多辛几何、变分积分器与非线性偏微分方程
Multisymplectic geometry, variational integrators, and nonlinear PDEs
AI总结：
本文基于多辛几何与变分原理，提出统一框架以构造保持辛性与诺特对称性的连续及离散力学/场论模型，并给出非线性正弦-戈登方程的数值验证。
摘要：
本文提出了一种面向连续与离散力学及场论的几何-变分方法。利用多辛几何，我们证明了基本几何结构的存在性以及它们沿解的保持可以直接从变分原理中获得。特别地，我们证明了通过作用函数的导数可以得到唯一的多辛结构，并利用该结构证明了辛性守恒和诺特定理的协变推广。随后，通过选择离散作用泛函，可以得到具有这些重要保持性质的自然偏微分方程离散化方案。在力学情形中，我们恢复了Veselov型变分辛积分器；而对于偏微分方程，我们获得了协变时空积分器，它们保持相应的离散多辛形式以及对应于对称性的离散动量映射。我们证明了沿无限维场空间的通常辛性概念可以通过时空分裂自然获得。我们方法的所有方面都通过非线性正弦-戈登方程进行了演示，包括计算结果以及与其他离散化方案的比较。
英文摘要：
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.
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