论文标题
动态PDE的时间涉及时间功能的拓扑费和保护法
Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs
论文作者
论文摘要
具有空间差异形式的动力学PDE具有涉及时间功能的保护定律。在一个空间维度中，这种保护定律被证明描述了$ x $独立的源/下沉的存在；在两个和更大的空间维度中，它们被证明描述了拓扑电荷。证明了两种应用。首先，拓扑电荷会产生相关的空间电势系统，从而可以找到给定动力学PDE的非本地保护定律和对称性。其次，当保守的密度涉及时间的任意功能以外的任意功能以外的函数本身时，其在任何给定的空间域上不可或缺的组成量都将减少到边界积分，在某些情况下，这可以对动态PDE的初始/边界数据进行限制。来自应用数学的非线性PDE的几个示例和可集成的系统理论用于说明这些新结果。
Dynamical PDEs that have a spatial divergence form possess conservation laws that involve an arbitrary function of time. In one spatial dimension, such conservation laws are shown to describe the presence of an $x$-independent source/sink; in two and more spatial dimensions, they are shown to describe a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing nonlocal conservation laws and symmetries to be found for a given dynamical PDE. Second,when a conserved density involves derivatives of an arbitrary function of time in addition to the function itself, its integral on any given spatial domain reduces to a boundary integral, which in some situations can place restrictions on initial/boundary data for which the dynamical PDE will be well-posed. Several examples of nonlinear PDEs from applied mathematics and integrable system theory are used to illustrate these new results.
