Schrödinger--Newton equation with spontaneous wave function collapse

Based on the assumption that the standard Schrödinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals has survived from the nineteen-eighties. The Schrödinger–Newton equation (1984) provides well-localized solitons for free macro-objects but lacks the mechanism how extended wave functions collapse on solitons. The gravity-related stochastic Schrödinger equation (1989) provides the spontaneous collapse but the resulting solitons undergo a tiny diffusion leading to an inconvenient steady increase of the kinetic energy. We propose the stochastic Schrödinger–Newton equation which contains the above two gravity-related modifications together. Then the wave functions of free macroscopic bodies will gradually and stochastically collapse to solitons which perform inertial motion without the momentum diffusion: conservation of momentum and energy is restored.