Reductions and exact solutions of nonlinear multidimensional Schrödinger equations

Using the canonical decomposition of an arbitrary subalgebra of the orthogonal algebra $AO(n)$, we describe the maximal subalgebras of rank $n$ and $n-1$ of the extended isochronous Galileo algebra, and also the maximal subalgebras of rank $n$ of the generalized extended classical Galileo algebra $A\widetilde G(1,n)$, the extended special Galileo algebra $A\widetilde G(2,n)$, and the extended complete Galileo algebra $A\widetilde G(3,n)$. Using the subalgebras of rank $n$, we construct ansatzes that reduce multidimensional Schrödinger equations to ordinary differential equations. Exact solutions of the Schrödinger equations are found from the solutions of the reduced equations.