  We evaluate differential distributions for the four-body $p p \to n n \pi^+
\pi^+$ reaction. The amplitude for the process is calculated in the Regge
approach including many diagrams. We make predictions for possible future
experiments at RHIC and LHC energies. Very large cross sections are found which
is partially due to interference of a few mechanisms. Presence of several
interfering mechanisms precludes extraction of the elastic $\pi^+ \pi^+$
scattering cross section. Absorption effects are estimated. Differential
distributions in pseudorapidity, rapidity, invariant two-pion mass,
transverse-momentum and energy distributions of neutrons are presented for
proton-proton collisions at $\sqrt{s}$ = 500 GeV (RHIC) and $\sqrt{s}$ = 0.9,
2.36 and 7 TeV (LHC). Cross sections with experimental cuts are presented.
