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

Tığlay, Feride

doi:10.1090/S0002-9947-07-04248-1

The Cauchy problem and integrability of a modified Euler-Poisson equation
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Trans. Amer. Math. Soc. 360 (2008), 1861-1877 Request permission

Abstract:

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

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