Nonlinear Dynamics of Quantum Systems and Soliton Theory

Abstract

We show that space-time evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a space-time dependence of a matrix element of fermionic systems related to the {\it Orthogonality Catastrophe} or {boundary states} with the $\tau$-function of the modified KP-hierarchy. The established relation allows to apply the apparatus of soliton theory to the study of non-linear aspects of quantum dynamics. We also describe a {\it bosonization in momentum space} - a representation of a fermion operator by a Bose field in the presence of a boundary state.