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We calculate the flux of non-thermal radiations from the supernova remnant (SNR) RX J1713.7−3946 in the context of the non-linear theory of particle acceleration at shocks, which allows us to take into account self-consistently the dynamical reaction of the accelerated particles, the generation of magnetic fields in the shock proximity and the dynamical reaction of the magnetic field on the plasma. When the fraction of particles which get accelerated is of the order of ∼10−4, we find that the strength of the magnetic field obtained as a result of streaming instability induced by cosmic rays is compatible with the interpretation of the X-ray emitting filaments being produced by strong synchrotron losses in G magnetic fields. The maximum energy of accelerated protons is GeV. If the X-ray filaments are explained in alternative ways, the constraint on the magnetic field downstream of the shock disappears and the HESS data can be marginally fitted with ICS of relativistic electrons off a complex population of photons, tailored to comprise cosmic microwave background and ambient infrared/optical photons. The fit, typically poor at the highest energies, requires a large density of target photons within the remnant; only a fraction of the order of ∼10−6 of the background particles gets accelerated; the local magnetic field is of the order of G and the maximum energy of protons is much lower than the knee energy. Current HESS gamma-ray observations combined with recent X-ray observations by Suzaku do not allow as yet to draw a definitive conclusion on whether RX J1713.7−3946 is an efficient cosmic ray accelerator, although at the present time a hadronic interpretation of HESS data seems more likely. We discuss the implications of our results for the GLAST gamma-ray telescope, which should be able to discriminate the two scenarios discussed above, by observing the shape of the gamma-ray spectrum at lower energies. [ABSTRACT FROM AUTHOR]