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 Schr\"odinger iZₜ+∇²Z=G(|Z|)Z equation in a plane with G(A)=γA^-δ-σA², 0<δ≤1.