SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 1758-1797, January 2020.
We consider a linearly unstable standing wave solution of a parabolic partial differential equation (PDE) on the real line and develop a high order method for polynomial approximation of the local unstable manifold. The unstable manifold describes the breakdown of the nonlinear wave after the loss of stability. Our method is based on the parameterization method for invariant manifolds and studies an invariance equation describing a local chart map. This invariance equation is a PDE posed on the product of a disk and the line. The dimension of the disk is equal to the Morse index of the wave. We develop a formal series solution for the invariance equation, and show that the coefficients of the series solve certain boundary value problems (BVPs) on the line. We solve these BVPs numerically to any desired order. The result is a polynomial describing the dynamics of the PDE in a macroscopic neighborhood of the unstable standing wave. The method is implemented for a number of example problems. Truncation/numerical errors are quantified via a posteriori indicators.

中文翻译：

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在线上驻波不稳定流形的参数化方法

SIAM应用动力系统杂志，第19卷第3期，第1758-1797页，2020年1月。
我们考虑实线上抛物线偏微分方程（PDE）的线性不稳定驻波解，并为局部不稳定流形的多项式逼近开发了一种高阶方法。不稳定流形描述了在失去稳定性之后非线性波的分解。我们的方法基于不变流形的参数化方法，并研究描述局部图表的不变方程。该不变性方程是置于磁盘和直线乘积上的PDE。圆盘的尺寸等于波的莫尔斯指数。我们为不变性方程开发了形式级数解，并表明该级数的系数可以解决直线上的某些边值问题（BVP）。我们以数字方式将这些BVP求解为所需顺序。结果是一个多项式，描述了不稳定驻波的宏观邻域中PDE的动力学。针对许多示例问题实施该方法。截断/数字错误通过后验指标进行量化。