The Klein-Gordon model (KG) $\square \varphi =P^{\prime }(|\varphi |)\frac{\varphi }{|\varphi |}$ is Lorenz invariant and has a finite wave speed, yet its localized modes, whether $Q$ balls or vortices, suffer from the same fundamental flaw as all other solitons—they extend indefinitely. Using the KG model as a case study, we demonstrate that appending the site potential, $P_a(|\varphi |)$, with a subquadratic part $P(|\varphi |)=b^2|\varphi {|}^{1+\delta }+P_a(|\varphi |)$, $0\le \delta <1$, induces particlelike modes with strictly compact support. These modes are robust and shorten in the direction of motion. Their interactions, which occur only on contact, are studied in two and three dimensions and are shown to span the whole range from being nearly elastic to plastic.

• Received 9 June 2009

DOI:https://doi.org/10.1103/PhysRevLett.104.034101