An action variable of the sine–Gordon model

It was conjectured that the classical bosonic string in AdS times a sphere has a special action variable which corresponds to the length of the operator on the field theory side. We discuss the analogous action variable in the sine–Gordon model. We explain the relation between this action variable and the Bäcklund transformations and show that the corresponding hidden symmetry acts on breathers by shifting their phase. It can be considered a nonlinear analogue of splitting the solution of the free field equations into positive- and negative-frequency parts.