Small amplitude limit of solitary waves for the Euler–Poisson system

Abstract

The one-dimensional Euler-Poisson system arises in the study of phenomena of plasma such as plasma solitons, plasma sheaths, and double layers. When the system is rescaled by the Gardner-Morikawa transformation, the rescaled system is known to be formally approximated by the Korteweg-de Vries (KdV) equation. In light of this, we show existence of solitary wave solutions of the Euler-Poisson system in the stretched moving frame given by the transformation, and prove that they converge to the solitary wave solution of the associated KdV equation as the small amplitude parameter tends to zero. Our results assert that the formal expansion for the rescaled system is mathematically valid in the presence of solitary waves and justify Sagdeev's formal approximation for the solitary wave solutions of the pressureless Euler-Poisson system. Our work extends to the isothermal case.

Linear Stability of Solitary Waves for the Isothermal Euler–Poisson System

“…where ε K > 0 is a critical value (see (2.6)) and √ 1 + K is called the ion sound speed in the context of plasma physics. 2 Furthermore, the authors of this paper show in [3] that (n c , u c , φ c ) converges to the rescaled solitary wave solution of the associated KdV equation as the amplitude parameter ε > 0 tends to zero. More specifically, in the Gardner-Morikawa scaling (also called as the KdV scaling)…”

“…Our main result follows from Proposition 1.2 and the Gearhart-Prüss stability theorem [44]. 3 Let P 0 be the spectral projection associated with the isolated eigenvalue λ = 0. Theorem 1.3.…”

“…The existence of the solitary wave solutions is shown in [8] (see also [3]) through a phase plane analysis of the reduced system (2.3). To be more precise, we define a small amplitude parameter ε > 0 by…”

“…x ∂ c Z c ds decays exponentially to zero as x → −∞, but it is merely bounded as x → +∞. Lastly, we briefly discuss the result of [3] which will be used throughout this paper.…”

“…Various analytical [3,21] and numerical [22,31] studies indicate that in certain physical regime, the KdV equation is a good approximation of the Euler-Poisson system (1.1). Moreover, as solutions of the KdV equation are dominated by their solitary waves [4,5,18,19,30,[34][35][36]42,52], this gives hope of a similar result for the Euler-Poisson system with more general initial data.…”