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.

We study the asymptotic linear stability of a two-parameter family of solitary waves for the isothermal Euler-Poisson system. When the linearized equations about the solitary waves are considered, the associated eigenvalue problem in L 2 space has a zero eigenvalue embedded in the neutral spectrum, i.e., there is no spectral gap. To resolve this issue, use is made of an exponentially weighted L 2 norm so that the essential spectrum is strictly shifted into the left-half plane, and this is closely related to the fact that solitary waves exist in the super-ion-sonic regime. Furthermore, in a certain long-wavelength scaling, we show that the Evans function for the Euler-Poisson system converges to that for the Korteweg-de Vries (KdV) equation as an amplitude parameter tends to zero, from which we deduce that the origin is the only eigenvalue on its natural domain with algebraic multiplicity two. We also show that the solitary waves are spectrally stable in L 2 space. Moreover, we discuss (in)stability of large amplitude solitary waves.

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We establish a reconstruction formula for the initial information of a density (pressure) from the time records of data measured on a sphere for a certain class of wave-type equations. We first derive a reconstruction formula of integral form, and by applying the theory of Fourier Bessel series for this, we also obtain a formula of discrete version, which is more robust and numerically advantageous in the presence of noises. The reconstruction formulas we propose in this work are explicit and applicable to a wider class of wave-type equations including the plasma-acoustic wave equations.

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