  We study a system of interacting triplons (the elementary excitations of a
valence-bond solid) described by an effective interacting boson model derived
within the bond-operator formalism. In particular, we consider the square
lattice spin-1/2 $J_1$-$J_2$ antiferromagnetic Heisenberg model, focus on the
intermediate parameter region, where a quantum paramagnetic phase sets in, and
consider the columnar valence-bond solid phase. Within the bond-operator
theory, the Heisenberg model is mapped into an effective boson model in terms
of triplet operators $t$. The effective boson model is studied at the harmonic
approximation and the energy of the triplons and the expansion of the triplon
operators $b$ in terms of the triplet operators $t$ are determined. Such an
expansion allows us to performed a second mapping, and therefore, determine an
effective interacting boson model in terms of the triplon operators $b$. We
then consider systems with a fixed number of triplons and determined the
ground-state energy and the spectrum of elementary excitations within a
mean-field approximation. We show that many-triplon states are stable, the
lowest-energy ones are constituted by a small number of triplons, and the
excitation gaps are finite. For $J_2=0.48 J_1$ and $J_2=0.52 J_1$, we also
calculate spin-spin and dimer-dimer correlation functions, dimer order
parameters, and the bipartite von Neumann entanglement entropy within our
mean-field formalism in order to determine the properties of the many-triplon
state as a function of the triplon number. We find that the spin and the dimer
correlations decay exponentially and that the entanglement entropy obeys an
area law, regardless the triplon number. Moreover, only for $J_2=0.48 J_1$, the
spin correlations indicate that the many-triplon states with large triplon
number might display a more homogeneous singlet pattern than the columnar
valence-bond solid.
