Effect of inner Lindblad resonance on spiral density waves propagation in disc galaxies: reflection over absorption

Interaction of spiral density waves with stars in the vicinity of the inner Lindblad resonance in galactic discs is investigated using the linear perturbation theory and the leading orders in the epicyclic and WKB approximations. In analogy with shear flows in hydrodynamics, we conjecture that a weak nonlinearity in a narrow resonance region modifies the standard (Landau-Lin) bypass rule of the singularity to the integration in the principal value sense. This indeed leads to the reflection of the spiral wave instead of absorption, but the detailed picture looks awkward: the intervals of the wave weakening alternate with the intervals of the wave growth, so that the net absorption is absent. Incidentally, we rectify the result concerning leading spiral waves obtained earlier for the standard bypass rule.