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This paper studies a discrete one-dimensional monatomie Klein-Gordon chain with only quartic nearest-neighbour interactions, in which the compact-like discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. Introducing the trying method, it proves that compact-like discrete breathers exist in this nonlinear system. It also discusses the linear stability of the compact-like discrete breathers, when the coefficient (β)of quartic on-site potential and the coupling constant (K4) of quartic interactive potential satisfy the given conditions,they are linearly stable.