Soliton collisions in the ion acoustic plasma equations. (English) Zbl 0934.35148

Li, Yi; Sattinger, D. H.

J. Math. Fluid Mech. 1, No. 1, 117-130 (1999).

The paper treats a standard system of equations governing the ion-acoustic waves in plasmas, \[ n_t + (nv)_x = 0,\quad v_t + \left(\tfrac 12 v^2 + \varphi\right)_x = 0,\quad \varphi_{xx}-\exp(\varphi) +n = 0, \] where \(\varphi\), \(n\), and \(v\) are the electrostatic potential, ion density, and ion velocity, respectively. This system has a family of soliton solutions parametrized by their velocity. In the paper, collisions of solitons with different velocities are simulated numerically, and the result is compared with exact two-soliton solutions to the Korteweg - de Vries (KdV) equation, to which the system of the equations for the ion-acoustic equations is equivalent in a certain asymptotic limit. On the basis of the simulations, it is concluded that, despite the nonintegrability of the system (on the contrary to the KdV equation), the collisions are almost perfectly elastic. The only inelastic effect revealed by the simulations is the generation of a very small radiative field component, with an amplitude four orders of magnitude smaller than the amplitudes of the colliding solitons. Moreover, the whole numerically found solution for the two-soliton collision is compared to a similar exact solution to the KdV equation. It is concluded that the KdV solutions provide a fairly accurate approximation for the collisions.

Reviewer: B.A.Malomed (Tel Aviv)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
76X05	Ionized gas flow in electromagnetic fields; plasmic flow

Keywords:

Korteweg-de Vries equation; integrability; ion-acoustic waves in plasmas; collisions of solitons; elastic; inelastic