Noether Theorems and Discrete Variational Integrators in Field Theory

The discrete analogue of the Noether-type identities in field theory is investigated by means of the difference discrete variational principle in which the difference is regarded as an entire geometric object. The discrete counterparts of the Noether theorems are obtained. It is proved that there exists the discrete version of the Noether conservation law in field theory. The discretization for the nonlinear Schrödinger equation is presented to illustrate the results.