  Intensive studies in the past decades have suggested that the heat
conductivity $\kappa$ diverges with the system size $L$ as $\kappa\sim
L^{\alpha}$ in one dimensional momentum conserving nonlinear lattices and the
value of $\alpha$ is universal. But in the Fermi-Pasta-Ulam-$\beta$ lattices
with next-nearest-neighbor interactions we find that $\alpha$ strongly depends
on $\gamma$, the ratio of the next-nearest-neighbor coupling to the
nearest-neighbor coupling. We relate the $\gamma$-dependent heat conduction to
the interactions between the long-wavelength phonons and the randomly
distributed discrete breathers. Our results provide an evidence to show that
the nonlinear excitations affect the heat transport.
