Nonlinear $\sigma$ model for the Dodd–Bullough equation

It is shown that the Dodd–Bullough equation is related to the nonlinear two-dimensional $\rm SL(3,R)$ sigma model in which a triplet of massless fields takes values on a sphere of three-dimensional unimodular affine space in the same way that the sine-Gordon equation is related to the $S^2$ sigma model, which describes a three-component field with values on a sphere of ordinary three-dimensional Euclidean space. Equations of motion are obtained for the $\rm SL(3,R)$ sigma model.