Miura transformation
= Miura transformation
{c}
{title2=$u=v^2+v_x$}
{wiki}
Direct differentiation gives
$$
u_t-6uu_x+u_{xxx}=(\partial_x+2v)(v_t-6v^2v_x+v_{xxx}).
$$
Consequently this transformation takes solutions of the <modified Korteweg-De Vries equation> to solutions of the <Korteweg-De Vries equation> in these sign conventions. Both summands in $u$ have the same scaling weight, so the modified-equation scaling induces $(x,t,u)\mapsto(e^sx,e^{3s}t,e^{-2s}u)$.