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Nonlinear waves in strongly interacting relativistic fluids

During the past decades the study of strongly interacting fluids experienced a tremendous progress. In the relativistic heavy ion accelerators, specially the RHIC and LHC colliders, it became possible to study not only fluids made of hadronic matter but also fluids of quarks and gluons. Part of the physics program of these machines is the observation of waves in this strongly interacting medium. From the theoretical point of view, these waves are often treated with li-nearized hydrodynamics. In this text we review the attempts to go beyond linearization. We show how to use the Reductive Perturbation Method to expand the equations of (ideal and viscous) relativistic hydrodynamics to obtain nonlinear wave equations. These nonlinear wave equations govern the evolution of energy density perturbations (in hot quark gluon plasma) or baryon density perturbations (in cold quark gluon plasma and nuclear matter). Different nonlinear wave equations, such as the breaking wave, Korteweg-de Vries and Burgers equations, are obtained from different equations of state (EOS). In nuclear matter, the Walecka EOS may lead to a KdV equation. We explore equations of state such as those extracted from the MIT Bag Model and from QCD in the mean field theory approach. Some of these equations are integrable and have analytical solitonic solutions. We derive these equations also in spherical and cylindrical coordinates. We extend the analysis to two and three dimensions to obtain the Kadomtsev-Petviashvili (KP) equation, which is the generalization of the KdV. The KP is also integrable and presents analytical solitonic solutions. In viscous relativistic hydrodynamics we have second order patial derivatives which physically represent dissipation terms. We present numerical solutions and their corresponding algorithms for the cases where the equations are not integrable.