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

Abstract

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in H s ( T m ) H^{s}(\mathbb {T}^{m}) when s > m / 2 + 1 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 D i f f ⋉ C ∞ ( T ) \mathrm {Diff} \ltimes C^{\infty }(\mathbb {T}) as a Hamiltonian equation, we concentrate on one space dimension ( m = 1 m=1 ) and show that the equation is bihamiltonian.