Invariant Manifolds and Dispersive Hamiltonian Evolution Equations

By means of certain dispersive PDEs (such as the nonlinear Klein-Gordon equation) we will exhibit a new family of phenomena related to the ground state solitons. These solitons are (exponentially) unstable, and one can construct stable, unstable, and center(-stable) manifolds associated with these ground states in the sense of hyperbolic dynamics. In terms of these invariant manifolds one can completely characterize the global dynamics of solutions whose energy exceeds that of the ground states by at most a small amount. In particular, we will establish a trichotomy in forward time giving either finite-time blow up, global forward existence and scattering to zero, or global existence and scattering to the ground states as all possibilities. It turns out that all nine sets consisting of all possible combinations of the forward/backward trichotomies arise. This extends the classical Payne-Sattinger picture (from 1975) which gives such a characterization at energies below that of the ground state; in the latter case the aforementioned (un)stable and center manifolds do not arise, since they require larger energy than that of the ground state. Our methods proceed by combining a perturbative analysis near the ground states with a global and variational analysis away from them. This work is joint with Kenji Nakanishi from Kyoto University, Japan.

通过某些色散偏微分方程（例如非线性克莱因 - 戈登方程），我们将展示一系列与基态孤子相关的新现象。这些孤子是（指数）不稳定的，并且可以在双曲动力学的意义下构造与这些基态相关的稳定、不稳定和中心（ - 稳定）流形。根据这些不变流形，可以完全刻画能量至多比基态能量稍高的解的全局动力学。特别地，我们将在正向时间建立一种三分法，给出有限时间爆破、全局正向存在且散射到零，或者全局存在且散射到基态作为所有可能的情况。结果表明，由正向/反向三分法的所有可能组合构成的所有九种情况都会出现。这扩展了经典的佩恩 - 萨廷格图景（1975年），该图景在能量低于基态时给出了这样的一种刻画；在后一种情况下，上述（不）稳定和中心流形不会出现，因为它们需要比基态更大的能量。我们的方法是通过将基态附近的微扰分析与远离基态的全局变分分析相结合来进行的。这项工作是与日本京都大学的中西贤治合作完成的。