On Miura transformations of evolution equations

The general Miura transformation (t,x,u(t,x)) to (s,y,v(s,y)): v=a(t,x,u,. . ., delta ^ru/ delta x^r), y=b(t,x,u,. . ., delta ^ru/ delta x^r), s=c(t,x,u,. . ., delta ^ru/ delta x^r) is considered which connects two evolution equations u_t=f(t,x,u,. . ., delta ⁿu/ delta xⁿ) and v_s=g(t,x,u,. . ., delta ^mu/ delta x^m). The conditions c=c(t) and m=n are proven to be necessary. It is shown that every Miura transformation, admitted by a constant separant equation u_t=f, consists of the following three transformations: (i) (t,x,u) to (t,x,w), where w=a(t,x,u,. . .,u_{x. . .x}); (ii) (t,x,w) to (t,y,v), where y=x and v=w, or y=w and v=w_x, or y=w_x and v=w_xx; (iii) a transformation of time s=c(t) and a contact transformation of (y,v). As an example, the Korteweg-de Vries equation is transformed to three new nonlinear equations, of which two have neither nontrivial algebra of generalized symmetries nor infinite set of conserved densities.