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Small amplitude limit of solitary waves for the Euler-Poisson system. (English) Zbl 1406.35256

Summary: 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
35Q31 Euler equations
76B25 Solitary waves for incompressible inviscid fluids
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35C08 Soliton solutions
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References:

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