Theory of nonmonotonic double layers

A simple graphic method of solving the Vlasov–Poisson system associated with nonlinear eigenvalue conditions for arbitrary potential structures is presented. A general analytic formulation for nonmonotonic double layers is presented and illustrated with some particular closed form solutions. This class of double layers satisfies the time stationary Vlasov–Poisson system while requiring a Sagdeev potential, which is a double-valued function of the physical potential. It follows that any distribution function having a density representation as any integer or noninteger power series of potential can never satisfy the nonmonotonic double-layer boundary conditions. A Korteweg–de Vries-like equation is found showing a relationship among the speed of the nonmonotonic double layer, its scale length, and its degree of asymmetry.