σmodels of nonlinear evolution equations

A systematic method is developed of constructing the $\sigma$ model associated with any given nonlinear evolution equation solvable by the inverse scattering method. The $\sigma$ model is obtained from the adjoint representation of the group associated with the Lax representation of the evolution equation. Bäcklund transformations for the $\sigma$ model and for the evolution equation are realized as gauge transformations. The complete integrability of the $\sigma$ model follows from Pohlmeyer's $R$ transformation which is systematically constructed in each case. The examples of the sine-Gordon, nonlinear Schrödinger, Korteweg-de Vries, and modified Korteweg-de Vries equations are discussed in detail.