The Korteweg de Vries-Burgers (KdV-B) (1+1) equation
\(\begin{aligned} \frac{\partial \psi }{\partial t} +a \, \psi \frac{\partial \psi }{\partial x} + b \, \frac{\partial ^{3}\psi }{\partial x^{3}} = c \, \frac{\partial ^{2} \psi }{\partial x^2} \,, \end{aligned}\) incorporating constant (real) coefficients representing nonlinearity (a), dispersion (b) and dissipation (c), is a long known paradigm in e.g. plasma physics, where it can be derived from plasma fluid-dynamical models, so that all coefficients depend parametrically on the plasma composition. For a positive dispersion coefficient b (value), which is the general case in beam-free electron-ion plasma, this PDE possesses analytical solutions representing “shock”-shaped traveling waves with a characteristic kink (or anti-kink) soliton-like profile, for negative (or positive, respectively) values of the nonlinearity coefficient (a). In a plasma context, these excitations represent a monotonic transition between two (different) asymptotic values of the electrostatic potential \(\phi\) , associated with a monopolar (i.e. bell-shaped) disturbance of the electric field (E). Contrary to widespread belief (based on a beam-free plasma description), an investigation of nonlinear electrostatic waves in beam-permeated plasmas reveals that the sign(s) of all (any) of the coefficients (a, b or c) may be reversed, independently from each other, depending on the beam velocity (value). In the light of this result, the analytical solutions have been reexamined in an effort to elucidate their applicability in plasma-physical scenarios (e.g., reconnection jets and other planetary plasma environments) in terms of the combined sign(s) of the various coefficients involved in the KdV-B equation. Different types of excitations are demonstrated to exist and the influence of the various coefficients on the solution’s propagation characteristics is examined.
