The concept of a classical integrable model with finitely many degrees of freedom dates back to Liouville. More recently, the notion of integrability has been extended to quantum-mechanical systems (examples include Gaudin models and Toda chains) and to field theories with infinitely many degrees of freedom. The prototypical example of an integrable field theory is the KdV equation, which models non-linear waves in shallow water. Other important examples include Toda field theories and non-linear sigma models.
An interesting question is how to quantize a given classical integrable field theory. In recent years many interesting dualities between integrable models have been discovered. Both the ODE/IM and AdS/CFT correspondences can be seen as dualities between classical and quantum integrable models.
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