  We develop a theoretical framework that predicts and fully characterizes the
diverse experimental observations of the nonlinear, combustion wave propagation
in a rotating detonation engine (RDE), including the nucleation and formation
of combustion pulses, the soliton-like interactions between these combustion
fronts, and the fundamental, underlying Hopf bifurcation to time-periodic
modulation of the waves. In this framework, the mode-locked structures are
classified as autosolitons, or stably-propagating nonlinear waves where the
local physics of nonlinearity, dispersion, gain, and dissipation exactly
balance. We find that the global dominant balance physics in the RDE combustion
chamber are dissipative and multi-scale in nature, with local fast scale (nano-
to microseconds) combustion balances generating the fundamental mode-locked
autosoliton state, while slow scale (milliseconds) gain-loss balances determine
the instabilities and structure of the total number of autosolitons. In this
manner, the global multi-scale balance physics give rise to the stable
structures - not exclusively the frontal dynamics prescribed by classical
detonation theory. Experimental observations and numerical models of the RDE
combustion chamber are in strong qualitative agreement with no parameter
tuning. Moreover, numerical continuation (computational bifurcation tracking)
of the RDE analog system establishes that a Hopf bifurcation of the steadily
propagating pulse train leads to the fundamental instability of the RDE, or
time-periodic modulation of the waves. Along branches of Hopf orbits in
parameter space exist a continuum of wave-pair interactions that exhibit
solitonic interactions of varying strength.
