  We consider splitting and stabilization of second-order solitons (2-soliton
breathers) in a model based on the nonlinear Schr\"{o}dinger equation (NLSE),
which includes a small quintic term, and weak resonant nonlinearity management
(NLM), i.e., time-periodic modulation of the cubic coefficient, at the
frequency close to that of shape oscillations of the 2-soliton. The model
applies to the light propagation in media with cubic-quintic optical
nonlinearities and periodic alternation of linear loss and gain, and to BEC,
with the self-focusing quintic term accounting for the weak deviation of the
dynamics from one-dimensionality, while the NLM can be induced by means of the
Feshbach resonance. We propose an explanation to the effect of the resonant
splitting of the 2-soliton under the action of the NLM. Then, using systematic
simulations and an analytical approach, we conclude that the weak quintic
nonlinearity with the self-focusing sign stabilizes the 2-soliton, while the
self-defocusing quintic nonlinearity accelerates its splitting. It is also
shown that the quintic term with the self-defocusing/focusing sign makes the
resonant response of the 2-soliton to the NLM essentially broader, in terms of
the frequency.
