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 produce a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing non-local conservation laws and symmetries to be found for a given dynamical PDE. This type of potential system has a different form and different gauge freedom compared to potential systems that arise from ordinary conservation laws. Second, when a topological charge arises from a conservation law whose conserved density is non-trivial off of solutions to the dynamical PDE, then this relation yields a constraint 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 results.

中文翻译：

---

涉及动态 PDE 的任意时间函数的拓扑电荷和守恒定律

具有空间发散形式的动态偏微分方程具有涉及任意时间函数的守恒定律。在一个空间维度中，这种守恒定律被用来描述与 x 无关的源/汇的存在；在两个或更多空间维度中，它们被证明会产生拓扑电荷。演示了两个应用程序。首先，拓扑电荷产生相关的空间势系统，允许为给定的动态 PDE 找到非局部守恒定律和对称性。与由普通守恒定律产生的潜在系统相比，这种类型的潜在系统具有不同的形式和不同的规范自由度。其次，当拓扑电荷产生于守恒密度与动力学 PDE 解无关的守恒定律时，那么这个关系会产生一个对初始/边界数据的约束，动态 PDE 将很好地提出。应用数学和可积系统理论中非线性偏微分方程的几个例子被用来说明这些结果。