  A non-stationary one-dimensional cavity can be described by the
time-dependent and multi-mode effective Hamiltonian of the so-called dynamical
Casimir effect. Due to the non-adiabatic boundary conditions imposed in one of
the cavity mirrors, this effect predicts the generation of real photons out of
vacuum fluctuations of the electromagnetic field. Such photon generation
strongly depends on the number of modes in the cavity and their intermode
couplings. Here, by using an algebraic approach, we show that for any set of
functions parameterizing the effective Hamiltonian, the corresponding
time-dependent Schr\"odinger equation admits an exact solution when the cavity
has one intermode interaction. With the exact time evolution operator, written
as a product of eleven exponentials, we obtain the average photon number in
each mode, a few relevant observables and some statistical properties for the
evolved vacuum state.
