We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.

翻译：我们考虑的是带有定期边界条件的参数正格罗登模式。我们证明这是一个基本的最高原则,它能先验地统一控制解决方案。在一维的例子中,我们对所有受约束的稳定状态进行分类,并展示出一些明确的解决办法。对于数字分解,我们使用第一级IMEX,第二级BDF2分解,而没有任何额外的稳定期限。我们严格地证明数字计划在几乎尖锐和相当轻微的时间步骤限制下的能源稳定性。我们证明,参数法罗-哥尔登模式与标准的Allen-Cahn方程式有着惊人的相似性,它具有双重井的双重潜力。