 31 July 2006
The Korteweg-de Vries (KdV) equation is
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 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.
