Bäcklund transformation ($v={\mathcal{B}}_a[u]$)

A parameter-dependent system of differential relations maps a solution of one differential equation to a solution of another, or to a new solution of the same equation. Compatibility of the relations supplies the equations of motion. Integration constants select a particular transformed field. The Sine-Gordon Bäcklund transformation is an example of an auto-transformation and also generates conserved currents.