Schrödinger Equation with Convolution Nonlinearity on Lie Groups and Commutative Homogeneous Spaces

The Schrödinger equation with nonlocal nonlinearity of convolution type on Lie groups and commutative homogeneous spaces is considered. It is shown that in the special case of an abelian group the Schrödinger equation admits a solution in the form of a superposition of non-interacting solitons. In the case of a commutative homogeneous space, a noncommutative reduction of the Schrödinger equation is carried out. A general solution in the particular case when the nonlinearity factorizes in the spatial variables is found.