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