 Modified Korteweg-De Vries equation

This nonlinear dispersive equation replaces the quadratic KdV nonlinearity by a cubic one. In the displayed sign convention, the Miura transformation maps its solutions to $u_t-6u{u}_x+u_{xxx}=0$. Its scaling symmetry is $(x,t,v)\mapsto (e^{s}x,e^{3s}t,e^{-s}v)$.