  Relativistic integrable field theories like the sine-Gordon equation have an
infinite set of conserved charges. In a light-front formalism these conserved
charges are closely related to the integrable modified KdV hierarchy at the
classical level. The latter hierarchy admits a family of symplectic structures
which we argue can be viewed as deformations of the relativistic sine-Gordon
symplectic structure. These deformed theories are integrable but no longer
relativistic and the basic excitations of the theory, the solitons, have an
interesting non-relativistic dispersion relation that in a certain limit
becomes the dispersion relation of dyonic giant magnons of string theory in the
AdS/CFT correspondence. We argue that the deformed classical theories can be
lifted to quantum theories when the sine-Gordon theory is embedded in a larger
theory that describes the string world-sheet sigma model in AdS(5)xS(5).
