一类Hamilton方程的Birkhoff-Lewis型定理
A Birkhoff--Lewis Type Theorem for Some Hamiltonian Pdes
摘要：
本文将Birkhoff-Lewis定理推广到一类半线性偏微分方程。由此证明了无穷多个在原点处具有大周期积累的周期轨道的存在性。这样的周期轨道分叉的共振有限维不变环面的四阶标准形的系统。除了标准的非共振和非简并假设，我们的主要结果是假设正则化的非线性。我们将主要定理应用于半线性梁方程和具有光滑非线性项的非线性薛定谔方程。
英文摘要：
In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schrödinger equation with smoothing nonlinearity.
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