Nonlinearity shapes lattice dynamics affecting vibrational spectrum,transport and thermalization phenomena. Beside breathers and solitons one findsthe third fundamental class of nonlinear modes -- $q$-breathers -- periodicorbits in nonlinear lattices, exponentially localized in the reciprocal modespace. To date, the studies of $q$-breathers have been confined to the cubicand quartic nonlinearity in the interaction potential. In this paper we studythe case of arbitrary nonlinearity index $\gamma$ in an acoustic chain. Weuncover qualitative difference in the scaling of delocalization and stabilitythresholds of $q$-breathers with the system size: there exists a critical index$\gamma^*=6$, below which both thresholds (in nonlinearity strength) tend tozero, and diverge when above. We also demonstrate that this critical indexvalue is decisive for the presence or absense of thermalization. For a genericinteraction potential the mode space localized dynamics is determined only bythe three lowest order nonlinear terms in the power series expansion.
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