Hamiltonian theory of integrable generalizations of the nonlinear Schrödinger equation

A method for constructing integrable systems and their Bäcklund transformations is proposed. The case of integrable generalizations of the nonlinear Schrödinger equation in the one-dimensional case and the possibility of extending the method to higher dimensions are discussed in detail. The existence of Bäcklund transformations of a definite type in the systems considered is used as a criterion of integrability. This leads to “gauge fixing” — the number of physically different integrable systems is strongly diminished. The method can be useful in constructing the admissible nonlinear terms in some models of quantum field theory, e.g., in Ginzburg-Landau functionals.