Superposition of small-amplitude nonlinear waves in collisionless plasmas

Single Bhatanagar-Gross-Krook (BGK) waves are exact undamped traveling wave solutions of the Vlasov-Poisson-Ampere equations. While recent studies demonstrate the existence of spatially periodic BGK waves near most physically relevant Vlasov equilibria, straightforward analysis suggests that, due to the unusual nonlinearity of the Vlasov-Poisson system, superposition does not yield a new solution even in the limit of small wave amplitude. We have shown, however, via investigation of the nonintegrable dynamics of charged particles in such superimposed fields, that this conclusion is erroneous: small-amplitude periodic BGK waves of sufficiently different velocity do satisfy a principle of superposition. This analytical result explains recent numerical calculations, which suggest that a large-amplitude electrostatic plasma wave suffering nonlinear Landau damping approaches a state well described by a superposition of two BGK waves.