Exact solutions for the quintic nonlinear Schrödinger equation with time and space

Using similarity transformations, the analytical solutions to the quintic nonlinear Schrödinger equation with potentials and nonlinearities depending both on time and space are constructed. The stability of resonance solitons is examined by direct numerical simulation. It is found that stable fundamental soliton states with (Formula presented.) and low-order soliton states with (Formula presented.) can be supported both by self-focusing and self-defocusing materials. Higher-order solitons are found unstable, however, displaying quasi-stable propagation over prolonged distances.