Near-equilibrium multiple-wave plasma states

We report results showing that spatially periodic Bernstein-Greene-Kruskal (BGK) waves, which are exact nonlinear traveling wave solutions of the Vlasov-Maxwell equations for collisionless plasmas, satisfy a nonlinear principle of superposition in the small-amplitude limit. For an electric potential consisting of N traveling waves, cphi(x,t)= ${\mathcal{J}}_{\mathit{i}=1\mathrm{}}^{\mathit{N}}$ ${\mathit{cphi}}^{\left(\mathit{i}\right)}$(x-${\nu }_{\mathit{i}}$t), where ${\nu }_{\mathit{i}}$ is the velocity of the ith wave and each wave amplitude ${\mathit{cphi}}^{\left(\mathit{i}\right)}$ is of order ε which is small, we first derive a set of quantities ${\mathit{scr}\mathit{Ē}}^{\left(\mathit{i}\right)}$(x,u,t) which are invariants through first order in ε for charged particle motion in this N-wave field. We then use these functions ${\mathit{scr}\mathit{Ē}}^{\left(\mathit{i}\right)}$(x,u,t) to construct smooth distribution functions for a multispecies plasma which satisfy the Vlasov equation through first order in ε uniformly over the entire x-u phase plane for all time. By integrating these distribution functions to obtain the charge and current densities, we also demonstrate that the Poisson and Ampère equations are satisfied to within errors that are O(${\mathrm{\epsilon }}^{3/2}$). Thus the constructed distribution functions and corresponding field describe a self-consistent superimposed N-wave solution that is accurate through first order in ε. The entire analysis explicates the notion of small-amplitude multiple-wave BGK states which, as recent numerical calculations suggest, is crucial in the proper description of the time-asymptotic state of a plasma in which a large-amplitude electrostatic wave undergoes nonlinear Landau damping.