The Cauchy problem and integrability of a modified Euler-Poisson equation

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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+2 and we improve the Sobolev index to s>3/2 for m=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∞() as a Hamiltonian equation, we concentrate to one space dimension (m=1) and show that the equation is bihamiltonian.