In fact, a transformation due to Gardner provides an algorithm for computing an infinite number of conserved densities of the KdV equation, which are connected to those of the so-called modified KdV equation through the Miura transformation v_x+v^2=u
(Tabor 1989, p. 291). The Korteweg-de Vries equation also exhibits Galilean invariance.
In the course of attempting to solve the KdV equation exactly, it was discovered that the equation has an infinite sequence of nontrivial conservation laws, which we shall presently define.
The so called turbulence closure problem has to do with the fact, that in fluid dynamics in principle an infinite number of conservation laws exist. Is there a connection? Can we write down a Korteweg-de Vries-like Equation, which generated the infinite conservation laws encountered in fluid dynamics?
- Lax(1990): The zero dispersion limit, a deterministic analogue of turbulence. read it at [http://books.google.com/books?hl=en&lr=&id=9ozj3Bs45kEC&oi=fnd&pg=PA53&ots=yTpf57ObXB&sig=DEACFXy2v5gQkNppN-rEmGaFoiA#v=onepage&q=&f=false]
- D.B. Fairlie (1996) : Equations with an infinite number of explicit Conservation Laws