Miura transformation ($u=v^2+v_x$)

Direct differentiation givesConsequently 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 $(x,t,u)\mapsto (e^{s}x,e^{3s}t,e^{-2s}u)$.