  Gamma-ray bursts (GRBs) are presumed to be powered by still unknown central
engines for the timescales in the range $1ms \sim$ a few s. We propose that the
GRB central engines would be a viable site for strong meson synchrotron
emission if they were the compact astrophysical objects such as neutron stars
or rotating black holes with extremely strong magnetic fields $H \sim10^{12} -
10^{17}G$ and if protons or heavy nuclei were accelerated to ultra-relativistic
energies of order $\sim 10^{12}-10^{22}eV$. We show that the charged scalar
mesons like $\pi^{\pm}$ and heavy vector mesons like $\rho$, which have several
decay modes onto $\pi^{\pm}$, could be emitted with high intensity a thousand
times larger than photons through strong couplings to ultra-relativistic
nucleons. These meson synchrotron emission processes eventually produce a burst
of very high-energy cosmic neutrinos with $10^{12} eV \leq E_{\nu}$. These
neutrinos are to be detected during the early time duration of short GRBs.
