Compactlike discrete breathers in systems with nonlinear and nonlocal dispersive terms

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Discrete breathers with purely anharmonic short-range interaction potentials localize super-exponentially becoming compact-like. We analyze their spatial localization properties and their dynamical stability. Several branches of solutions are identified. One of them connects to the well-known Page and Sievers-Takeno lattice modes, another one connects with the compacton solutions of Rosenau. The absence of linear dispersion allows for extremely long-lived time-quasiperiodic localized excitations. Adding long-range anharmonic interactions leads to an extreme case of competition between length scales defining the spatial breather localization. We show that short- and long-range interaction terms competition results in the appearance of several characteristic cross-over lengths and essentially breaks the concept of compactness of the corresponding discrete breathers.