Formation of Double Layers: Shocklike Solutions of an mKdV-Equation

Small amplitude double layers (DLs) in a plasma with a suitable electron distribution may be identifed with shocklike solutions of a modified Korteweg-deVries (mKdV) equation. A thought experiment for the formation of such DLs is specified to clarify the physical constraints and to demonstrate the emergence of a DL from an initial disturbance. A scattering formulation of the mKdV initial value problem may be diagonalized to give a pair of Schrödinger equations with a scattering potential satisfying the ordinary KdV equation. The initial value problem can then be treated using Khruslov's generalisation of the inverse scattering method which allows a difference in the asymptotic values of the potential. A necessary and sufficient condition for the emergence of a shocklike soliton (DL) is found. The existence or otherwise of a soliton (wave) train and of a finite number of isolated solitons may also be determined from the scattering properties of the initial potential.