Symmetry breaking interactions for the Schrodinger equation in three-dimensional space-time

A systematic study of symmetry breaking for the nonlinear Schrodinger equation i psi(t) + Delta psi = F(x, y, t, psi, psi*) with Delta being the two dimensional Laplace operator is presented. The free panicle equation that corresponds to F = 0 is known to be invariant under the nine dimensional Schrodinger group Sch(2). Tn this Letter. using the existing subalgebra classification of the Schrodinger algebra, we construct the most general interaction term F(x, y, t, psi, psi*) for each subgroup. Thus, while the symmetry group of the equation is reduced from Sch(2) to the: considered subgroup, invariance under the remaining subgroup still allows us to find the group theoretical properties of the corresponding modified nonlinear equations which are good candidates to be solvable models. We list all the results obtained in tables. (C) 2000 Elsevier Science B.V. All rights reserved.