  Moving self-localized vibration of large size (vibrational soliton) in
monatomic chain with cubic and quartic anharmonicities is considered. Two types
of motion of such vibrations are found: the subsonic and the supersonic ones.
First type of motion is possible for chains with hard quartic anharmonicity
supposing that cubic anharmonicity is weak. This motion was earlier observed in
numerical simulations. For the second type of motion the cubic anharmonicity
plays the decisive role: the moving mode is stabilized by co-moving deformation
of the lattice caused by this anharmonicity. The quartic anharmonicity in this
case may be absent or even soft.
