Phonons mimicking doubly special relativity kinematics

Collective excitations (phonons) in barotropic, irrotational, inviscid fluids exhibit an effective Lorentz invariance, where the sound speed plays the role of the invariant speed of light in special relativity. By carefully selecting the interaction potentials, we explicitly construct two hydrodynamic models in which phonons obey doubly special relativistic kinematics, with the analogue Planck scale emerging from non-Newtonian behavior at high energies. Specifically, we demonstrate that elastic storage leads to an approximate realization of Amelino-Camelia's scenario, while the Magueijo-Smolin model naturally emerges in the presence of elastic restoring forces.