Traveling wave solutions of a hybrid KdV-Burgers equation with arbitrary real coefficients in relation with beam-permeated multi-ion plasma fluids

Abstract

The Korteweg de Vries-Burgers (KdV-B) (1+1) equation [Formula: see text]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 ϕ, 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.

Figure 1

[Sub-case I(a)] (a) A “standard” polarity shock wave (anti-kink) and (b) the associated electric field (E) are depicted in the plane, for (arbitrarily chosen) fixed values . Propagation towards the right (i.e. in the positive direction of the axis) is assumed.

Figure 2

[Sub-case I(b)] (a) An “inverse” polarity shock wave (kink-soliton like profile) and (b) the associated electric field (E) are depicted in the plane, for arbitrary fixed values and . Propagation towards the right (i.e. in the positive direction of the axis) is assumed.

Figure 3

[Sub-case I(c)] (a) A kink-shaped shock excitation (for ) and (b) its corresponding electric field (E) are depicted in the plane. Upon a space reversal (), the left panels (a,b) give their place to the right panels (c, d). Arbitrary fixed values and have been considered.

Figure 4

[Sub-case I(d)] (a) An anti-kink and (b) its corresponding electric field (E) are depicted in the plane. Upon a space reversal(), the left panels (a, b) take the form of the right panels (c, d). Arbitrary fixed values and have been considered.

Figure 5

[Sub-case II(a)] (a) A shock wave (kink soliton) for and (b) the corresponding electric field (E) in - plane for fixed ,.

Figure 6

[Sub-case II(b)] (a) A shock wave (anti-kink soliton) and (b) the corresponding electric field (E) are depicted in the plane. We have considered the arbitrary fixed values and , in these plots.

Figure 7

[Sub-case II(c)] Traveling waves: (a) a shock wave (antikink), for , and the corresponding (negative pulse) for the electric field E, are depicted in the plane. Upon a space reversal (), the left panels (a, b) (for left-ward propagation) cede their place to the right panels (c, d), where propagation takes place in the forward direction, i.e. to the right of the axis. We have taken and , in these plots.

Figure 8

3D profiles of (a) shock wave (anti-kink) and corresponding electric field (E) in - plane; the space reversal () for panels (a, b) is showed in (c, d) for fixed .