Asymptotic state of nonlinear Landau damping in one-dimensional plasma

In this work, the asymptotic state of nonlinear Landau damping in one-dimensional plasma has been examined using a quasi-linear model and a second-order symplectic integrator. The dispersion relation of the plateau distribution function for the steady-state solution of the quasi-linear mode is extended to the complex plane and compared with the nonlinear simulation. We determine that the asymptotic state of the collisionless plasma is a multi-wave BGK structure. This structure is characterized by multiple vortices in phase space, which correspond to distinct peaks in the frequency-wavenumber (ω, k) spectrum of the electric field

💡 Research Summary

The paper investigates the long‑time asymptotic state of nonlinear Landau damping in a one‑dimensional, collisionless electron plasma. Building on quasi‑linear theory, the authors introduce a “plateau” distribution function that flattens the velocity distribution over a narrow interval around a chosen phase velocity. By analytically continuing this distribution into the complex velocity plane, they derive a full dispersion relation ε(p,k)=0 with p=ω_R+iω_I. Numerical evaluation of this relation reveals that when the phase velocity of the underlying Vlasov mode lies inside the plateau, the spectrum contains three distinct families of solutions: (i) a nearly undamped mode with ω_I≈0 located close to the real axis, and (ii) two pairs of weakly damped modes situated near the edges of the plateau. If the plateau does not encompass the Vlasov phase velocity, only the two weakly damped branches survive.

To test these predictions, the authors perform high‑resolution, long‑duration Vlasov‑Poisson simulations using a second‑order symplectic integrator with cubic‑spline interpolation. Two sets of initial conditions are examined. First, a plateau distribution is perturbed with an infinitesimal sinusoidal electric field (amplitude α=10⁻⁵, wavenumber k₁=0.4 λ_D⁻¹). The early evolution follows linear Landau damping, but after ≈100 ω_p⁻¹ the field amplitude rebounds and settles into a quasi‑periodic “wave packet” with a period of roughly 500 ω_p⁻¹. When the plateau’s central velocity matches the linear phase velocity, the packet is stable and its Fourier spectrum displays a central peak (the nearly undamped mode) flanked by symmetric sidebands (the weakly damped edge modes). If the plateau is displaced, the packet decays slowly, consistent with the weakly damped solutions of the dispersion relation.

Second, a standard Maxwellian distribution is perturbed with a finite‑amplitude sinusoid (β=0.15). The initial decay proceeds faster than linear theory (≈1.5× the Landau rate) due to nonlinear mode coupling. Trapped particles then amplify the field, and the system evolves toward a complex, multi‑wave BGK state. Phase‑space diagnostics reveal a large vortex (a broad plateau) near the original Langmuir phase velocity, together with a cascade of smaller vortices at low velocities. The electric‑field spectrum shows the primary Langmuir frequency, symmetric sidebands (reflecting the beating of the two dominant wave packets), and a series of low‑frequency peaks that correspond to the low‑velocity vortices.

Overall, the study demonstrates that the asymptotic state of nonlinear Landau damping is not a simple single‑wave BGK structure but a superposition of several weakly damped waves—a multi‑wave BGK configuration. The presence of a plateau in the distribution function gives rise to an additional nearly undamped mode, while the edges of the plateau generate weakly damped modes that manifest as sidebands in the electric‑field spectrum. These findings extend the traditional picture of a two‑wave counter‑propagating BGK mode, highlighting the importance of multi‑mode interactions and trapped‑particle dynamics in shaping the long‑term behavior of collisionless plasmas. The work also suggests that similar multi‑wave BGK states may arise in higher‑dimensional or magnetized settings, offering a pathway for future investigations into plasma turbulence and wave–particle interactions.