Phase Space Hydrodynamics of Equivalent Nonlinear Systems: Experimental and Computational Observations

The one‐dimensional Vlasov equation describes the behavior of an incompressible self‐interacting classical fluid which moves in the $\mathrm{(q,\; p)}$ phase plane. This type of phase fluid occurs in many physical problems and its hydrodynamic properties can be examined from a general point of view. A characteristic feature with initially unstable spatially homogeneous configurations is the development of stable nonlinear phase structures. Such examples occur as the result of the gravitational Jeans instability, or the two‐stream and negative‐mass instabilities of charged‐particle beams. These structures can be related to one another by extending a duality principle due to Dory. The stable cavities in phase space which have been observed in numerical calculations on the two‐stream instability are compared with stable proton clusters which develop from the negative‐mass instability in the mirror experiment DCX‐1.