Institute of Mathematics of NAS of Ukraine,
3 Tereshchenkivska Street, 01601 Kyiv-4, UKRAINE
e-mail: rop@imath.kiev.ua

Group classification of nonlinear Schroedinger equations with potential

Abstract:
Preliminary group classification in the class of nonlinear Schroedinger equations with potential is carried out within the framework of the new approach to problems of group classification. Namely, we describe all possible nonlinearities depending on the wave function and its complex conjugate and derive the conditions for the potentials, for which the corresponding nonlinear Schroedinger equations with potential admit nontrivial symmetries. (Here a symmetry is considered trivial if it belongs to the group generated by the translations and the rotations of the space variables and the translations of the time variable.)
The main ingredient of the proposed approach is to investigate compatibility of the determining equations for arbitrary elements, which follows from the invariance criteria to be met by nontrivial symmetries. Group classification is carried out completely in some important particular cases of the class of equations under consideration, for example, when the potential vanishes or has the form as in the case of harmonic oscillator. As a result, we recover the already known invariant nonlinear Schroedinger equations and, most importantly, construct a number of new ones.