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

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