This work investigates the collective motion of kaons in heavy ion reactions at SIS energies (about 1--2 GeV/nucleon). A radial collective flow of ${K}^{+}$ mesons is predicted to exist in central Au + Au collisions, which manifests in a characteristic ``shoulder-arm'' shape of the transverse mass spectrum of the midrapidity ${K}^{+}$ mesons. The ${K}^{+}$ radial flow arises from the repulsive ${K}^{+}$ mean field in nuclear matter. In spite of a strong reabsorption and rescattering the attractive ${K}^{\ensuremath{-}}$ mean field leads as well to a collective radial flow of ${K}^{\ensuremath{-}}$ mesons. The ${K}^{\ensuremath{-}}$ radial flow, however, is different from that of ${K}^{+}$ mesons and can be observed by a characteristic ``concave'' structure of the transverse mass spectrum of the ${K}^{\ensuremath{-}}$ mesons emitted at midrapidity. The kaon radial flows can therefore serve as a novel tool for the investigation of kaon properties in dense nuclear matter.