Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations.

Stability of the soliton solutions to the fourth-order nonlinear Schr\"odinger equations with arbitrary power nonlinearities in different space dimensions is studied. Necessary and sufficient conditions of the stability with respect to small perturbations are found. The results obtained represent also necessary conditions of the stabilization of self-focusing and collapse by high-order dispersion.