The coupled equations: wave + self-gravity

Closing the loop with Poisson

The wave moves in the gravitational potential $\Phi$; but $\Phi$ is generated by the wave's own mass density $\rho=m|\psi|^2$ through Poisson's equation. Together:

This is the Schrödinger–Poisson (SP) system. The nonlinearity — $\psi$ sources $\Phi$, which acts back on $\psi$ — is what makes it produce self-bound solitons and rich structure rather than trivial spreading.

Conserved quantities

SP conserves total mass $M=\int m|\psi|^2\,d^3x$ (from the continuity equation), total energy (kinetic + gravitational + quantum), and momentum and angular momentum. These are not decoration — they are the primary numerical diagnostics: a simulation that fails to conserve mass or energy is not to be trusted.

Solitons as stationary solutions

Stationary states $\psi=e^{-i\gamma t/\hbar}\phi(r)$ of SP are the solitons (figure) — the self-gravitating ground states where quantum pressure balances gravity (Topic 5). Every FDM halo settles a soliton into its centre; the surrounding halo is the time-dependent, interfering part of the same $\psi$.

Why it must be solved numerically

SP has no general closed-form solution — the nonlinear gravitational coupling defeats analytic methods except for the stationary soliton and linear perturbations. Structure formation therefore demands numerical evolution.