The Cauchy Problem and Integrability of a Modified Euler-poisson Equation

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in Hs(Tm) when s > m/2 + 1. We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space Diff C∞(T) as a Hamiltonian equation, we concentrate on one space dimension (m = 1) and show that the equation is bihamiltonian. In this paper we study the periodic Cauchy problem for the modified EulerPoisson equation (mEP) (mEP) ∂tn+ div(nv) = 0, ∂tv + (v.∇)v + gradΛ−2n = 0, x ∈ T, t ∈ R, as well as its Hamiltonian structure and integrability. The equation (mEP) is related to the Euler-Poisson equation (1) ∂tn+ div(nv) = 0, ∂tv + (v.∇)v + gradφ = 0, ∆φ− e + n = 0, x ∈ T, t ∈ R, which describes the fluctuations in the ion density of a two-component plasma of positively charged ions and negatively charged electrons (therefore it is also called ion acoustic plasma equation [LiSat]). Linearizing the operator N(φ) = e−∆φ at φ = 0 in the Euler-Poisson equation (1), we obtain the local form of the modified Euler-Poisson equation (mEP): (2) ∂tn+ div(nv) = 0, ∂tv + (v.∇)v + gradφ = 0, ∆φ− φ+ n = 0, x ∈ T, t ∈ R. Equation (2), like the Euler-Poisson equation (1), admits an approximation which preserves dispersion and leads to KdV (see Remark 1). Inverting the operator Λ := I −∆, we write the system in (2) in the nonlocal form (mEP). Besides its relation to the Euler-Poisson equation (1), the modified Euler-Poisson equation is also remarkable for its bihamiltonian structure in one space dimension that we describe here. Received by the editors October 15, 2004 and, in revised form, June 9, 2005 and October 13, 2005. 2000 Mathematics Subject Classification. Primary 35Q53, 35Q05, 35A10, 37K65. c ©2007 American Mathematical Society Reverts to public domain 28 years from publication