Momentum Flux for Small-Amplitude Oscillons

An oscillon in scalar field theory in $(1+1)$-dimensions is an oscillating, localised, almost periodic classical solution that slowly radiates and decays to the vacuum solution. We calculate its momentum flux integrated over a period, which would be zero for an exact breather. For an oscillon of small amplitude $\text{eps}$, truncated at order ${\text{eps}}^N$, we show that the integrated momentum flux is of leading order ${\text{eps}}^{N+4}$ for even $N$ and ${\text{eps}}^{N+5}$ for odd $N$, except for $N=1$, where it is of order ${\text{eps}}^4$. It is therefore very small.