Propagation of nonlinear perturbations in a quasineutral collisionless plasma

The Russo–Smereka kinetic equation: Conservation laws, reductions and numerical solutions

“…In the theory of quasineutral collisionless plasma flows the following result is known [10,25]: during the evolution the kinetic roll-over of the distribution function is possible (the formation of two peaks of the distribution function which originally had a single peak). Let us establish a similar property for the considered kinetic model for a bubbly flow.…”

“…1). [7,10], in the plane (Z 1 , Z 2 ) we construct a closed contour C consisting of the contours C + and C − . The contour C + is given parametrically by the equations…”

Classical mechanical systems with one-and-a-half degrees of freedom and Vlasov kinetic equation

“…. , (5) where Φ(F ) is an appropriate rapidly decreasing at infinities p → ±∞ function such that the integrals are finite. Certainly we can obtain the coefficients A k (x, t) in (4) as residues at infinity A k (x, t) = 1 2πi p k F (x, t, p) dp or choose another deformation of the contour in this integral.…”

“…Certainly we can obtain the coefficients A k (x, t) in (4) as residues at infinity A k (x, t) = 1 2πi p k F (x, t, p) dp or choose another deformation of the contour in this integral. See Appendix A for more detail on possible relations of the coefficients in the asymptotic expansion (4) and the integrals (5).…”

“…However, equation (3) is obviously invariant w.r.t. any point transformations F (x, t, p) → f (F (x, t, p)) with arbitrary function f of one variable, while all moments (5) are preserved after the appropriate change of Φ(F ). In all the cases treated below one can easily find the modified f (F (x, t, p)) with the required asymptotic expansion (4).…”