Transverse collective instabilities induced by the beam-coupling impedance of the accelerator structure lead to beam quality degradation and pose a major limitation to the machine performance. Landau damping, a powerful stabilising mechanism that can be employed against various types of instabilities, is present in the transverse planes when there is a betatron frequency spread among the beam particles. Traditional approaches use octupole magnets to introduce betatron detuning with the transverse particle oscillation amplitudes. Their damping efficiency depends on the transverse geometric beam emittances which decrease with increasing beam energy and brightness. For the Future Circular Collider (FCC) they may hence no longer be the most suitable instability mitigation tool. Within the framework of this PhD thesis a novel approach to Landau damping is studied from the theoretical, numerical, and experimental points-of-view. The novelty of the method is to introduce the betatron frequency spread through detuning with the longitudinal instead of the transverse amplitudes. This is motivated by the fact that in typical high-energy proton machines the longitudinal emittance is several orders of magnitude larger compared to the transverse ones. Two equivalent detuning schemes are considered: a radio-frequency (rf) quadrupole cavity and nonlinear chromaticity. The first achievement of the project is the development of the Vlasov theory for nonlinear chromaticity to provide the analytical foundation for the novel Landau damping technique. The formalism is validated successfully against the circulant matrix model and the PyHEADTAIL tracking code. Based on the new theory, two beam dynamics effects introduced by detuning with longitudinal amplitude are identified: Landau damping and a change of the effective impedance altering the head-tail instability formation mechanism. Second, the first numerical proof-of-concept of an rf quadrupole for Landau damping is realised in PyHEADTAIL. A two-family scheme for rf quadrupoles is also evaluated for FCC operational scenarios demonstrating an improved overall damping performance of the device. In particular, the required active magnetic length of the rf quadrupole is significantly shorter compared to octupole elements. Third, the numerical models and the theory are validated against measurements in the Large Hadron Collider (LHC) and the Super Proton Synchrotron (SPS) at CERN. In the two machines, the second-order chromaticity is successfully enhanced using a sextupole and an octupole scheme respectively and the measured nonlinear optics parameters are shown to be consistent with MAD-X calculations. The stabilisation of single bunches by means of a betatron frequency spread produced by nonlinear chromaticity is demonstrated in the LHC which marks the first experimental proof of the novel Landau damping method. The measurements are in good agreement with detailed PyHEADTAIL simulations. In particular, the two effects predicted by the theory are consistently observed in both experiments and simulations confirming a thorough understanding of the involved beam dynamics.