Compactification of Nonlinear Patterns and Waves

We present a nonlinear mechanism(s) which may be an alternative to a missing wave speed: it induces patterns with a compact support and sharp fronts which propagate with a finite speed. Though such mechanism may emerge in a variety of physical contexts, its mathematical characterization is universal, very simple, and given via a sublinear substrate (site) force. Its utility is shown studying a Klein-Gordon -utt+[Φ'(ux)]x=P'(u) equation, where Φ'(σ)=σ+βσ3 and endowed with a subquadratic site potential P(u)∼|1-u2|α+1, 0≤α<1, and the Schroedinger iZt+∇2Z=G(|Z|)Z equation in a plane with G(A)=γA-δ-σA2, 0<δ≤1