First Observation of Dispersive Shock Waves in an Electron Beam

Abstract

Dispersive shock waves (DSWs) are expanding nonlinear wave trains that arise when dispersion regularizes a steepening front, a phenomenon observed in fluids, plasmas, optics, and superfluids. Here we report the first experimental observation of DSWs in an intense electron beam, using the University of Maryland Electron Ring (UMER). A localized induction-cell perturbation produced a negative density pulse that evolved into a leading soliton-like peak followed by an expanding train of oscillations. The leading peak satisfied soliton scaling laws for width^2 vs inverse amplitude and velocity vs amplitude, while the total wave-train width increased linearly with time, consistent with Korteweg--de Vries (KdV) predictions. Successive peaks showed decreasing amplitude and velocity toward the trailing edge, in agreement with dispersive shock ordering. These results demonstrate that intense charged particle beams provide a new laboratory platform for studying dispersive hydrodynamics, extending nonlinear wave physics into the high-intensity beam regime.

Figures (4)

• Figure 1: Numerical KdV solution for a negative initial perturbation on a periodic domain $x \in [-\pi, \pi]$, computed using a pseudo-spectral spatial method and a fourth-order Runge--Kutta time stepper. Profiles are shown in $100\ \mu\text{s}$ increments, increasing upward in time.
• Figure 2: Simulated beam current evolution using WARP for turns 5--14, stacked with 0.0075 A upward offsets. Early turns are omitted while the beam relaxes in phase space; the DSW forms once the initial negative density perturbation steepens.
• Figure 3: Measured current perturbation (A, background-subtracted) vs time (s) at successive revolutions in UMER, showing the formation of a dispersive shock wave from an initial negative density perturbation. The initial beam current was approximately 30 mA. Turn number increases from bottom to top. The width of the DSW at each turn was measured between the red dots, with the blue dotted lines fitted to these points to illustrate the linear expansion of the DSW.
• Figure 4: Three-panel figure illustrating key features of the dispersive shock wave data: a) Squared width of the leading peak vs inverse amplitude, showing linear scaling consistent with soliton behavior. Reduced $\chi^2$ = 0.51. b) Velocity of the leading peak vs amplitude, showing linear scaling consistent with soliton behavior. Reduced $\chi^2$ = 1.16. c) DSW width versus turn number, showing linear expansion. Width as measured between blue lines in \ref{['data']}. Width of DSW expands linearly, as predicted with a reduced $\chi^2$ of 0.297.