Nonlinear Travelling Wave Solutions Research Articles

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}^{\mathit{N}}$${\mathit{cphi}}^{(\mathit{i})}$(x-${\ensuremath{\nu}}_{\mathit{i}}$t), where ${\ensuremath{\nu}}_{\mathit{i}}$ is the velocity of the ith wave and each wave amplitude ${\mathit{cphi}}^{(\mathit{i})}$ is of order \ensuremath{\epsilon} which is small, we first derive a set of quantities ${\mathit{scr}\mathit{\ifmmode \bar{E}\else \={E}\fi{}}\phantom{\rule{0ex}{0ex}}}^{(\mathit{i})}$(x,u,t) which are invariants through first order in \ensuremath{\epsilon} for charged particle motion in this N-wave field. We then use these functions ${\mathit{scr}\mathit{\ifmmode \bar{E}\else \={E}\fi{}}}^{(\mathit{i})}$(x,u,t) to construct smooth distribution functions for a multispecies plasma which satisfy the Vlasov equation through first order in \ensuremath{\epsilon} 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 Amp\`ere equations are satisfied to within errors that are O(${\mathrm{\ensuremath{\epsilon}}}^{3/2}$). Thus the constructed distribution functions and corresponding field describe a self-consistent superimposed N-wave solution that is accurate through first order in \ensuremath{\epsilon}. 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.

Superposition of nonlinear plasma waves.

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. The analysis explicates the notion of superimposed BGK waves which, as recent numerical calculations suggest, is crucial in the proper description of the time-asymptotic state of a plasma when a large amplitude electrostatic wave undergoes nonlinear Landau damping.

Soliton solutions for free-electron-laser applications.

A new class of nonlinear traveling-wave solutions of the relativistic cold-fluid model are presented for a system consisting of a relativistic cold electron beam propagating through a helical wiggler magnetic field. The solutions, which are in the form of isolated soliton pulses of coupled electromagnetic and plasma waves, are obtained numerically. They represent possible nonlinear saturated states of the free-electron-laser instability and may also have useful applications in particle acceleration schemes.

Propagation of quasi-simple waves in a compressible rotating atmosphere

A class of self-propagating linear and nonlinear travelling wave solutions for compressible rotating fluid is studied using both numerical and analytical techiques. It is shown that, in general, a three dimensional linear wave is not periodic. However, for some range of wave numbers depending on rotation, horizontally propagating waves are periodic. When the rotation ω is equal to\(\sqrt {(\gamma - 1)/(4\gamma )}\), all horizontal waves are periodic. Here, γ is the ratio of specific heats. The analytical study is based on phase space analysis. It reveals that the quasi-simple waves are periodic only in some plane, even when the propagation is horizontal, in contrast to the case of non-rotating flows for which there is a single parameter family of periodic solutions provided the waves propagate horizontally. A classification of the singular points of the governing differential equations for quasi-simple waves is also appended.